  
  [1X9 [33X[0;0YAmbiguous Class Fusions in the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YDate: January 11th, 2004[133X
  
  [33X[0;0YThis is a collection of examples showing how class fusions between character
  tables  can  be  determined  using  the [5XGAP[105X system [GAP24]. In each of these
  examples,  the fusion is [13Xambiguous[113X in the sense that the character tables do
  not determine it up to table automorphisms. Our strategy is to compute first
  all  possibilities  with  the  [5XGAP[105X function [2XPossibleClassFusions[102X ([14XReference:
  PossibleClassFusions[114X),  and  then  to  use  either other character tables or
  information  about  the  groups for excluding some of these candidates until
  only one (orbit under table automorphisms) remains.[133X
  
  [33X[0;0YThe  purpose  of  this writeup is twofold. On the one hand, the computations
  are  documented  this  way.  On  the  other hand, the [5XGAP[105X code shown for the
  examples  can  be  used as test input for automatic checking of the data and
  the  functions  used;  therefore, each example ends with a comparison of the
  result  with  the  fusion that is actually stored in the [5XGAP[105X Character Table
  Library [Bre25].[133X
  
  [33X[0;0YThe  examples  use  the  [5XGAP[105X  Character Table Library, so we first load this
  package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "ctbllib", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.1 [33X[0;0YSome [5XGAP[105X[101X[1X Utilities[133X[101X
  
  [33X[0;0YThe  function  [10XSetOfComposedClassFusions[110X  takes  two  list of class fusions,
  where the first list consists of fusions between the character tables of the
  groups  [22XH[122X  and [22XG[122X, say, and the second list consists of class fusions between
  the character tables of the groups [22XU[122X and [22XH[122X, say; the return value is the set
  of  compositions  of  each map in the first list with each map in the second
  list (via [2XCompositionMaps[102X ([14XReference: CompositionMaps[114X)).[133X
  
  [33X[0;0YNote  that  the  returned  list  may  be  a  proper subset of the set of all
  possible  class  fusions  between  [22XU[122X  and  [22XG[122X,  which  can  be  computed with
  [2XPossibleClassFusions[102X ([14XReference: PossibleClassFusions[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetOfComposedClassFusions:= function( hfusg, ufush )[127X[104X
    [4X[25X>[125X [27X    local result, map1, map2;[127X[104X
    [4X[25X>[125X [27X    result:= [];;[127X[104X
    [4X[25X>[125X [27X    for map2 in hfusg do[127X[104X
    [4X[25X>[125X [27X      for map1 in ufush do[127X[104X
    [4X[25X>[125X [27X        AddSet( result, CompositionMaps( map2, map1 ) );[127X[104X
    [4X[25X>[125X [27X      od;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X    return result;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  
  [1X9.2 [33X[0;0YFusions Determined by Factorization through Intermediate Subgroups[133X[101X
  
  [33X[0;0YThis  situation  clearly  occurs  only for nonmaximal subgroups. Interesting
  examples are Sylow normalizers.[133X
  
  
  [1X9.2-1 [33X[0;0Y[22XCo_3N5 → Co_3[122X[101X[1X (September 2002)[133X[101X
  
  [33X[0;0YLet [22XH[122X be the Sylow [22X5[122X normalizer in the sporadic simple group [22XCo_3[122X. The class
  fusion  of [22XH[122X into [22XCo_3[122X is not uniquely determined by the character tables of
  the two groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xco3:= CharacterTable( "Co3" );[127X[104X
    [4X[28XCharacterTable( "Co3" )[128X[104X
    [4X[25Xgap>[125X [27Xh:= CharacterTable( "Co3N5" );[127X[104X
    [4X[28XCharacterTable( "5^(1+2):(24:2)" )[128X[104X
    [4X[25Xgap>[125X [27Xhfusco3:= PossibleClassFusions( h, co3 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( h, hfusco3, co3 ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  [22XH[122X  is  not  maximal  in [22XCo_3[122X, we look at those maximal subgroups of [22XCo_3[122X
  whose order is divisible by that of [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmx:= Maxes( co3 );[127X[104X
    [4X[28X[ "McL.2", "HS", "U4(3).(2^2)_{133}", "M23", "3^5:(2xm11)", [128X[104X
    [4X[28X  "2.S6(2)", "U3(5).3.2", "3^1+4:4s6", "2^4.a8", "L3(4).D12", [128X[104X
    [4X[28X  "2xm12", "2^2.[2^7*3^2].S3", "s3xpsl(2,8).3", "a4xs5" ][128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( mx, CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, x -> Size( x ) mod Size( h ) = 0 );[127X[104X
    [4X[28X[ CharacterTable( "McL.2" ), CharacterTable( "HS" ), [128X[104X
    [4X[28X  CharacterTable( "U3(5).3.2" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording to the [5XAtlas[105X (see [CCN+85, pp. 34 and 100]), [22XH[122X occurs as the Sylow
  [22X5[122X normalizer in [22XU_3(5).3.2[122X and in [22XMcL.2[122X; however, [22XH[122X is not a subgroup of [22XHS[122X,
  since  otherwise  [22XH[122X  would  be  contained  in  subgroups  of  type  [22XU_3(5).2[122X
  (see [CCN+85,  p.  80]), but the only possible subgroups in these groups are
  too small (see [CCN+85, p. 34]).[133X
  
  [33X[0;0YWe  compute  the  possible class fusions from [22XH[122X into [22XMcL.2[122X and from [22XMcL.2[122X to
  [22XCo_3[122X, and then form the compositions of these maps.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmax:= filt[1];;[127X[104X
    [4X[25Xgap>[125X [27Xhfusmax:= PossibleClassFusions( h, max );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxfusco3:= PossibleClassFusions( max, co3 );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( maxfusco3, hfusmax );;[127X[104X
    [4X[25Xgap>[125X [27XLength( comp );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xreps:= RepresentativesFusions( h, comp, co3 );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 8, 8, 7, 9, 10, 11, 17, 17, 19, 19, 22, 23, 27, 27, [128X[104X
    [4X[28X      30, 33, 34, 40, 40, 40, 40, 42 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  factoring through a maximal subgroup of type [22XMcL.2[122X determines the fusion
  from [22XH[122X to [22XCo_3[122X uniquely up to table automorphisms.[133X
  
  [33X[0;0YAlternatively,  we  can  use  the group [22XU_3(5).3.2[122X as intermediate subgroup,
  which leads to the same result.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmax:= filt[3];;[127X[104X
    [4X[25Xgap>[125X [27Xhfusmax:= PossibleClassFusions( h, max );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxfusco3:= PossibleClassFusions( max, co3 );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( maxfusco3, hfusmax );;[127X[104X
    [4X[25Xgap>[125X [27Xreps2:= RepresentativesFusions( h, comp, co3 );;[127X[104X
    [4X[25Xgap>[125X [27Xreps2 = reps;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we  compare  the result with the map that is stored on the library
  table of [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( h, co3 ) in reps;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.2-2 [33X[0;0Y[22X31:15 → B[122X[101X[1X (March 2003)[133X[101X
  
  [33X[0;0YThe  Sylow  [22X31[122X normalizer [22XH[122X in the sporadic simple group [22XB[122X has the structure
  [22X31:15[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xh:= CharacterTable( "31:15" );;[127X[104X
    [4X[25Xgap>[125X [27Xhfusb:= PossibleClassFusions( h, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( h, hfusb, b ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  determine  the  correct  fusion  using the fact that [22XH[122X is contained in a
  (maximal) subgroup of type [22XTh[122X in [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xth:= CharacterTable( "Th" );;[127X[104X
    [4X[25Xgap>[125X [27Xhfusth:= PossibleClassFusions( h, th );;[127X[104X
    [4X[25Xgap>[125X [27Xthfusb:= PossibleClassFusions( th, b );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( thfusb, hfusth );;[127X[104X
    [4X[25Xgap>[125X [27XLength( comp );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xreps:= RepresentativesFusions( h, comp, b );[127X[104X
    [4X[28X[ [ 1, 145, 146, 82, 82, 19, 82, 7, 19, 82, 82, 19, 7, 82, 19, 82, 82 [128X[104X
    [4X[28X     ] ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( h, b ) in reps;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.2-3 [33X[0;0Y[22XSuzN3 → Suz[122X[101X[1X (September 2002)[133X[101X
  
  [33X[0;0YThe  class fusion from the Sylow [22X3[122X normalizer into the sporadic simple group
  [22XSuz[122X is not uniquely determined by the character tables of these groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xh:= CharacterTable( "SuzN3" );[127X[104X
    [4X[28XCharacterTable( "3^5:(3^2:SD16)" )[128X[104X
    [4X[25Xgap>[125X [27Xsuz:= CharacterTable( "Suz" );[127X[104X
    [4X[28XCharacterTable( "Suz" )[128X[104X
    [4X[25Xgap>[125X [27Xhfussuz:= PossibleClassFusions( h, suz );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( h, hfussuz, suz ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  [22XH[122X  is  not  maximal  in [22XSuz[122X, we try to factorize the fusion through a
  suitable maximal subgroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( suz ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, x -> Size( x ) mod Size( h ) = 0 );[127X[104X
    [4X[28X[ CharacterTable( "3_2.U4(3).2_3'" ), CharacterTable( "3^5:M11" ), [128X[104X
    [4X[28X  CharacterTable( "3^2+4:2(2^2xa4)2" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe group [22X3_2.U_4(3).2_3^'[122X does not admit a fusion from [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPossibleClassFusions( h, filt[1] );[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YDefinitely  [22X3^5:M_11[122X contains a group isomorphic with [22XH[122X, because the Sylow [22X3[122X
  normalizer  in  [22XM_11[122X  has the structure [22X3^2:SD_16[122X; using [22X3^2+4:2(2^2 × A_4)2[122X
  would  lead  to the same result as we get below. We compute the compositions
  of possible class fusions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmax:= filt[2];;[127X[104X
    [4X[25Xgap>[125X [27Xhfusmax:= PossibleClassFusions( h, max );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxfussuz:= PossibleClassFusions( max, suz );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( maxfussuz, hfusmax );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( h, comp, suz );[127X[104X
    [4X[28X[ [ 1, 2, 2, 4, 5, 4, 5, 5, 5, 5, 5, 6, 9, 9, 14, 15, 13, 16, 16, 14, [128X[104X
    [4X[28X      15, 13, 13, 13, 16, 15, 14, 16, 16, 16, 21, 21, 23, 22, 29, 29, [128X[104X
    [4X[28X      29, 38, 39 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo the factorization determines the fusion map up to table automorphisms. We
  check that this map is equal to the stored one.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( h, suz ) in repr;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.2-4 [33X[0;0Y[22XF_{3+}N5 → F_{3+}[122X[101X[1X (March 2002)[133X[101X
  
  [33X[0;0YThe  class fusion from the table of the Sylow [22X5[122X normalizer [22XH[122X in the sporadic
  simple group [22XF_{3+}[122X into [22XF_{3+}[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf3p:= CharacterTable( "F3+" );;[127X[104X
    [4X[25Xgap>[125X [27Xh:= CharacterTable( "F3+N5" );;[127X[104X
    [4X[25Xgap>[125X [27Xhfusf3p:= PossibleClassFusions( h, f3p );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( h, hfusf3p, f3p ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22XH[122X  is not maximal in [22XF_{3+}[122X, so we look for tables of maximal subgroups that
  can contain [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( f3p ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, x -> Size( x ) mod Size( h ) = 0 );[127X[104X
    [4X[28X[ CharacterTable( "Fi23" ), CharacterTable( "2.Fi22.2" ), [128X[104X
    [4X[28X  CharacterTable( "(3xO8+(3):3):2" ), CharacterTable( "O10-(2)" ), [128X[104X
    [4X[28X  CharacterTable( "(A4xO8+(2).3).2" ), CharacterTable( "He.2" ), [128X[104X
    [4X[28X  CharacterTable( "F3+M14" ), CharacterTable( "(A5xA9):2" ) ][128X[104X
    [4X[25Xgap>[125X [27Xpossfus:= List( filt, x -> PossibleClassFusions( h, x ) );[127X[104X
    [4X[28X[ [  ], [  ], [  ], [  ], [128X[104X
    [4X[28X  [ [ 1, 69, 110, 12, 80, 121, 4, 72, 113, 11, 11, 79, 79, 120, 120, [128X[104X
    [4X[28X          3, 71, 11, 79, 23, 91, 112, 120, 132, 29, 32, 97, 100, 37, [128X[104X
    [4X[28X          37, 105, 105, 139, 140, 145, 146, 155, 155, 156, 156, 44, [128X[104X
    [4X[28X          44, 167, 167, 48, 48, 171, 171, 57, 57, 180, 180, 66, 66, [128X[104X
    [4X[28X          189, 189 ], [128X[104X
    [4X[28X      [ 1, 69, 110, 12, 80, 121, 4, 72, 113, 11, 11, 79, 79, 120, [128X[104X
    [4X[28X          120, 3, 71, 11, 79, 23, 91, 112, 120, 132, 29, 32, 97, 100, [128X[104X
    [4X[28X          37, 37, 105, 105, 140, 139, 146, 145, 156, 156, 155, 155, [128X[104X
    [4X[28X          44, 44, 167, 167, 48, 48, 171, 171, 57, 57, 180, 180, 66, [128X[104X
    [4X[28X          66, 189, 189 ] ], [  ], [  ], [  ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that from the eight possible classes of maximal subgroups in [22XF_{3+}[122X
  that  might  contain [22XH[122X, only the group of type [22X(A_4 × O_8^+(2).3).2[122X admits a
  class  fusion  from [22XH[122X. Hence we can compute the compositions of the possible
  fusions  from  [22XH[122X  into  this group with the possible fusions from this group
  into [22XF_{3+}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmax:= filt[5];[127X[104X
    [4X[28XCharacterTable( "(A4xO8+(2).3).2" )[128X[104X
    [4X[25Xgap>[125X [27Xhfusmax:= possfus[5];;[127X[104X
    [4X[25Xgap>[125X [27Xmaxfusf3p:= PossibleClassFusions( max, f3p );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( maxfusf3p, hfusmax );;[127X[104X
    [4X[25Xgap>[125X [27XLength( comp );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( h, comp, f3p );[127X[104X
    [4X[28X[ [ 1, 2, 4, 12, 35, 54, 3, 3, 16, 9, 9, 11, 11, 40, 40, 2, 3, 9, 11, [128X[104X
    [4X[28X      35, 36, 13, 40, 90, 7, 22, 19, 20, 43, 43, 50, 50, 8, 8, 23, [128X[104X
    [4X[28X      23, 46, 46, 47, 47, 10, 10, 9, 9, 10, 10, 11, 11, 26, 26, 28, [128X[104X
    [4X[28X      28, 67, 67, 68, 68 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we check whether the map stored in the table library is correct.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( h, f3p ) in repr;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  we  did [13Xnot[113X determine the class fusion from the maximal subgroup
  [22X(A_4   ×   O_8^+(2).3).2[122X   into   [22XF_{3+}[122X  up  to  table  automorphisms  (see
  Section [14X9.3-2[114X  for this problem), since also the ambiguous result was enough
  for computing the fusion from [22XH[122X into [22XF_{3+}[122X.[133X
  
  
  [1X9.3   [33X[0;0YFusions   Determined  Using  Commutative  Diagrams  Involving  Smaller[101X
  [1XSubgroups[133X[101X
  
  [33X[0;0YIn  each  of  the following examples, the class fusion of a (not necessarily
  maximal)  subgroup  [22XM[122X  of  a  group  [22XG[122X into [22XG[122X is determined by considering a
  proper  subgroup  [22XU[122X  of [22XM[122X whose class fusion into [22XG[122X can be computed, perhaps
  using another subgroup [22XS[122X of [22XG[122X that also contains [22XU[122X.[133X
  
                                  G
                                 ╱ ╲
                                ╱   ╲
                               M     S
                                ╲   ╱
                                 ╲ ╱
                                  .
                                  │
                                  U
                                  │
                                  │
                                  1
  
  
  [1X9.3-1 [33X[0;0Y[22XBN7 → B[122X[101X[1X (March 2002)[133X[101X
  
  [33X[0;0YLet  [22XH[122X  be  a  Sylow  [22X7[122X normalizer in the sporadic simple group [22XB[122X. The class
  fusion of [22XH[122X into [22XB[122X is not uniquely determined by the character tables of the
  two groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );[127X[104X
    [4X[28XCharacterTable( "B" )[128X[104X
    [4X[25Xgap>[125X [27Xh:= CharacterTable( "BN7" );[127X[104X
    [4X[28XCharacterTable( "BN7" )[128X[104X
    [4X[25Xgap>[125X [27Xhfusb:= PossibleClassFusions( h, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( h, hfusb, b ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  us  consider  a  maximal  subgroup of the type [22XTh[122X in [22XB[122X (cf. [CCN+85, p.
  217]).  By [CCN+85,  p.  177],  the  Sylow  [22X7[122X  normalizers in [22XTh[122X are maximal
  subgroups  of  [22XTh[122X  and  have  the  structure [22X7^2:(3 × 2S_4)[122X. Let [22XU[122X be such a
  subgroup.[133X
  
  [33X[0;0YNote  that  the only maximal subgroups of [22XTh[122X whose order is divisible by the
  order  of  a  Sylow  [22X7[122X  subgroup of [22XB[122X have the types [22X^3D_4(2).3[122X and [22X7^2:(3 ×
  2S_4)[122X,  and  the Sylow [22X7[122X normalizers in the former groups have the structure
  [22X7^2:(3 × 2A_4)[122X, cf. [CCN+85, p. 89].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumber( Factors( Size( b ) ), x -> x = 7 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xth:= CharacterTable( "Th" );[127X[104X
    [4X[28XCharacterTable( "Th" )[128X[104X
    [4X[25Xgap>[125X [27XFiltered( Maxes( th ), x -> Size( CharacterTable( x ) ) mod 7^2 = 0 );[127X[104X
    [4X[28X[ "3D4(2).3", "7^2:(3x2S4)" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  class fusion of [22XU[122X into [22XB[122X via [22XTh[122X is uniquely determined by the character
  tables of these groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xthn7:= CharacterTable( "ThN7" );[127X[104X
    [4X[28XCharacterTable( "7^2:(3x2S4)" )[128X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( PossibleClassFusions( th, b ),[127X[104X
    [4X[25X>[125X [27X              PossibleClassFusions( thn7, th ) );[127X[104X
    [4X[28X[ [ 1, 31, 7, 7, 5, 28, 28, 17, 72, 72, 6, 6, 7, 28, 27, 27, 109, [128X[104X
    [4X[28X      109, 17, 45, 45, 72, 72, 127, 127, 127, 127 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe condition that the class fusion of [22XU[122X into [22XB[122X factors through [22XH[122X determines
  the class fusion of [22XH[122X into [22XB[122X up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xthn7fush:= PossibleClassFusions( thn7, h );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( hfusb, x ->[127X[104X
    [4X[25X>[125X [27X              ForAny( thn7fush, y -> CompositionMaps( x, y ) in comp ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( h, filt, b ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we  compare  the result with the map that is stored on the library
  table of [22XH[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( h, b ) in filt;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.3-2 [33X[0;0Y[22X(A_4 × O_8^+(2).3).2 → Fi_24^'[122X[101X[1X (November 2002)[133X[101X
  
  [33X[0;0YThe  class  fusion  of  the maximal subgroup [22XM ≅ (A_4 × O_8^+(2).3).2[122X of [22XG =
  Fi_24^'[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "(A4xO8+(2).3).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "F3+" );;[127X[104X
    [4X[25Xgap>[125X [27Xmfust:= PossibleClassFusions( m, t );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( m, mfust, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( repr );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  first observe that the elements of order three in the normal subgroup of
  type [22XA_4[122X in [22XM[122X lie in the class [10X3A[110X of [22XFi_24^'[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa4inm:= Filtered( ClassPositionsOfNormalSubgroups( m ),[127X[104X
    [4X[25X>[125X [27X                     n -> Sum( SizesConjugacyClasses( m ){ n } ) = 12 );[127X[104X
    [4X[28X[ [ 1, 69, 110 ] ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( m ){ a4inm[1] };[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XList( repr, map -> map[110] );[127X[104X
    [4X[28X[ 4, 4 ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( t ){ [ 1 .. 4 ] };[127X[104X
    [4X[28X[ 1, 2, 2, 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet us take one such element [22Xg[122X, say. Its normalizer [22XS[122X in [22XG[122X has the structure
  [22X(3  ×  O_8^+(3).3).2[122X; this group is maximal in [22XG[122X, and its character table is
  available in [5XGAP[105X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "F3+N3A" );[127X[104X
    [4X[28XCharacterTable( "(3xO8+(3):3):2" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  intersection  [22XN_M(g)  =  S  ∩  M[122X  contains a subgroup [22XU[122X of the type [22X3 ×
  O_8^+(2).3[122X,  and  in  the following we compute the class fusions of [22XU[122X into [22XS[122X
  and [22XM[122X, and then utilize the fact that only those class fusions from [22XM[122X into [22XG[122X
  are  possible whose composition with the class fusion from [22XU[122X into [22XM[122X equals a
  composition of class fusions from [22XU[122X into [22XS[122X and from [22XS[122X into [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "Cyclic", 3 ) * CharacterTable( "O8+(2).3" );[127X[104X
    [4X[28XCharacterTable( "C3xO8+(2).3" )[128X[104X
    [4X[25Xgap>[125X [27Xufuss:= PossibleClassFusions( u, s );;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= PossibleClassFusions( u, m );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( sfust, ufuss );;[127X[104X
    [4X[25Xgap>[125X [27XLength( comp );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( mfust,[127X[104X
    [4X[25X>[125X [27X    x -> ForAny( ufusm, map -> CompositionMaps( x, map ) in comp ) );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( m, filt, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( repr );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, t ) in repr;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo the class fusion from [22XM[122X into [22XG[122X is determined up to table automorphisms by
  the commutative diagram.[133X
  
  
  [1X9.3-3 [33X[0;0Y[22XA_6 × L_2(8).3 → Fi_24^'[122X[101X[1X (November 2002)[133X[101X
  
  [33X[0;0YThe  class  fusion of the maximal subgroup [22XM ≅ A_6 × L_2(8).3[122X of [22XG = Fi_24^'[122X
  is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "A6xL2(8):3" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "F3+" );;[127X[104X
    [4X[25Xgap>[125X [27Xmfust:= PossibleClassFusions( m, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( m, mfust, t ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  will  use  the fact that the direct factor of the type [22XA_6[122X in [22XM[122X contains
  elements in the class [10X3A[110X of [22XG[122X. This fact can be shown as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdppos:= ClassPositionsOfDirectProductDecompositions( m );[127X[104X
    [4X[28X[ [ [ 1, 12 .. 67 ], [ 1 .. 11 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XList( dppos[1], l -> Sum( SizesConjugacyClasses( t ){ l } ) );[127X[104X
    [4X[28X[ 17733424133316996808705, 4545066196775803392 ][128X[104X
    [4X[25Xgap>[125X [27XList( dppos[1], l -> Sum( SizesConjugacyClasses( m ){ l } ) );[127X[104X
    [4X[28X[ 360, 1512 ][128X[104X
    [4X[25Xgap>[125X [27X3Apos:= Position( OrdersClassRepresentatives( t ), 3 );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27X3Ainm:= List( mfust, map -> Position( map, 3Apos ) );[127X[104X
    [4X[28X[ 23, 23, 23, 23, 34, 34, 34, 34 ][128X[104X
    [4X[25Xgap>[125X [27XForAll( 3Ainm, x -> x in dppos[1][1] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince the normalizer of an element of order three in [22XA_6[122X has the form [22X3^2:2[122X,
  such  a  [10X3A[110X  element  in  [22XM[122X  contains  a subgroup [22XU[122X of the structure [22X3^2:2 ×
  L_2(8).3[122X  which  is  contained  in  the  [10X3A[110X normalizer [22XS[122X in [22XG[122X, which has the
  structure [22X(3 × O_8^+(3).3).2[122X.[133X
  
  [33X[0;0Y(Note  that  all  classes  in  the  [22X3^2:2[122X  type  group are rational, and its
  character  table  is  available  in the [5XGAP[105X Character Table Library with the
  identifier [10X"3^2:2"[110X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "3^2:2" ) * CharacterTable( "L2(8).3" );[127X[104X
    [4X[28XCharacterTable( "3^2:2xL2(8).3" )[128X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "F3+N3A" );[127X[104X
    [4X[28XCharacterTable( "(3xO8+(3):3):2" )[128X[104X
    [4X[25Xgap>[125X [27Xufuss:= PossibleClassFusions( u, s );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( sfust, ufuss );;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= PossibleClassFusions( u, m );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( mfust,[127X[104X
    [4X[25X>[125X [27X              map -> ForAny( ufusm,[127X[104X
    [4X[25X>[125X [27X                         map2 -> CompositionMaps( map, map2 ) in comp ) );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( m, filt, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( repr );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, t ) in repr;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.3-4 [33X[0;0Y[22X(3^2:D_8 × U_4(3).2^2).2 → B[122X[101X[1X (June 2007)[133X[101X
  
  [33X[0;0YLet  [22XG[122X  be  a  maximal  subgroup of the type [22X(3^2:D_8 × U_4(3).2^2).2[122X in the
  sporadic simple group [22XB[122X, cf. [CCN+85, p. 217]. Computing the class fusion of
  [22XG[122X  into  [22XB[122X  just from the character tables of the two groups takes extremely
  long. So we use additional information.[133X
  
  [33X[0;0YAccording  to [CCN+85,  p.  217],  [22XG[122X is the normalizer in [22XB[122X of an elementary
  abelian  group  [22X⟨ x, y ⟩[122X of order [22X9[122X, with [22Xx, y[122X in the class [10X3A[110X of [22XB[122X, and [22XN =
  N_B(⟨  x  ⟩)[122X has the structure [22XS_3 × Fi_22.2[122X. The intersection [22XG ∩ N[122X has the
  structure [22XS_3 × S_3 × U_4(3).2^2[122X, which is the direct product of [22XS_3[122X and the
  normalizer in [22XFi_22.2[122X of a [10X3A[110X element of [22XFi_22.2[122X, see [CCN+85, p. 163]. Thus
  we may use that the class fusions from [22XG ∩ N[122X into [22XB[122X through [22XG[122X or [22XN[122X coincide.[133X
  
  [33X[0;0YThe  class  fusion  from  [22XN[122X  into  [22XB[122X is uniquely determined by the character
  tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= CharacterTable( "BN3A" );[127X[104X
    [4X[28XCharacterTable( "S3xFi22.2" )[128X[104X
    [4X[25Xgap>[125X [27Xnfusb:= PossibleClassFusions( n, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( nfusb );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xnfusb:= nfusb[1];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  computation  of  the  class  fusion  from  [22XG  ∩  N[122X into [22XN[122X is sped up by
  computing  first  the  class  fusion  modulo the direct factor [22XS_3[122X, and then
  lifting these fusion maps.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfi222:= CharacterTable( "Fi22.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xfi222n3a:= CharacterTable( "S3xU4(3).(2^2)_{122}" );;[127X[104X
    [4X[25Xgap>[125X [27Xs3:= CharacterTable( "S3" );;[127X[104X
    [4X[25Xgap>[125X [27Xinter:= s3 * fi222n3a;;[127X[104X
    [4X[25Xgap>[125X [27Xintermods3fusnmods3:= PossibleClassFusions( fi222n3a, fi222 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( intermods3fusnmods3 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( fi222n3a, intermods3fusnmods3, fi222 ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  get two equivalent possibilities, and need to consider only one of them.
  For  lifting  it  to  a  map between [22XG ∩ N[122X and [22XN[122X, the safe way is to use the
  fusion   map  between  the  two  factors  for  computing  an  approximation.
  (Additionally,  we  could  interpret  the  known maps as fusions between two
  subgroups,  and  use  this for improving the approximation, but in this case
  the speedup is not worth the effort.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinterfusn:= CompositionMaps( InverseMap( GetFusionMap( n, fi222 ) ),[127X[104X
    [4X[25X>[125X [27X       CompositionMaps( intermods3fusnmods3[1],[127X[104X
    [4X[25X>[125X [27X           GetFusionMap( inter, fi222n3a ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xinterfusn:= PossibleClassFusions( inter, n,[127X[104X
    [4X[25X>[125X [27X       rec( fusionmap:= interfusn, quick:= true ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( interfusn );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  lift  is  unique.  Since  we lift a class fusion to direct products, we
  could  also  [21Xextend[121X the fusion directly. But note that this would assume the
  ordering of classes in character tables of direct products. This alternative
  would work as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnccl:= NrConjugacyClasses( fi222 );;[127X[104X
    [4X[25Xgap>[125X [27Xinterfusn[1] = Concatenation( List( [ 0 .. 2 ],[127X[104X
    [4X[25X>[125X [27X                      i -> intermods3fusnmods3[1] + i * nccl ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  compute  the  class fusions from [22XG ∩ N[122X to [22XG[122X. We get two equivalent
  solutions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtblg:= CharacterTable( "BM14" );[127X[104X
    [4X[28XCharacterTable( "(3^2:D8xU4(3).2^2).2" )[128X[104X
    [4X[25Xgap>[125X [27Xinterfusg:= PossibleClassFusions( inter, tblg );;[127X[104X
    [4X[25Xgap>[125X [27XLength( interfusg );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( inter, interfusg, tblg ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  approximation  of the class fusion from [22XG[122X to [22XB[122X is computed by composing
  the known maps. Because we have chosen one of the two possible maps from [22XG ∩
  N[122X  to  [22XN[122X, here we consider the two possibilities. From these approximations,
  we compute the possible class fusions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinterfusb:= CompositionMaps( nfusb, interfusn[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xapprox:= List( interfusg,[127X[104X
    [4X[25X>[125X [27X       map -> CompositionMaps( interfusb, InverseMap( map ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xgfusb:= Set( Concatenation( List( approx,[127X[104X
    [4X[25X>[125X [27X                    map -> PossibleClassFusions( tblg, b,[127X[104X
    [4X[25X>[125X [27X                               rec( fusionmap:= map ) ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( gfusb );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( tblg, gfusb, b ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we  compare the result with the class fusion that is stored on the
  library table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( tblg, b ) in gfusb;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.3-5 [33X[0;0Y[22X7^1+4:(3 × 2.S_7) → M[122X[101X[1X (May 2009)[133X[101X
  
  [33X[0;0YThe  class fusion of the maximal subgroup [22XU[122X of type [22X7^1+4:(3 × 2.S_7)[122X of the
  Monster group [22XM[122X into [22XM[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtblu:= CharacterTable( "7^(1+4):(3x2.S7)" );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= PossibleClassFusions( tblu, m );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( tblu, ufusm, m ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  subgroup  [22XU[122X  contains  a  Sylow  [22X7[122X-subgroup  of [22XM[122X, and the only maximal
  subgroups  of  [22XM[122X  with this property are the class of [22XU[122X and another class of
  subgroups,  of  the  type  [22X7^2+1+2:GL_2(7)[122X.  Moreover, it turns out that the
  Sylow  [22X7[122X  normalizers  in the subgroups in both classes have the same order,
  hence they are the Sylow [22X7[122X normalizers in [22XM[122X.[133X
  
  [33X[0;0YFor    that,    we   use   representations   from   the   [5XAtlas[105X   of   Group
  Representations [WWT+], and access these representations via the [5XGAP[105X package
  [5XAtlasRep[105X ([WPN+22]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "atlasrep", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xg1:= AtlasGroup( "7^(2+1+2):GL2(7)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs1:= SylowSubgroup( g1, 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xn1:= Normalizer( g1, s1 );;[127X[104X
    [4X[25Xgap>[125X [27Xg2:= AtlasGroup( "7^(1+4):(3x2.S7)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs2:= SylowSubgroup( g2, 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xn2:= Normalizer( g2, s2 );;[127X[104X
    [4X[25Xgap>[125X [27XSize( n1 ) = Size( n2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X( Size( m ) / Size( s1 ) ) mod 7 <> 0;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  let  [22XN[122X be a Sylow [22X7[122X normalizer in [22XU[122X, and choose a subgroup [22XS[122X of the type
  [22X7^2+1+2:GL_2(7)[122X that contains [22XN[122X.[133X
  
  [33X[0;0YWe compute the character table of [22XN[122X. Computing the possible class fusions of
  [22XN[122X into [22XM[122X directly yields two possibilities, but the class fusion of [22XN[122X into [22XM[122X
  via [22XS[122X is uniquely determined by the character tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbln:= CharacterTable( Image( IsomorphismPcGroup( n1 ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtbls:= CharacterTable( "7^(2+1+2):GL2(7)" );;[127X[104X
    [4X[25Xgap>[125X [27Xnfusm:= PossibleClassFusions( tbln, m );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( tbln, nfusm, m ) );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xnfuss:= PossibleClassFusions( tbln, tbls );;[127X[104X
    [4X[25Xgap>[125X [27Xsfusm:= PossibleClassFusions( tbls, m );;[127X[104X
    [4X[25Xgap>[125X [27Xnfusm:= SetOfComposedClassFusions( sfusm, nfuss );;[127X[104X
    [4X[25Xgap>[125X [27XLength( nfusm );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  use  the  condition  that  the  class fusions from [22XN[122X into [22XM[122X factors
  through  [22XU[122X.  This  determines  the  class  fusion  of  [22XU[122X  into [22XM[122X up to table
  automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnfusu:= PossibleClassFusions( tbln, tblu );;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= Filtered( ufusm, map2 -> ForAny( nfusu, [127X[104X
    [4X[25X>[125X [27X       map1 -> CompositionMaps( map2, map1 ) in nfusm ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( tblu, ufusm, m ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet [22XC[122X be the centralizer in [22XU[122X of the normal subgroup of order [22X7[122X; note that [22XC[122X
  is  the  [10X7B[110X  centralizer  on  [22XM[122X.  We can use the information about the class
  fusion  of  [22XU[122X into [22XM[122X for determining the class fusion of [22XC[122X into [22XM[122X. The class
  fusion  of [22XC[122X into [22XM[122X is not determined by the character tables, but the class
  fusion  of  [22XC[122X  into  [22XU[122X  is determined up to table automorphisms, so the same
  holds for the class fusion of [22XC[122X into [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtblc:= CharacterTable( "MC7B" );                             [127X[104X
    [4X[28XCharacterTable( "7^1+4.2A7" )[128X[104X
    [4X[25Xgap>[125X [27Xcfusm:= PossibleClassFusions( tblc, m );;             [127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( tblc, cfusm, m ) );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xcfusu:= PossibleClassFusions( tblc, tblu );;[127X[104X
    [4X[25Xgap>[125X [27Xcfusm:= SetOfComposedClassFusions( ufusm, cfusu );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( tblc, cfusm, m ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  
  [1X9.3-6 [33X[0;0Y[22X3^7.O_7(3):2 → Fi_24[122X[101X[1X (November 2010)[133X[101X
  
  [33X[0;0YThe  class  fusion  of  the maximal subgroup [22XM ≅ 3^7.O_7(3):2[122X of [22XG = Fi_24 =
  F_{3+}.2[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "3^7.O7(3):2" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "F3+.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xmfust:= PossibleClassFusions( m, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( m, mfust, t ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  will  use  the fact that the elementary abelian normal subgroup of order
  [22X3^7[122X  in [22XM[122X contains an element [22Xx[122X, say, in the class [10X3A[110X of [22XG[122X. This fact can be
  shown as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnsg:= ClassPositionsOfNormalSubgroups( m );[127X[104X
    [4X[28X[ [ 1 ], [ 1 .. 4 ], [ 1 .. 158 ], [ 1 .. 291 ] ][128X[104X
    [4X[25Xgap>[125X [27XSum( SizesConjugacyClasses( m ){ nsg[2] } );[127X[104X
    [4X[28X2187[128X[104X
    [4X[25Xgap>[125X [27X3^7;[127X[104X
    [4X[28X2187[128X[104X
    [4X[25Xgap>[125X [27Xrest:= Set( mfust, map -> map{ nsg[2] } );[127X[104X
    [4X[28X[ [ 1, 4, 5, 6 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( rest, l -> ClassNames( t, "Atlas" ){ l } );[127X[104X
    [4X[28X[ [ "1A", "3A", "3B", "3C" ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  normalizer  [22XS[122X  of  [22X⟨  x ⟩[122X in [22XG[122X has the form [22XS_3 × O_8^+(3):S_3[122X, and the
  order of [22XU = S ∩ M = N_M( ⟨ x ⟩)[122X is [22X53059069440[122X, so [22XU[122X has index [22X3360[122X in [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "F3+.2N3A" );[127X[104X
    [4X[28XCharacterTable( "S3xO8+(3):S3" )[128X[104X
    [4X[25Xgap>[125X [27XPowerMap( m, 2 )[4];[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xsize_u:= 2 * SizesCentralizers( m )[ 2 ];[127X[104X
    [4X[28X53059069440[128X[104X
    [4X[25Xgap>[125X [27XSize( s ) / size_u;[127X[104X
    [4X[28X3360[128X[104X
  [4X[32X[104X
  
  [33X[0;0YUsing  the  list  of  maximal  subgroups  of  [22XO_8^+(3)[122X, we see that only the
  maximal  subgroups  of  the  type  [22X3^6:L_4(3)[122X  have  index  dividing [22X3360[122X in
  [22XO_8^+(3)[122X.  (There  are three classes of such subgroups.) This implies that [22XU[122X
  contains a subgroup of the type [22XS_3 × 3^6:L_4(3)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo8p3:= CharacterTable( "O8+(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( o8p3 ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( mx, x -> 3360 mod Index( o8p3, x ) = 0 );[127X[104X
    [4X[28X[ CharacterTable( "3^6:L4(3)" ), CharacterTable( "O8+(3)M8" ), [128X[104X
    [4X[28X  CharacterTable( "O8+(3)M9" ) ][128X[104X
    [4X[25Xgap>[125X [27XList( filt, x -> Index( o8p3, x ) );[127X[104X
    [4X[28X[ 1120, 1120, 1120 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  compute  the  possible  class  fusions from [22XU[122X into [22XM[122X and [22XS[122X in two steps,
  because  this  is  faster.  First  the  possible  class  fusions from [22XU^'' ≅
  3^6:L_4(3)[122X  into  [22XM[122X  and  [22XS[122X are computed, and then these fusions are used to
  derive approximations for the fusions from [22XU[122X into [22XM[122X and [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xuu:= filt[1];;[127X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "Symmetric", 3 ) * uu;[127X[104X
    [4X[28XCharacterTable( "Sym(3)x3^6:L4(3)" )[128X[104X
    [4X[25Xgap>[125X [27Xuufusm:= PossibleClassFusions( uu, m );;[127X[104X
    [4X[25Xgap>[125X [27XLength( uufusm );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27Xapprox:= List( uufusm, map -> CompositionMaps( map,[127X[104X
    [4X[25X>[125X [27X                  InverseMap( GetFusionMap( uu, u ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= Concatenation( List( approx, map ->[127X[104X
    [4X[25X>[125X [27X       PossibleClassFusions( u, m, rec( fusionmap:= map ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ufusm );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27Xuufuss:= PossibleClassFusions( uu, s );;[127X[104X
    [4X[25Xgap>[125X [27XLength( uufuss );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27Xapprox:= List( uufuss, map -> CompositionMaps( map,[127X[104X
    [4X[25X>[125X [27X             InverseMap( GetFusionMap( uu, u ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xufuss:= Concatenation( List( approx, map ->[127X[104X
    [4X[25X>[125X [27X  PossibleClassFusions( u, s, rec( fusionmap:= map ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ufuss );[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  the  possible  class  fusions  from  [22XS[122X  into  [22XG[122X,  and  the
  compositions of these maps with the possible class fusions from [22XU[122X into [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( sfust, ufuss );;[127X[104X
    [4X[25Xgap>[125X [27XLength( comp );[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  turns  out that only one orbit of the possible class fusions from [22XM[122X to [22XG[122X
  is compatible with these possible class fusions from [22XU[122X to [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( mfust, map2 -> ForAny( ufusm, map1 ->[127X[104X
    [4X[25X>[125X [27X       CompositionMaps( map2, map1 ) in comp ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( filt );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( m, filt, t ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe class fusion stored in the [5XGAP[105X Character Table Library is one of them.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, t ) in filt;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.3-7 [33X[0;0Y[22X^2E_6(2)N3C → ^2E_6(2)[122X[101X[1X (January 2019)[133X[101X
  
  [33X[0;0YLet  [22XG  = ^2E_6(2)[122X, and [22Xg ∈ G[122X in the conjugacy class [10X3C[110X. Using a permutation
  representation  of  [22XG[122X,  Frank  Lübeck  has computed a representation and the
  character table of the maximal subgroup [22XN = N_G(⟨ g ⟩)[122X of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "2E6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xpos3CinG:= Position( ClassNames( t ), "3c" );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27Xn:= CharacterTable( "2E6(2)N3C" );;[127X[104X
    [4X[25Xgap>[125X [27Xnclasses:= SizesConjugacyClasses( n );;[127X[104X
    [4X[25Xgap>[125X [27Xpos3CinN:= Filtered( [ 1 .. NrConjugacyClasses( n ) ],[127X[104X
    [4X[25X>[125X [27X                        i -> nclasses[i] = 2 );[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27Xnfust:= PossibleClassFusions( n, t );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( nfust, x -> x[ pos3CinN[1] ] = pos3CinG );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize( n ) = 2 * SizesCentralizers( t )[ pos3CinG ];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XForAll( Irr( n ), x -> IsInt( x[ pos3CinN[1] ] ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe class fusion of [22XN[122X in [22XG[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrep:= RepresentativesFusions( n, nfust, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( rep );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe use the fact that [22Xg[122X is contained in a subgroup [22XS ≅ Fi_22[122X of [22XG[122X, [22X...[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "Fi22" );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( sfust, x -> x[6] = pos3CinG );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xpos3CinS:= 6;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22X...[122X  and  that [22XU = N_S(⟨ g ⟩) ≅ 3^1+6:2^3+4:3^2:2[122X is a maximal subgroup of [22XS[122X
  whose character table is available. Thus [22XU ≤ N[122X, of index four.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( Maxes( s )[11] );[127X[104X
    [4X[28XCharacterTable( "3^(1+6):2^(3+4):3^2:2" )[128X[104X
    [4X[25Xgap>[125X [27Xuclasses:= SizesConjugacyClasses( u );;[127X[104X
    [4X[25Xgap>[125X [27Xpos3CinU:= Filtered( [ 1 .. NrConjugacyClasses( u ) ],[127X[104X
    [4X[25X>[125X [27X                        i -> uclasses[i] = 2 );[127X[104X
    [4X[28X[ 2 ][128X[104X
    [4X[25Xgap>[125X [27Xufuss:= PossibleClassFusions( u, s );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( ufuss, x -> x[ pos3CinU[1] ] = pos3CinS );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize( n ) / Size( u );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YComposing  the  class  fusions  of  [22XU[122X  in  [22XN[122X and [22XN[122X in [22XG[122X must be equal to the
  composition  of  the  class  fusions  of [22XU[122X in [22XS[122X and [22XS[122X in [22XG[122X. This reduces the
  number of candidates for the fusion of [22XN[122X in [22XG[122X from four to two.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xufusn:= PossibleClassFusions( u, n );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( sfust, ufuss );;[127X[104X
    [4X[25Xgap>[125X [27Xgood:= Filtered( nfust, map2 -> ForAny( ufusn,[127X[104X
    [4X[25X>[125X [27X              map1 -> CompositionMaps( map2, map1 ) in comp ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( good );[127X[104X
    [4X[28X1728[128X[104X
    [4X[25Xgap>[125X [27Xgoodrep:= RepresentativesFusions( n, good, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( goodrep );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  use  the  fact  that  [22Xg[122X  and  thus  [22XN[122X  is invariant under an outer
  automorphism  [22Xα[122X,  say,  of  order three of [22XG[122X. Note that such an automorphism
  acts  nontrivially  on  the  conjugacy classes of [22XG[122X, for example because the
  class  fusion  of  [22XG[122X  into  [22XG.3  = ⟨ G, α ⟩[122X shows the existence of orbits of
  length  three,  and  that the permutation action of [22Xα[122X on the classes of [22XG[122X is
  given  by  the  unique  subgroup  of  order  three  in  the  group  of table
  automorphisms of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtfust3:= GetFusionMap( t, CharacterTable( "2E6(2).3" ) );;[127X[104X
    [4X[25Xgap>[125X [27XNumber( InverseMap( tfust3 ), IsList );[127X[104X
    [4X[28X14[128X[104X
    [4X[25Xgap>[125X [27Xautt:= AutomorphismsOfTable( t );;[127X[104X
    [4X[25Xgap>[125X [27Xord3:= Filtered( autt, x -> Order( x ) = 3 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ord3 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xalpha:= ord3[1];;[127X[104X
    [4X[25Xgap>[125X [27Xpos3CinG ^ alpha = pos3CinG;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  character  table  of [22XN[122X has [22X26[122X table automorphisms of order three. We do
  not  know  which of them (or perhaps the identity permutation) is induced by
  the  restriction [22Xα_N[122X of [22Xα[122X to [22XN[122X, but the embedding [22Xι: N → G[122X satisfies [22Xα ∘ ι =
  ι  ∘  α_N[122X,  and  we  can  check each fusion candidate for the existence of a
  candidate for [22Xα_N[122X such that this relation holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xautn:= AutomorphismsOfTable( n );;[127X[104X
    [4X[25Xgap>[125X [27Xord3:= Filtered( autn, x -> Order( x ) = 3 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ord3 );[127X[104X
    [4X[28X26[128X[104X
    [4X[25Xgap>[125X [27XAdd( ord3, () );[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( rep, map -> ForAny( ord3, beta ->[127X[104X
    [4X[25X>[125X [27XOnTuples( map, alpha ) = Permuted( map, beta ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( filt );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAgain,  the  number  of  candidates for the fusion of [22XN[122X in [22XG[122X is reduced from
  four to two. Moreover, we are lucky because only one candidate satifies also
  the first criterion we have checked.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinter:= Intersection( good, filt );[127X[104X
    [4X[28X[ [ 1, 7, 5, 6, 7, 2, 3, 4, 27, 30, 24, 32, 25, 26, 9, 11, 12, 13, [128X[104X
    [4X[28X      10, 14, 19, 19, 19, 16, 17, 18, 21, 58, 61, 62, 67, 68, 69, 57, [128X[104X
    [4X[28X      72, 59, 75, 76, 77, 78, 79, 80, 64, 65, 66, 60, 81, 82, 5, 6, [128X[104X
    [4X[28X      7, 6, 7, 7, 7, 7, 6, 7, 6, 7, 7, 24, 25, 27, 26, 28, 30, 29, [128X[104X
    [4X[28X      31, 32, 31, 32, 32, 32, 32, 31, 32, 31, 32, 51, 52, 52, 52, 52, [128X[104X
    [4X[28X      74, 76, 77, 77, 75, 74, 76, 74, 75, 99, 100, 101, 102, 4, 20, [128X[104X
    [4X[28X      29, 31, 32, 36, 36, 42, 42, 39, 40, 41, 49, 49, 49, 49, 49, 49, [128X[104X
    [4X[28X      71, 112, 112, 114, 115, 116 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  class  fusion  stored  in  the  [5XGAP[105X  Character  Table  Library  is this
  candidate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( n, t ) = inter[1];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XRemark:[113X[133X
  
  [33X[0;0YNote  that  the  structure of [22XN[122X is [22X3^1+6:2^3+6:3^2:2[122X, as is stated in [Nor].
  The structure [22X3^1+6.2^3+6.(S_3 × 3)[122X claimed in the [5XAtlas[105X [CCN+85, p. 191] is
  wrong,  as  we can read off for example from the fact that [22XN[122X has exactly two
  linear characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength( LinearCharacters( n ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X9.4 [33X[0;0YFusions Determined Using Commutative Diagrams Involving Factor Groups[133X[101X
  
  
  [1X9.4-1 [33X[0;0Y[22X3.A_7 → 3.Suz[122X[101X[1X (December 2010)[133X[101X
  
  [33X[0;0YThe  maximal  subgroups  of type [22XA_7[122X in the sporadic simple Suzuki group [22XSuz[122X
  lift  to  groups  of the type [22X3.A_7[122X in [22X3.Suz[122X. This can be seen from the fact
  that [22X3.Suz[122X does not admit a class fusion from [22XA_7[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Suz" );;[127X[104X
    [4X[25Xgap>[125X [27X3t:= CharacterTable( "3.Suz" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "A7" );;[127X[104X
    [4X[25Xgap>[125X [27X3s:= CharacterTable( "3.A7" );;[127X[104X
    [4X[25Xgap>[125X [27XPossibleClassFusions( s, 3t );[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe class fusion of [22X3.A_7[122X into [22X3.Suz[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X3sfus3t:= PossibleClassFusions( 3s, 3t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( 3sfus3t );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XRepresentativesFusions( 3s, 3sfus3t, 3t );[127X[104X
    [4X[28X[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48, [128X[104X
    [4X[28X      49, 50, 48, 49, 50 ], [128X[104X
    [4X[28X  [ 1, 11, 12, 4, 36, 37, 13, 16, 23, 82, 83, 32, 100, 101, 44, 38, [128X[104X
    [4X[28X      41, 48, 112, 116, 48, 115, 113 ] ][128X[104X
    [4X[25Xgap>[125X [27XClassPositionsOfCentre( 3t );[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  the  possible fusions in the second orbit avoid the centre of
  [22X3.Suz[122X. Since the preimages in [22X3.Suz[122X of the [22XA_7[122X type subgroups of [22XSuz[122X contain
  the centre of [22X3.Suz[122X, we know that the class fusion of these preimages belong
  to  the first orbit. This can be formalized by checking the commutativity of
  the diagram of fusions between [22X3.A_7[122X, [22X3.Suz[122X, and their factors [22XA_7[122X and [22XSuz[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( sfust );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( 3sfus3t, map -> CompositionMaps( GetFusionMap( 3t, t ),[127X[104X
    [4X[25X>[125X [27X                                        map )[127X[104X
    [4X[25X>[125X [27X              = CompositionMaps( sfust[1], GetFusionMap( 3s, s ) ) );[127X[104X
    [4X[28X[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48, [128X[104X
    [4X[28X      49, 50, 48, 49, 50 ], [128X[104X
    [4X[28X  [ 1, 3, 2, 7, 9, 8, 16, 16, 26, 28, 27, 32, 34, 33, 47, 47, 47, 48, [128X[104X
    [4X[28X      50, 49, 48, 50, 49 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  the  class fusion of maximal [22X3.A_7[122X type subgroups of [22X3.Suz[122X is determined
  up  to  table  automorphisms. One of these fusions is stored on the table of
  [22X3.A_7[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRepresentativesFusions( 3s, filt, 3t );[127X[104X
    [4X[28X[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48, [128X[104X
    [4X[28X      49, 50, 48, 49, 50 ] ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( 3s, 3t ) in filt;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso  the class fusions in the other orbit belong to subgroups of type [22X3.A_7[122X
  in   [22X3.Suz[122X.   Note   that   [22XSuz[122X  contains  maximal  subgroups  of  the  type
  [22X3_2.U_4(3).2_3^'[122X  (see [CCN+85,  p.  131]),  and  the  [22XA_7[122X type subgroups of
  [22XU_4(3)[122X  (see [CCN+85, p. 52]) lift to groups of the type [22X3.A_7[122X in [22X3_2.U_4(3)[122X
  because  [22X3_2.U_4(3)[122X does not admit a class fusion from [22XA_7[122X. The preimages in
  [22X3.Suz[122X of the [22X3.A_7[122X type subgroups of [22XSuz[122X have the structure [22X3 × 3.A_7[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "3_2.U4(3)" );;[127X[104X
    [4X[25Xgap>[125X [27XPossibleClassFusions( s, u );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( 3s, u ) );[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  
  [1X9.4-2 [33X[0;0Y[22XS_6 → U_4(2)[122X[101X[1X (September 2011)[133X[101X
  
  [33X[0;0YThe  simple  group [22XG = U_4(2)[122X contains a maximal subgroup [22XU[122X of type [22XS_6[122X. The
  class fusion from [22XU[122X to [22XG[122X is unique up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "S6" );[127X[104X
    [4X[28XCharacterTable( "A6.2_1" )[128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U4(2)" );[127X[104X
    [4X[28XCharacterTable( "U4(2)" )[128X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );[127X[104X
    [4X[28X[ [ 1, 3, 6, 7, 9, 10, 3, 2, 9, 16, 15 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 6, 9, 10, 2, 3, 9, 15, 16 ] ][128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( s, sfust, t ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn the double cover [22X2.G[122X of [22XG[122X, [22XU[122X lifts to the double cover [22X2.U[122X of [22XU[122X (which is
  unique  up  to isomorphism). Also the class fusion from [22X2.U[122X to [22X2.G[122X is unique
  up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2t:= CharacterTable( "2.U4(2)" );[127X[104X
    [4X[28XCharacterTable( "2.U4(2)" )[128X[104X
    [4X[25Xgap>[125X [27X2s:= CharacterTable( "2.A6.2_1" );[127X[104X
    [4X[28XCharacterTable( "2.A6.2_1" )[128X[104X
    [4X[25Xgap>[125X [27X2sfus2t:= PossibleClassFusions( 2s, 2t );[127X[104X
    [4X[28X[ [ 1, 2, 4, 11, 12, 9, 10, 15, 16, 17, 3, 4, 15, 24, 25, 26, 26 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 11, 12, 9, 10, 15, 16, 17, 3, 4, 15, 25, 24, 26, 26 ] ][128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( 2s, 2sfus2t, 2t ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHowever,  the  two  possible  fusions  from [22X2.U[122X to [22X2.G[122X are lifts of the same
  class fusion from [22XU[122X to [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2sfuss:= GetFusionMap( 2s, s );[127X[104X
    [4X[28X[ 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 11 ][128X[104X
    [4X[25Xgap>[125X [27X2tfust:= GetFusionMap( 2t, t );;[127X[104X
    [4X[25Xgap>[125X [27Xinduced:= Set( 2sfus2t, x -> CompositionMaps( 2tfust,[127X[104X
    [4X[25X>[125X [27X     CompositionMaps( x, InverseMap( 2sfuss ) ) ) );[127X[104X
    [4X[28X[ [ 1, 3, 7, 6, 9, 10, 2, 3, 9, 15, 16 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  point  is that the outer automorphism of [22XS_6[122X that makes the two fusions
  from  [22XU[122X  to  [22XG[122X  equivalent  does  not lift to [22X2.U[122X, and that we have silently
  assumed a fixed factor fusion from [22X2.U[122X to [22XU[122X. Note that composing this factor
  fusion  with  the  automorphism  of  [22XU[122X would also yield a factor fusion, and
  w. r. t. the  commutative  diagram  involving  this factor fusion, the other
  possible  class  fusion  from [22XU[122X to [22XG[122X is induced by the possible fusions from
  [22X2.U[122X to [22X2.G[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xauts:= AutomorphismsOfTable( s );[127X[104X
    [4X[28XGroup([ (3,4)(7,8)(10,11) ])[128X[104X
    [4X[25Xgap>[125X [27Xother:= OnTuples( 2sfuss, GeneratorsOfGroup( auts )[1] );[127X[104X
    [4X[28X[ 1, 1, 2, 4, 4, 3, 3, 5, 6, 6, 8, 7, 9, 11, 11, 10, 10 ][128X[104X
    [4X[25Xgap>[125X [27XSet( 2sfus2t, x -> CompositionMaps( 2tfust,[127X[104X
    [4X[25X>[125X [27X     CompositionMaps( x, InverseMap( other ) ) ) );[127X[104X
    [4X[28X[ [ 1, 3, 6, 7, 9, 10, 3, 2, 9, 16, 15 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  library table of [22XU[122X stores the class fusion to [22XG[122X that is compatible with
  the stored factor fusion from [22X2.U[122X to [22XU[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) in induced;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.5  [33X[0;0YFusions  Determined  Using  Commutative  Diagrams Involving Automorphic[101X
  [1XExtensions[133X[101X
  
  
  [1X9.5-1 [33X[0;0Y[22XU_3(8).3_1 → ^2E_6(2)[122X[101X[1X (December 2010)[133X[101X
  
  [33X[0;0YAccording  to  the  [5XAtlas[105X  (see [CCN+85,  p.  191]),  the group [22XG = ^2E_6(2)[122X
  contains  a maximal subgroup [22XU[122X of the type [22XU_3(8).3_1[122X. The class fusion of [22XU[122X
  into [22XG[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "U3(8).3_1" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "2E6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( sfust );[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( s, sfust, t ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  automorphic extension [22XG.2 = ^2E_6(2).2[122X of [22XG[122X, the subgroup [22XU[122X extends
  to  a  group  [22XU.2[122X  of  the type [22XU_3(8).6[122X (again, see  [CCN+85, p. 191]). The
  class fusion of [22XU.2[122X into [22XG.2[122X is unique up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( "U3(8).6" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "2E6(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xs2fust2:= PossibleClassFusions( s2, t2 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( s2fust2 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( s2, s2fust2, t2 ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOnly  half  of  the possible class fusions from [22XU[122X into [22XG[122X are compatible with
  the embeddings of [22XU[122X into [22XG.2[122X via [22XU.2[122X and [22XG[122X, and the compatible maps form one
  orbit under table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsfuss2:= PossibleClassFusions( s, s2 );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( s2fust2, sfuss2 );;[127X[104X
    [4X[25Xgap>[125X [27Xtfust2:= PossibleClassFusions( t, t2 );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( sfust, map -> ForAny( tfust2,[127X[104X
    [4X[25X>[125X [27X              map2 -> CompositionMaps( map2, map ) in comp ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( filt );[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( s, filt, t ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  us  see  which classes of [22XU[122X and [22XG[122X are involved in the disambiguation of
  the  class  fusion. The [21Xgood[121X fusion candidates differ from the excluded ones
  on  the classes at the positions [22X31[122X to [22X36[122X: Under all possible class fusions,
  two  pairs  of  classes are mapped to the classes [22X81[122X and [22X82[122X of [22XG[122X; from these
  classes,  the  excluded  maps  fuse classes at odd positions with classes at
  even positions, whereas the [21Xgood[121X class fusions do not have this property.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSet( filt, x -> x{ [ 31 .. 36 ] } );[127X[104X
    [4X[28X[ [ 74, 74, 81, 82, 81, 82 ], [ 74, 74, 82, 81, 82, 81 ], [128X[104X
    [4X[28X  [ 81, 82, 74, 74, 81, 82 ], [ 81, 82, 81, 82, 74, 74 ], [128X[104X
    [4X[28X  [ 82, 81, 74, 74, 82, 81 ], [ 82, 81, 82, 81, 74, 74 ] ][128X[104X
    [4X[25Xgap>[125X [27XSet( Difference( sfust, filt ), x -> x{ [ 31 .. 36 ] } );[127X[104X
    [4X[28X[ [ 74, 74, 81, 82, 82, 81 ], [ 74, 74, 82, 81, 81, 82 ], [128X[104X
    [4X[28X  [ 81, 82, 74, 74, 82, 81 ], [ 81, 82, 82, 81, 74, 74 ], [128X[104X
    [4X[28X  [ 82, 81, 74, 74, 81, 82 ], [ 82, 81, 81, 82, 74, 74 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNone  of  the  possible  class  fusions  from  [22XU[122X to [22XU.2[122X fuses classes at odd
  positions in the interval from [22X31[122X to [22X36[122X with classes at even positions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSet( sfuss2, x -> x{ [ 31 .. 36 ] } );[127X[104X
    [4X[28X[ [ 28, 29, 30, 31, 30, 31 ], [ 29, 28, 31, 30, 31, 30 ], [128X[104X
    [4X[28X  [ 30, 31, 28, 29, 30, 31 ], [ 30, 31, 30, 31, 28, 29 ], [128X[104X
    [4X[28X  [ 31, 30, 29, 28, 31, 30 ], [ 31, 30, 31, 30, 29, 28 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis suffices to exclude the [21Xbad[121X fusion candidates because no further fusion
  of the relevant classes of [22XG[122X happens in [22XG.2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( tfust2, x -> x{ [ 74, 81, 82 ] } );[127X[104X
    [4X[28X[ [ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ], [128X[104X
    [4X[28X  [ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ], [128X[104X
    [4X[28X  [ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(The same holds for the fusion of the relevant classes of [22XU.2[122X in [22XG.2[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( s2fust2, x -> x{ [ 28 .. 31 ] } );[127X[104X
    [4X[28X[ [ 65, 65, 70, 71 ], [ 65, 65, 71, 70 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we check that a correct map is stored on the library table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) in filt;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.5-2 [33X[0;0Y[22XL_3(4).2_1 → U_6(2)[122X[101X[1X (December 2010)[133X[101X
  
  [33X[0;0YAccording to the [5XAtlas[105X (see [CCN+85, p. 115]), the group [22XG = U_6(2)[122X contains
  a maximal subgroup [22XU[122X of the type [22XL_3(4).2_1[122X. The class fusion of [22XU[122X into [22XG[122X is
  ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L3(4).2_1" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "U6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( sfust );[127X[104X
    [4X[28X27[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( s, sfust, t ) );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the automorphic extension [22XG.3 = U_6(2).3[122X of [22XG[122X, the subgroup [22XU[122X extends to
  a  group  [22XU.3[122X of the type [22XL_3(4).6[122X (again, see  [CCN+85, p. 115]). The class
  fusion of [22XU.3[122X into [22XG.3[122X is unique up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs3:= CharacterTable( "L3(4).6" );;[127X[104X
    [4X[25Xgap>[125X [27Xt3:= CharacterTable( "U6(2).3" );;[127X[104X
    [4X[25Xgap>[125X [27Xs3fust3:= PossibleClassFusions( s3, t3 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( s3fust3 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( s3, s3fust3, t3 ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHere  the argument used in Section [14X9.5-1[114X does not work, because all possible
  class fusions from [22XU[122X into [22XG[122X are compatible with the embeddings of [22XU[122X into [22XG.3[122X
  via [22XU.3[122X and [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsfuss3:= PossibleClassFusions( s, s3 );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= SetOfComposedClassFusions( s3fust3, sfuss3 );;[127X[104X
    [4X[25Xgap>[125X [27Xtfust3:= PossibleClassFusions( t, t3 );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust = Filtered( sfust, map -> ForAny( tfust3,[127X[104X
    [4X[25X>[125X [27X               map2 -> CompositionMaps( map2, map ) in comp ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YConsider  the  elements  of  order  four  in [22XU[122X. There are three such classes
  inside [22XU^' ≅ L_3(4)[122X, which fuse to one class of [22XU.3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( s );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 4, 4, 5, 7, 2, 4, 6, 8, 8, 8 ][128X[104X
    [4X[25Xgap>[125X [27Xsfuss3;[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 10, 10 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThese  classes  of  [22XU[122X  fuse  into some of the classes [22X10[122X to [22X12[122X of [22XG[122X. In [22XG.3[122X,
  these three classes fuse into one class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSet( sfust, map -> map{ [ 4 .. 6 ] } );[127X[104X
    [4X[28X[ [ 10, 10, 10 ], [ 10, 10, 11 ], [ 10, 10, 12 ], [ 10, 11, 10 ], [128X[104X
    [4X[28X  [ 10, 11, 11 ], [ 10, 11, 12 ], [ 10, 12, 10 ], [ 10, 12, 11 ], [128X[104X
    [4X[28X  [ 10, 12, 12 ], [ 11, 10, 10 ], [ 11, 10, 11 ], [ 11, 10, 12 ], [128X[104X
    [4X[28X  [ 11, 11, 10 ], [ 11, 11, 11 ], [ 11, 11, 12 ], [ 11, 12, 10 ], [128X[104X
    [4X[28X  [ 11, 12, 11 ], [ 11, 12, 12 ], [ 12, 10, 10 ], [ 12, 10, 11 ], [128X[104X
    [4X[28X  [ 12, 10, 12 ], [ 12, 11, 10 ], [ 12, 11, 11 ], [ 12, 11, 12 ], [128X[104X
    [4X[28X  [ 12, 12, 10 ], [ 12, 12, 11 ], [ 12, 12, 12 ] ][128X[104X
    [4X[25Xgap>[125X [27XSet( tfust3, map -> map{ [ 10 .. 12 ] } );[127X[104X
    [4X[28X[ [ 10, 10, 10 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis means that the automorphism [22Xα[122X of [22XG[122X that is induced by the action of [22XG.3[122X
  permutes  the classes [22X10[122X to [22X12[122X of [22XG[122X transitively. The fact that [22XU[122X extends to
  [22XU.3[122X  in  [22XG.3[122X means that [22XU[122X is invariant under [22Xα[122X. This implies that [22XU[122X contains
  either  no  elements from the classes [22X10[122X to [22X12[122X or elements from all of these
  classes.  The  possible  class fusions from [22XU[122X to [22XG[122X satisfying this condition
  form one orbit under table automprhisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFiltered( sfust, map -> Intersection( map, [ 10 .. 12 ] ) = [] );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( sfust, map -> IsSubset( map, [ 10 .. 12 ] ) );[127X[104X
    [4X[28X[ [ 1, 3, 7, 10, 11, 12, 15, 24, 4, 14, 23, 26, 27, 28 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 10, 12, 11, 15, 24, 4, 14, 23, 26, 28, 27 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 11, 10, 12, 15, 24, 4, 14, 23, 27, 26, 28 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 11, 12, 10, 15, 24, 4, 14, 23, 27, 28, 26 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 12, 10, 11, 15, 24, 4, 14, 23, 28, 26, 27 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 12, 11, 10, 15, 24, 4, 14, 23, 28, 27, 26 ] ][128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( s, filt, t ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we check that a correct map is stored on the library table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) in filt;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.6 [33X[0;0YConditions Imposed by Brauer Tables[133X[101X
  
  [33X[0;0YThe  examples  in this section show that symmetries can be broken as soon as
  the  class  fusions between two ordinary tables shall be compatible with the
  corresponding  Brauer  character  tables. More precisely, we assume that the
  class  fusion from each Brauer table to its ordinary table is already fixed;
  choosing  these  fusions  consistently  can  be  a  nontrivial task, solving
  so-called  [21Xgenerality  problems[121X  may  require  the  construction  of certain
  modules, similar to the arguments used in [14X9.6-3[114X below.[133X
  
  
  [1X9.6-1 [33X[0;0Y[22XL_2(16).4 → J_3.2[122X[101X[1X (January 2004)[133X[101X
  
  [33X[0;0YIt can happen that Brauer tables decide ambiguities of class fusions between
  the  corresponding  ordinary  tables. An easy example is the class fusion of
  [22XL_2(16).4[122X into [22XJ_3.2[122X. The ordinary tables admit four possible class fusions,
  of which two are essentially different.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L2(16).4" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J3.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( s, t );[127X[104X
    [4X[28X[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ][128X[104X
    [4X[25Xgap>[125X [27XRepresentativesFusions( s, fus, t );[127X[104X
    [4X[28X[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YUsing Brauer tables, we will see that just one fusion is admissible.[133X
  
  [33X[0;0YWe  can  exclude  two possible fusions by the fact that their images all lie
  inside  the normal subgroup [22XJ_3[122X, but [22XJ_3[122X does not contain a subgroup of type
  [22XL_2(16).4[122X; so still one orbit of length two remains.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xj3:= CharacterTable( "J3" );;[127X[104X
    [4X[25Xgap>[125X [27XPossibleClassFusions( s, j3 );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( j3, t );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, [128X[104X
    [4X[28X  17, 17 ][128X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( fus,[127X[104X
    [4X[25X>[125X [27X         x -> not IsSubset( ClassPositionsOfDerivedSubgroup( t ), x ) );[127X[104X
    [4X[28X[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ], [128X[104X
    [4X[28X  [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  the  remaining  wrong  fusion  is  excluded  by the fact that the table
  automorphism  of [22XJ_3.2[122X that swaps the two classes of element order [22X17[122X –which
  swaps  two  of  the  possible  class fusions– does not live in the [22X2[122X-modular
  table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsmod2:= s mod 2;;[127X[104X
    [4X[25Xgap>[125X [27Xtmod2:= t mod 2;;[127X[104X
    [4X[25Xgap>[125X [27Xadmissible:= [];;  modmap:= "dummy";;[127X[104X
    [4X[25Xgap>[125X [27Xfor map in filt do[127X[104X
    [4X[25X>[125X [27X     modmap:= CompositionMaps( InverseMap( GetFusionMap( tmod2, t ) ),[127X[104X
    [4X[25X>[125X [27X                  CompositionMaps( map, GetFusionMap( smod2, s ) ) );[127X[104X
    [4X[25X>[125X [27X     if not fail in Decomposition( Irr( smod2 ),[127X[104X
    [4X[25X>[125X [27X           List( Irr( tmod2 ), chi -> chi{ modmap } ), "nonnegative" ) then[127X[104X
    [4X[25X>[125X [27X       AddSet( admissible, map );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xadmissible;[127X[104X
    [4X[28X[ [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  test  of  all  available  Brauer  tables is implemented in the function
  [10XCTblLib.Test.Decompositions[110X of the [5XGAP[105X Character Table Library ([Bre25]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCTblLib.Test.Decompositions( s, fus, t ) = admissible;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that [22Xp[122X-modular tables alone determine the class fusion uniquely; in
  fact the primes [22X2[122X and [22X3[122X suffice for that.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) in admissible;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XRemark:[113X[133X
  
  [33X[0;0YIn  May 2015,  the  [22X19[122X-modular character table of [22XJ_3[122X has been corrected, by
  swapping  the  two  classes  of  element order [22X17[122X. Since the class fusion of
  [22XL_2(16).4[122X  into  [22XJ_3.2[122X  is  uniquely  determined  by the [22X2[122X-modular tables of
  [22XL_2(16).4[122X and [22XJ_3.2[122X and since this class fusion has been compatible with the
  previous  version  of  the  [22X19[122X-modular table of [22XJ_3[122X, the correction does not
  affect the above arguments.[133X
  
  
  [1X9.6-2 [33X[0;0Y[22XL_2(17) → S_8(2)[122X[101X[1X (July 2004)[133X[101X
  
  [33X[0;0YThe  class  fusion  of  the  maximal  subgroup  [22XM ≅ L_2(17)[122X of [22XG = S_8(2)[122X is
  ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "L2(17)" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S8(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xmfust:= PossibleClassFusions( m, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( m, mfust, t ) );[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  Brauer  tables  for  [22XM[122X  and  [22XG[122X  determine  the class fusion up to table
  automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= CTblLib.Test.Decompositions( m, mfust, t );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( m, filt, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( repr );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( m, t ) in repr;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.6-3 [33X[0;0Y[22XL_2(19) → J_3[122X[101X[1X (April 2003)[133X[101X
  
  [33X[0;0YIt  can  happen  that  Brauer tables impose conditions such that ambiguities
  arise which are not visible if one considers only ordinary tables.[133X
  
  [33X[0;0YThe class fusion between the ordinary character tables of [22XL_2(19)[122X and [22XJ_3[122X is
  unique up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L2(19)" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J3" );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );[127X[104X
    [4X[28X[ [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 10, 11, 12, 14, 13, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 10, 11, 12, 14, 13, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 21, 20 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfusreps:= RepresentativesFusions( s, sfust, t );[127X[104X
    [4X[28X[ [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 20, 21 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  Galois  automorphism that permutes the three classes of element order [22X9[122X
  in  the  tables of ([22XL_2(19)[122X and) [22XJ_3[122X does not live in characteristic [22X19[122X. For
  example,  the  unique  irreducible  Brauer  character  of  degree [22X110[122X in the
  [22X19[122X-modular table of [22XJ_3[122X is [22Xφ_3[122X, and the value of this character on the class
  [10X9A[110X is [10X-1+2y9+&4[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtmod19:= t mod 19;[127X[104X
    [4X[28XBrauerTable( "J3", 19 )[128X[104X
    [4X[25Xgap>[125X [27Xdeg110:= Filtered( Irr( tmod19 ), phi -> phi[1] = 110 );[127X[104X
    [4X[28X[ Character( BrauerTable( "J3", 19 ),[128X[104X
    [4X[28X  [ 110, -2, 5, 2, 2, 0, 0, 1, 0, [128X[104X
    [4X[28X      -2*E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6-2*E(9)^7, [128X[104X
    [4X[28X      E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6+E(9)^7, [128X[104X
    [4X[28X      E(9)^2+E(9)^3+2*E(9)^4+2*E(9)^5+E(9)^6+E(9)^7, -2, -2, -1, 0, [128X[104X
    [4X[28X      0, E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12[128X[104X
    [4X[28X         +E(17)^14, [128X[104X
    [4X[28X      E(17)+E(17)^2+E(17)^4+E(17)^8+E(17)^9+E(17)^13+E(17)^15+E(17)^16[128X[104X
    [4X[28X     ] ) ][128X[104X
    [4X[25Xgap>[125X [27X9A:= Position( OrdersClassRepresentatives( tmod19 ), 9 );[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27Xdeg110[1][ 9A ];[127X[104X
    [4X[28X-2*E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6-2*E(9)^7[128X[104X
    [4X[25Xgap>[125X [27XAtlasIrrationality( "-1+2y9+&4" ) = deg110[1][ 9A ];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  turns  out  that  four  of  the  twelve  possible  class fusions are not
  compatible with the [22X19[122X-modular tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsmod19:= s mod 19;[127X[104X
    [4X[28XBrauerTable( "L2(19)", 19 )[128X[104X
    [4X[25Xgap>[125X [27Xcompatible:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor map in sfust do[127X[104X
    [4X[25X>[125X [27X     comp:= CompositionMaps( InverseMap( GetFusionMap( tmod19, t ) ),[127X[104X
    [4X[25X>[125X [27X     CompositionMaps( map, GetFusionMap( smod19, s ) ) );[127X[104X
    [4X[25X>[125X [27X     rest:= List( Irr( tmod19 ), phi -> phi{ comp } );[127X[104X
    [4X[25X>[125X [27X     if not fail in Decomposition( Irr( smod19 ), rest, "nonnegative" ) then[127X[104X
    [4X[25X>[125X [27X       Add( compatible, map );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xcompatible;[127X[104X
    [4X[28X[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 21, 20 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 21, 20 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YMoreover,  the subgroups of those table automorphisms of the ordinary tables
  that  leave  the set of compatible fusions invariant make two orbits on this
  set.  Indeed,  the two orbits belong to essentially different decompositions
  of the restriction of [22Xφ_3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xreps:= RepresentativesFusions( s, compatible, t );[127X[104X
    [4X[28X[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcompatiblemod19:= List( reps, map -> CompositionMaps([127X[104X
    [4X[25X>[125X [27X       InverseMap( GetFusionMap( tmod19, t ) ),[127X[104X
    [4X[25X>[125X [27X       CompositionMaps( map, GetFusionMap( smod19, s ) ) ) );[127X[104X
    [4X[28X[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14 ], [128X[104X
    [4X[28X  [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14 ] ][128X[104X
    [4X[25Xgap>[125X [27Xrest:= List( compatiblemod19, map -> Irr( tmod19 )[3]{ map } );;[127X[104X
    [4X[25Xgap>[125X [27Xdec:= Decomposition( Irr( smod19 ), rest, "nonnegative" );[127X[104X
    [4X[28X[ [ 0, 0, 1, 2, 1, 2, 2, 1, 0, 1 ], [ 0, 2, 0, 2, 0, 1, 2, 0, 2, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( Irr( smod19 ), phi -> phi[1] );[127X[104X
    [4X[28X[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order  to  decide  which  class  fusion  is  correct, we take the matrix
  representation of [22XJ_3[122X that affords [22Xφ_3[122X, restrict it to [22XL_2(19)[122X, which is the
  second  maximal  subgroup  of  [22XJ_3[122X, and compute the composition factors. For
  that,    we    use    a    representation    from   the   [5XAtlas[105X   of   Group
  Representations [WWT+],   and   access  it  via  the  [5XGAP[105X  package  [5XAtlasRep[105X
  ([WPN+22]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "atlasrep", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xprog:= AtlasProgram( "J3", "maxes", 2 );[127X[104X
    [4X[28Xrec( groupname := "J3", identifier := [ "J3", "J3G1-max2W1", 1 ], [128X[104X
    [4X[28X  program := <straight line program>, size := 3420, [128X[104X
    [4X[28X  standardization := 1, subgroupname := "L2(19)", version := "1" )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,[127X[104X
    [4X[25X>[125X [27X              Dimension, 110 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( gens );[127X[104X
    [4X[28Xrec( charactername := "110a", constituents := [ 3 ], [128X[104X
    [4X[28X  contents := "core", dim := 110, [128X[104X
    [4X[28X  generators := [ < immutable compressed matrix 110x110 over GF(19) >,[128X[104X
    [4X[28X      < immutable compressed matrix 110x110 over GF(19) > ], [128X[104X
    [4X[28X  groupname := "J3", id := "", [128X[104X
    [4X[28X  identifier := [ "J3", [ "J3G1-f19r110B0.m1", "J3G1-f19r110B0.m2" ], [128X[104X
    [4X[28X      1, 19 ], repname := "J3G1-f19r110B0", repnr := 35, [128X[104X
    [4X[28X  ring := GF(19), size := 50232960, standardization := 1, [128X[104X
    [4X[28X  type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27Xrestgens:= ResultOfStraightLineProgram( prog.program, gens.generators );[127X[104X
    [4X[28X[ < immutable compressed matrix 110x110 over GF(19) >, [128X[104X
    [4X[28X  < immutable compressed matrix 110x110 over GF(19) > ][128X[104X
    [4X[25Xgap>[125X [27Xmodule:= GModuleByMats( restgens, GF( 19 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfacts:= SMTX.CollectedFactors( module );;[127X[104X
    [4X[25Xgap>[125X [27XLength( facts );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27XList( facts, x -> x[1].dimension );[127X[104X
    [4X[28X[ 5, 7, 9, 11, 13, 15, 19 ][128X[104X
    [4X[25Xgap>[125X [27XList( facts, x -> x[2] );[127X[104X
    [4X[28X[ 1, 2, 1, 2, 2, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  means that there are seven pairwise nonisomorphic composition factors,
  the  smallest  one  of  dimension five. In other words, the first of the two
  maps  is  the correct one. Let us check whether this map equals the one that
  is stored on the library table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) = reps[1];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XRemark:[113X[133X
  
  [33X[0;0YIn  May 2015,  the  [22X19[122X-modular character table of [22XJ_3[122X has been corrected, by
  swapping  the  two  classes  of  element  order  [22X17[122X.  This affects the above
  computations only in one place, where the values of the character [10Xdeg110[110X are
  shown.[133X
  
  
  [1X9.7 [33X[0;0YFusions Determined by Information about the Groups[133X[101X
  
  [33X[0;0YIn  the  examples  in  this  section,  character  theoretic arguments do not
  suffice  for  determining the class fusions. So we use computations with the
  groups  in  question  or information about these groups beyond the character
  table, and perhaps additionally character theoretic arguments.[133X
  
  [33X[0;0YThe   group   representations   are   taken   from   the   [5XAtlas[105X   of  Group
  Representations [WWT+]  and  are  accessed  via  the  [5XGAP[105X  package  [5XAtlasRep[105X
  ([WPN+22]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "atlasrep", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-1 [33X[0;0Y[22XU_3(3).2 → Fi_24^'[122X[101X[1X (November 2002)[133X[101X
  
  [33X[0;0YThe  group  [22XG = Fi_24^'[122X contains a maximal subgroup [22XH[122X of type [22XU_3(3).2[122X. From
  the  character  tables  of  [22XG[122X and [22XH[122X, one gets a lot of essentially different
  possibilities  (and  additionally  this  takes  quite some time). We use the
  description of [22XH[122X as the normalizer in [22XG[122X of a [22XU_3(3)[122X type subgroup containing
  elements  in  the  classes  [10X2B[110X,  [10X3D[110X,  [10X3E[110X,  [10X4C[110X,  [10X4C[110X,  [10X6J[110X,  [10X7B[110X,  [10X8C[110X,  and  [10X12M[110X
  (see [BN95]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "F3+" );[127X[104X
    [4X[28XCharacterTable( "F3+" )[128X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "U3(3).2" );[127X[104X
    [4X[28XCharacterTable( "U3(3).2" )[128X[104X
    [4X[25Xgap>[125X [27Xtnames:= ClassNames( t, "ATLAS" );[127X[104X
    [4X[28X[ "1A", "2A", "2B", "3A", "3B", "3C", "3D", "3E", "4A", "4B", "4C", [128X[104X
    [4X[28X  "5A", "6A", "6B", "6C", "6D", "6E", "6F", "6G", "6H", "6I", "6J", [128X[104X
    [4X[28X  "6K", "7A", "7B", "8A", "8B", "8C", "9A", "9B", "9C", "9D", "9E", [128X[104X
    [4X[28X  "9F", "10A", "10B", "11A", "12A", "12B", "12C", "12D", "12E", [128X[104X
    [4X[28X  "12F", "12G", "12H", "12I", "12J", "12K", "12L", "12M", "13A", [128X[104X
    [4X[28X  "14A", "14B", "15A", "15B", "15C", "16A", "17A", "18A", "18B", [128X[104X
    [4X[28X  "18C", "18D", "18E", "18F", "18G", "18H", "20A", "20B", "21A", [128X[104X
    [4X[28X  "21B", "21C", "21D", "22A", "23A", "23B", "24A", "24B", "24C", [128X[104X
    [4X[28X  "24D", "24E", "24F", "24G", "26A", "27A", "27B", "27C", "28A", [128X[104X
    [4X[28X  "29A", "29B", "30A", "30B", "33A", "33B", "35A", "36A", "36B", [128X[104X
    [4X[28X  "36C", "36D", "39A", "39B", "39C", "39D", "42A", "42B", "42C", [128X[104X
    [4X[28X  "45A", "45B", "60A" ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( s );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 2, 4, 6, 8, 12, 12 ][128X[104X
    [4X[25Xgap>[125X [27Xsfust:= List( [ "1A", "2B", "3D", "3E", "4C", "4C", "6J", "7B", "8C",[127X[104X
    [4X[25X>[125X [27X                   "12M" ], x -> Position( tnames, x ) );[127X[104X
    [4X[28X[ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50 ][128X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t, rec( fusionmap:= sfust ) );[127X[104X
    [4X[28X[ [ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50, 3, 9, 23, 28, 43, 43 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50, 3, 11, 23, 28, 50, 50 ] ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( s );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 2, 4, 6, 8, 12, 12 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  still  have  two possibilities, which differ on the outer classes of
  element order [22X4[122X and [22X12[122X.[133X
  
  [33X[0;0YOur  idea  is  to take a subgroup [22XU[122X of [22XH[122X that contains such elements, and to
  compute  the  possible  class  fusions  of  [22XU[122X  into [22XG[122X, via the factorization
  through a suitable maximal subgroup [22XM[122X of [22XG[122X.[133X
  
  [33X[0;0YWe  take  [22XU  =  N_H(⟨ g ⟩)[122X where [22Xg[122X is an element in the first class of order
  three elements of [22XH[122X; this is a maximal subgroup of [22XH[122X, of order [22X216[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMaxes( s );[127X[104X
    [4X[28X[ "U3(3)", "3^(1+2):SD16", "L3(2).2", "2^(1+4).S3", "4^2:D12" ][128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( s );[127X[104X
    [4X[28X[ 12096, 192, 216, 18, 96, 32, 24, 7, 8, 12, 48, 48, 6, 8, 12, 12 ][128X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( Maxes( s )[2] );;[127X[104X
    [4X[25Xgap>[125X [27Xufuss:= GetFusionMap( u, s );[127X[104X
    [4X[28X[ 1, 2, 11, 3, 4, 5, 12, 7, 13, 9, 9, 15, 16, 10 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YCandidates for [22XM[122X are those subgroups of [22XG[122X that contain elements in the class
  [10X3D[110X of [22XG[122X whose centralizer is the full [10X3D[110X centralizer in [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X3Dcentralizer:= SizesCentralizers( t )[7];[127X[104X
    [4X[28X153055008[128X[104X
    [4X[25Xgap>[125X [27Xcand:= [];;                                                               [127X[104X
    [4X[25Xgap>[125X [27Xfor name in Maxes( t ) do[127X[104X
    [4X[25X>[125X [27X     m:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     mfust:= GetFusionMap( m, t );        [127X[104X
    [4X[25X>[125X [27X     if ForAny( [ 1 .. Length( mfust ) ],                    [127X[104X
    [4X[25X>[125X [27X         i -> mfust[i] = 7 and SizesCentralizers( m )[i] = 3Dcentralizer )   [127X[104X
    [4X[25X>[125X [27X     then[127X[104X
    [4X[25X>[125X [27X       Add( cand, m );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xcand;[127X[104X
    [4X[28X[ CharacterTable( "3^7.O7(3)" ), [128X[104X
    [4X[28X  CharacterTable( "3^2.3^4.3^8.(A5x2A4).2" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  these two groups [22XM[122X, we show that the possible class fusions from [22XU[122X to [22XG[122X
  via [22XM[122X factorize through [22XH[122X only if the second possible class fusion from [22XH[122X to
  [22XG[122X is chosen.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpossufust:= List( sfust, x -> CompositionMaps( x, ufuss ) );[127X[104X
    [4X[28X[ [ 1, 3, 3, 7, 8, 11, 9, 22, 23, 28, 28, 43, 43, 50 ], [128X[104X
    [4X[28X  [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ][128X[104X
    [4X[25Xgap>[125X [27Xm:= cand[1];;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= PossibleClassFusions( u, m );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ufusm );[127X[104X
    [4X[28X242[128X[104X
    [4X[25Xgap>[125X [27Xcomp:= List( ufusm, x -> CompositionMaps( GetFusionMap( m, t ), x ) );;[127X[104X
    [4X[25Xgap>[125X [27XIntersection( possufust, comp );[127X[104X
    [4X[28X[ [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ][128X[104X
    [4X[25Xgap>[125X [27Xm:= cand[2];;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= PossibleClassFusions( u, m );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ufusm );                        [127X[104X
    [4X[28X256[128X[104X
    [4X[25Xgap>[125X [27Xcomp:= List( ufusm, x -> CompositionMaps( GetFusionMap( m, t ), x ) );;   [127X[104X
    [4X[25Xgap>[125X [27XIntersection( possufust, comp );[127X[104X
    [4X[28X[ [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  we  check  that  the correct fusion is stored in the [5XGAP[105X Character
  Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) = sfust[2];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-2 [33X[0;0Y[22XL_2(13).2 → Fi_24^'[122X[101X[1X (September 2002)[133X[101X
  
  [33X[0;0YThe  class fusion of maximal subgroups [22XU[122X of type [22XL_2(13).2[122X in [22XG = Fi_24^'[122X is
  ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "F3+" );;[127X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "L2(13).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( u, t );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( u, fus, t );;[127X[104X
    [4X[25Xgap>[125X [27XLength( repr );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn [LW91,  p.  155],  it is stated that [22XU^'[122X contains elements in the classes
  [10X2B[110X,  [10X3D[110X,  and  [10X7B[110X  of  [22XG[122X.  (Note  that  the  two conjugacy classes of groups
  isomorphic to [22XU[122X have the same class fusion because the outer automorphism of
  [22XG[122X fixes the relevant classes.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( repr, x -> t.2b in x and t.3d in x and t.7b in x );[127X[104X
    [4X[28X[ [ 1, 3, 7, 22, 25, 25, 25, 51, 3, 9, 43, 43, 53, 53, 53 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 22, 25, 25, 25, 51, 3, 11, 50, 50, 53, 53, 53 ] ][128X[104X
    [4X[25Xgap>[125X [27XClassNames( t ){ [ 43, 50 ] };[127X[104X
    [4X[28X[ "12f", "12m" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we have to decide whether [22XU[122X contains elements in the class [10X12F[110X or in [10X12M[110X
  of [22XG[122X.[133X
  
  [33X[0;0YThe order [22X12[122X elements in question lie inside subgroups of type [22X13 : 12[122X in [22XU[122X.
  These  subgroups  are  clearly  contained  in the Sylow [22X13[122X normalizers of [22XG[122X,
  which  are  contained  in maximal subgroups of type [22X(3^2:2 × G_2(3)).2[122X in [22XG[122X;
  the class fusion of the latter groups is unique up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpos:= Position( OrdersClassRepresentatives( t ), 13 );[127X[104X
    [4X[28X51[128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( t )[ pos ];[127X[104X
    [4X[28X234[128X[104X
    [4X[25Xgap>[125X [27XClassOrbit( t, pos );[127X[104X
    [4X[28X[ 51 ][128X[104X
    [4X[25Xgap>[125X [27Xcand:= [];;                                                         [127X[104X
    [4X[25Xgap>[125X [27Xfor name in Maxes( t ) do[127X[104X
    [4X[25X>[125X [27X     m:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     pos:= Position( OrdersClassRepresentatives( m ), 13 );[127X[104X
    [4X[25X>[125X [27X     if pos <> fail and                                             [127X[104X
    [4X[25X>[125X [27X        SizesCentralizers( m )[ pos ] = 234                         [127X[104X
    [4X[25X>[125X [27X        and ClassOrbit( m, pos ) = [ pos ] then[127X[104X
    [4X[25X>[125X [27X       Add( cand, m );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xcand;[127X[104X
    [4X[28X[ CharacterTable( "(3^2:2xG2(3)).2" ) ][128X[104X
    [4X[25Xgap>[125X [27Xs:= cand[1];;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  no  [22X13:12[122X type subgroup is contained in the derived subgroup of [22X(3^2:2 ×
  G_2(3)).2[122X, we look at the elements of order [22X12[122X in the outer half.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xder:= ClassPositionsOfDerivedSubgroup( s );;[127X[104X
    [4X[25Xgap>[125X [27Xouter:= Difference( [ 1 .. NrConjugacyClasses( s ) ], der );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27Ximgs:= Set( Flat( List( sfust, x -> x{ outer } ) ) );[127X[104X
    [4X[28X[ 2, 3, 10, 11, 15, 17, 18, 19, 21, 22, 26, 44, 45, 49, 50, 52, 62, [128X[104X
    [4X[28X  83, 87, 98 ][128X[104X
    [4X[25Xgap>[125X [27Xt.12f in imgs;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xt.12m in imgs;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  [22XL_2(13).2  ∖  L_2(13)[122X does not contain [10X12F[110X elements of [22XG[122X, i. e., we have
  determined the class fusion of [22XU[122X in [22XG[122X.[133X
  
  [33X[0;0YFinally,  we check whether the correct fusion is stored in the [5XGAP[105X Character
  Table Library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( u, t ) = filt[2];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-3 [33X[0;0Y[22XM_11 → B[122X[101X[1X (April 2009)[133X[101X
  
  [33X[0;0YThe  sporadic  simple group [22XB[122X contains a maximal subgroup [22XM[122X of the type [22XM_11[122X
  whose class fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xm11:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27Xm11fusb:= PossibleClassFusions( m11, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( m11fusb );[127X[104X
    [4X[28X31[128X[104X
    [4X[25Xgap>[125X [27XCompositionMaps( ClassNames( b, "ATLAS" ), Parametrized( m11fusb ) );[127X[104X
    [4X[28X[ "1A", [ "2B", "2D" ], [ "3A", "3B" ], [128X[104X
    [4X[28X  [ "4B", "4E", "4G", "4H", "4J" ], [ "5A", "5B" ], [128X[104X
    [4X[28X  [ "6C", "6E", "6H", "6I", "6J" ], [128X[104X
    [4X[28X  [ "8B", "8E", "8G", "8J", "8K", "8L", "8M", "8N" ], [128X[104X
    [4X[28X  [ "8B", "8E", "8G", "8J", "8K", "8L", "8M", "8N" ], "11A", "11A" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording  to [Wil93a,  Thm.  12.1],  [22XM[122X contains no [10X5A[110X elements of [22XB[122X. By the
  proof of [Wil99, Prop. 4.1], the involutions in any [22XS_5[122X type subgroup [22XU[122X of [22XM[122X
  lie  in  the  class [10X2C[110X or [10X2D[110X of [22XB[122X, and since the possible class fusions of [22XM[122X
  computed above admit only involutions in the class [10X2B[110X or [10X2D[110X, all involutions
  of  [22XU[122X  lie  in  the class [10X2D[110X. Again by the proof of [Wil99, Prop. 4.1], [22XU[122X is
  contained in a maximal subgroup of type [22XTh[122X in [22XB[122X.[133X
  
  [33X[0;0YNow  we use the embedding of [22XU[122X into [22XB[122X via [22XM[122X and [22XTh[122X for determining the class
  fusion  of  [22XM[122X  into  [22XB[122X.  The  class  fusion  of the embedding of [22XU[122X via [22XTh[122X is
  uniquely determined.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xth:= CharacterTable( "Th" );;[127X[104X
    [4X[25Xgap>[125X [27Xs5:= CharacterTable( "S5" );;[127X[104X
    [4X[25Xgap>[125X [27Xs5fusth:= PossibleClassFusions( s5, th );[127X[104X
    [4X[28X[ [ 1, 2, 4, 8, 2, 7, 11 ] ][128X[104X
    [4X[25Xgap>[125X [27Xthfusb:= PossibleClassFusions( th, b );;[127X[104X
    [4X[25Xgap>[125X [27Xs5fusb:= Set( thfusb, x -> CompositionMaps( x, s5fusth[1] ) );[127X[104X
    [4X[28X[ [ 1, 5, 7, 19, 5, 17, 29 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlso  the  class fusion of [22XU[122X into [22XM[122X is unique, and this determines the class
  fusion of [22XM[122X into [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs5fusm11:= PossibleClassFusions( s5, m11 );[127X[104X
    [4X[28X[ [ 1, 2, 3, 5, 2, 4, 6 ] ][128X[104X
    [4X[25Xgap>[125X [27Xm11fusb:= Filtered( m11fusb,[127X[104X
    [4X[25X>[125X [27X                 map -> CompositionMaps( map, s5fusm11[1] ) = s5fusb[1] );[127X[104X
    [4X[28X[ [ 1, 5, 7, 17, 19, 29, 45, 45, 54, 54 ] ][128X[104X
    [4X[25Xgap>[125X [27XCompositionMaps( ClassNames( b, "ATLAS" ), m11fusb[1] );[127X[104X
    [4X[28X[ "1A", "2D", "3B", "4J", "5B", "6J", "8N", "8N", "11A", "11A" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Using  the information that the [22XM_10[122X type subgroups of [22XM[122X are also contained
  in  [22XTh[122X type subgroups would not have helped us, since these subgroups do not
  contain elements of order [22X6[122X, and two possibilities would have remained.)[133X
  
  
  [1X9.7-4 [33X[0;0Y[22XL_2(11):2 → B[122X[101X[1X (April 2009)[133X[101X
  
  [33X[0;0YThe  sporadic  simple  group  [22XB[122X  contains  a  maximal subgroup [22XL[122X of the type
  [22XL_2(11):2[122X whose class fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xl:= CharacterTable( "L2(11).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xlfusb:= PossibleClassFusions( l, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( lfusb );[127X[104X
    [4X[28X16[128X[104X
    [4X[25Xgap>[125X [27XCompositionMaps( ClassNames( b, "ATLAS" ), Parametrized( lfusb ) );[127X[104X
    [4X[28X[ "1A", [ "2B", "2D" ], [ "3A", "3B" ], [ "5A", "5B" ], [128X[104X
    [4X[28X  [ "5A", "5B" ], [ "6C", "6H", "6I", "6J" ], "11A", [ "2C", "2D" ], [128X[104X
    [4X[28X  [ "4D", "4E", "4F", "4G", "4H", "4J" ], [ "10C", "10E", "10F" ], [128X[104X
    [4X[28X  [ "10C", "10E", "10F" ], [128X[104X
    [4X[28X  [ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12P", "12Q", [128X[104X
    [4X[28X      "12R", "12S" ], [128X[104X
    [4X[28X  [ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12P", "12Q", [128X[104X
    [4X[28X      "12R", "12S" ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording  to [Wil93a,  Thm.  12.1],  [22XL[122X contains no [10X5A[110X elements of [22XB[122X. By the
  proof  of [Wil99,  Prop.  4.1], [22XB[122X contains exactly one class of [22XL_2(11)[122X type
  subgroups  with  this  property.  Hence  the subgroup [22XU[122X of index two in [22XL[122X is
  contained  in a maximal subgroup [22XM[122X of type [22XM_11[122X in [22XB[122X, whose class fusion was
  determined in Section [14X9.7-3[114X.[133X
  
  [33X[0;0YIn  the same way as we proceeded in Section [14X9.7-3[114X, we use the embedding of [22XU[122X
  into [22XB[122X via [22XL[122X and [22XM[122X for determining the class fusion of [22XL[122X into [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "L2(11)" );;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= PossibleClassFusions( u, m );;[127X[104X
    [4X[25Xgap>[125X [27Xmfusb:= GetFusionMap( m, b );;[127X[104X
    [4X[25Xgap>[125X [27Xufusb:= Set( ufusm, x -> CompositionMaps( mfusb, x ) );[127X[104X
    [4X[28X[ [ 1, 5, 7, 19, 19, 29, 54, 54 ] ][128X[104X
    [4X[25Xgap>[125X [27Xufusl:= PossibleClassFusions( u, l );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 5, 6, 7, 7 ], [ 1, 2, 3, 5, 4, 6, 7, 7 ] ][128X[104X
    [4X[25Xgap>[125X [27Xlfusb:= Filtered( lfusb, [127X[104X
    [4X[25X>[125X [27X             map2 -> ForAny( ufusl, [127X[104X
    [4X[25X>[125X [27X                       map1 -> CompositionMaps( map2, map1 ) in ufusb ) );[127X[104X
    [4X[28X[ [ 1, 5, 7, 19, 19, 29, 54, 5, 15, 53, 53, 73, 73 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X9.7-5 [33X[0;0Y[22XL_3(3) → B[122X[101X[1X (April 2009)[133X[101X
  
  [33X[0;0YThe sporadic simple group [22XB[122X contains a maximal subgroup [22XT[122X of the type [22XL_3(3)[122X
  whose class fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "L3(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xtfusb:= PossibleClassFusions( t, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( tfusb );[127X[104X
    [4X[28X36[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording  to [Wil99,  Section  9],  [22XT[122X  contains  a  subgroup  [22XU[122X of the type
  [22X3^2:2S_4[122X  that  is  contained  also  in  a  maximal  subgroup  [22XM[122X of the type
  [22X3^2.3^3.3^6.(S_4  × 2S_4)[122X. So we throw away the possible fusions from [22XT[122X to [22XB[122X
  that  are not compatible with the compositions of the embeddings of [22XU[122X into [22XB[122X
  via [22XT[122X and [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "3^2.3^3.3^6.(S4x2S4)" );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= PSL(3,3);;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= MaximalSubgroupClassReps( g );;[127X[104X
    [4X[25Xgap>[125X [27Xu:= First( mx, x -> Size( x ) = 432 );;[127X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( u );;[127X[104X
    [4X[25Xgap>[125X [27Xufusm:= PossibleClassFusions( u, m );;[127X[104X
    [4X[25Xgap>[125X [27Xufust:= PossibleClassFusions( u, t );;[127X[104X
    [4X[25Xgap>[125X [27Xmfusb:= GetFusionMap( m, b );;[127X[104X
    [4X[25Xgap>[125X [27Xufusb:= Set( ufusm, map -> CompositionMaps( mfusb, map ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtfusb:= Filtered( tfusb, map -> ForAny( ufust,[127X[104X
    [4X[25X>[125X [27X       map2 -> CompositionMaps( map, map2 ) in ufusb ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtfusb;[127X[104X
    [4X[28X[ [ 1, 5, 6, 7, 12, 27, 41, 41, 75, 75, 75, 75 ], [128X[104X
    [4X[28X  [ 1, 5, 7, 6, 12, 28, 41, 41, 75, 75, 75, 75 ], [128X[104X
    [4X[28X  [ 1, 5, 7, 7, 12, 28, 41, 41, 75, 75, 75, 75 ], [128X[104X
    [4X[28X  [ 1, 5, 7, 7, 12, 29, 41, 41, 75, 75, 75, 75 ], [128X[104X
    [4X[28X  [ 1, 5, 7, 7, 17, 29, 45, 45, 75, 75, 75, 75 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  use  that  [22XT[122X  does  not contain [10X4E[110X elements of [22XB[122X (again see [Wil99,
  Section  9]).  Thus  the  last  of  the five candidates is the correct class
  fusion.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XClassNames( b, "ATLAS" ){ [ 12, 17 ] };[127X[104X
    [4X[28X[ "4E", "4J" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe check that this map is stored on the library table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( t, b ) = tfusb[5];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-6 [33X[0;0Y[22XL_2(17).2 → B[122X[101X[1X (March 2004)[133X[101X
  
  [33X[0;0YThe  sporadic  simple  group  [22XB[122X  contains  a  maximal subgroup [22XU[122X of the type
  [22XL_2(17).2[122X whose class fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "L2(17).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xufusb:= PossibleClassFusions( u, b );[127X[104X
    [4X[28X[ [ 1, 5, 7, 15, 42, 42, 47, 47, 47, 91, 4, 30, 89, 89, 89, 89, 97, [128X[104X
    [4X[28X      97, 97 ], [128X[104X
    [4X[28X  [ 1, 5, 7, 15, 44, 44, 46, 46, 46, 91, 5, 29, 90, 90, 90, 90, 96, [128X[104X
    [4X[28X      96, 96 ], [128X[104X
    [4X[28X  [ 1, 5, 7, 15, 44, 44, 47, 47, 47, 91, 5, 29, 90, 90, 90, 90, 95, [128X[104X
    [4X[28X      95, 95 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording  to [Wil99, Prop. 11.1], [22XU[122X contains elements in the classes [10X8M[110X and
  [10X9A[110X of [22XB[122X. This determines the fusion map.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnames:= ClassNames( b, "ATLAS" );;[127X[104X
    [4X[25Xgap>[125X [27Xpos:= List( [ "8M", "9A" ], x -> Position( names, x ) );[127X[104X
    [4X[28X[ 44, 46 ][128X[104X
    [4X[25Xgap>[125X [27Xufusb:= Filtered( ufusb, map -> IsSubset( map, pos ) );[127X[104X
    [4X[28X[ [ 1, 5, 7, 15, 44, 44, 46, 46, 46, 91, 5, 29, 90, 90, 90, 90, 96, [128X[104X
    [4X[28X      96, 96 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe check that this map is stored on the library table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( u, b ) = ufusb[1];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-7 [33X[0;0Y[22XL_2(49).2_3 → B[122X[101X[1X (June 2006)[133X[101X
  
  [33X[0;0YThe  sporadic  simple  group  [22XB[122X contains a class of maximal subgroups of the
  type  [22XL_2(49).2_3[122X  (a  non-split  extension of [22XL_2(49)[122X, see [Wil93b, Theorem
  2]).  Let [22XU[122X be such a subgroup. The class fusion of [22XU[122X in [22XB[122X is not determined
  by the character tables of [22XU[122X and [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "L2(49).2_3" );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xufusb:= PossibleClassFusions( u, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( u, ufusb, b ) );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xufusb;[127X[104X
    [4X[28X[ [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128, [128X[104X
    [4X[28X      128, 128, 15, 71, 71, 89, 89, 89, 89 ], [128X[104X
    [4X[28X  [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128, [128X[104X
    [4X[28X      128, 128, 17, 72, 72, 89, 89, 89, 89 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  show  that  the  fusion  is  determined by the embeddings of the Sylow [22X7[122X
  normalizer  [22XN[122X,  say, of [22XU[122X into [22XU[122X and into the Sylow [22X7[122X normalizer of [22XB[122X. (Note
  that  the  fusion  of  the  latter  group  into  [22XB[122X  has  been  determined in
  Section [14X9.3-1[114X.)[133X
  
  [33X[0;0YFor  that,  we  compute the character table of [22XN[122X from a representation of [22XU[122X.
  Note  that  [22XU[122X  is  a  non-split extension of the simple group [22XL_2(49)[122X by the
  product  of  a  diagonal automorphism and a field automorphism. In [Wil93b],
  the structure of [22XN[122X is described as [22X7^2:(3 × Q_16)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SL( 2, 49 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfGroup( g );;[127X[104X
    [4X[25Xgap>[125X [27Xf:= GF(49);;[127X[104X
    [4X[25Xgap>[125X [27Xmats:= List( gens, x -> IdentityMat( 4, f ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. Length( gens ) ] do[127X[104X
    [4X[25X>[125X [27X     mats[i]{ [ 1, 2 ] }{ [ 1, 2 ] }:= gens[i];[127X[104X
    [4X[25X>[125X [27X     mats[i]{ [ 3, 4 ] }{ [ 3, 4 ] }:= List( gens[i],[127X[104X
    [4X[25X>[125X [27X                                             x -> List( x, y -> y^7 ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xfieldaut:= PermutationMat( (1,3)(2,4), 4, f );;[127X[104X
    [4X[25Xgap>[125X [27Xdiagaut:= IdentityMat( 4, f );;[127X[104X
    [4X[25Xgap>[125X [27Xdiagaut[1][1]:= Z(49);;[127X[104X
    [4X[25Xgap>[125X [27Xdiagaut[3][3]:= Z(49)^7;;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( Concatenation( mats, [ fieldaut * diagaut ] ) );;[127X[104X
    [4X[25Xgap>[125X [27Xv:= [ 1, 0, 0, 0 ] * Z(7)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, v, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xact:= Action( g, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= Normalizer( act, SylowSubgroup( act, 7 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xntbl:= CharacterTable( n );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  the  possible  class  fusions of [22XN[122X into [22XB[122X, via the Sylow [22X7[122X
  normalizer in [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbn7:= CharacterTable( "BN7" );;[127X[104X
    [4X[25Xgap>[125X [27Xnfusbn7:= PossibleClassFusions( ntbl, bn7 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( ntbl, nfusbn7, bn7 ) );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27Xnfusb:= SetOfComposedClassFusions( PossibleClassFusions( bn7, b ),[127X[104X
    [4X[25X>[125X [27X                                      nfusbn7 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( ntbl, nfusb, b ) );[127X[104X
    [4X[28X5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlthough  there  are  several  possibilities,  this information is enough to
  exclude one of the possible fusions of [22XU[122X into [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnfusu:= PossibleClassFusions( ntbl, u );;[127X[104X
    [4X[25Xgap>[125X [27XLength( nfusu );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( ufusb,[127X[104X
    [4X[25X>[125X [27X             x -> ForAny( nfusu, y -> CompositionMaps( x, y ) in nfusb ) );[127X[104X
    [4X[28X[ [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128, [128X[104X
    [4X[28X      128, 128, 17, 72, 72, 89, 89, 89, 89 ] ][128X[104X
    [4X[25Xgap>[125X [27XClassNames( b, "ATLAS" ){ filt[1] };[127X[104X
    [4X[28X[ "1A", "2D", "3B", "4H", "5B", "6I", "7A", "8K", "8K", "12Q", "24L", [128X[104X
    [4X[28X  "24L", "25A", "25A", "25A", "25A", "25A", "4J", "12R", "12R", [128X[104X
    [4X[28X  "16G", "16G", "16G", "16G" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  the  class  fusion of [22XU[122X into [22XB[122X can be described by the property that the
  elements  of  order  four inside and outside the simple subgroup [22XL_2(49)[122X are
  not conjugate in [22XB[122X.[133X
  
  [33X[0;0YWe check that the correct map is stored on the library table.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( u, b ) in filt;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  us confirm that the two groups of the types [22XL_2(49).2_1[122X and [22XL_2(49).2_2[122X
  cannot occur as subgroups of [22XB[122X. First we show that [22XL_2(49).2_1[122X is isomorphic
  with  PGL[22X(2,49)[122X,  an  extension  of  [22XL_2(49)[122X by a diagonal automorphism, and
  [22XL_2(49).2_2[122X is an extension by a field automorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses( u );  NrConjugacyClasses( act );[127X[104X
    [4X[28X24[128X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "L2(49).2_1" );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( Concatenation( mats, [ diagaut ] ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, v, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xact:= Action( g, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XSize(act );[127X[104X
    [4X[28X117600[128X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses( u );  NrConjugacyClasses( act );[127X[104X
    [4X[28X51[128X[104X
    [4X[28X51[128X[104X
    [4X[25Xgap>[125X [27Xu:= CharacterTable( "L2(49).2_2" );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( Concatenation( mats, [ fieldaut ] ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= Orbit( g, v, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27Xact:= Action( g, orb, OnLines );;[127X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses( u );  NrConjugacyClasses( act );[127X[104X
    [4X[28X27[128X[104X
    [4X[28X27[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group  [22XL_2(49).2_1[122X  can  be  excluded because no class fusion into [22XB[122X is
  possible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPossibleClassFusions( CharacterTable( "L2(49).2_1" ), b );[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [22XL_2(49).2_2[122X,  it  is not that easy. We would get several possible class
  fusions  into  [22XB[122X.  However,  the  Sylow [22X7[122X normalizer of [22XL_2(49).2_2[122X does not
  admit a class fusion into the Sylow [22X7[122X normalizer of [22XB[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xn:= Normalizer( act, SylowSubgroup( act, 7 ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( PossibleClassFusions( CharacterTable( n ), bn7 ) );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-8 [33X[0;0Y[22X2^3.L_3(2) → G_2(5)[122X[101X[1X (January 2004)[133X[101X
  
  [33X[0;0YThe  Chevalley  group  [22XG  = G_2(5)[122X contains a maximal subgroup [22XU[122X of the type
  [22X2^3.L_3(2)[122X whose class fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "G2(5)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "2^3.L3(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xsfust:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27XRepresentativesFusions( s, sfust, t );[127X[104X
    [4X[28X[ [ 1, 2, 2, 5, 6, 4, 13, 16, 17, 15, 15 ], [128X[104X
    [4X[28X  [ 1, 2, 2, 5, 6, 4, 14, 16, 17, 15, 15 ] ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( s );[127X[104X
    [4X[28X[ 1, 2, 2, 4, 4, 3, 6, 8, 8, 7, 7 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  the  question  is whether [22XU[122X contains elements in the class [10X6B[110X or [10X6C[110X of [22XG[122X
  (position  [22X13[122X or [22X14[122X in the [5XAtlas[105X table). We use a permutation representation
  of  [22XG[122X,  restrict  it  to  [22XU[122X,  and compute the centralizer in [22XG[122X of a suitable
  element of order [22X6[122X in [22XU[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "G2(5)" );;[127X[104X
    [4X[25Xgap>[125X [27Xu:= AtlasSubgroup( "G2(5)", 7 );;[127X[104X
    [4X[25Xgap>[125X [27XSize( u );[127X[104X
    [4X[28X1344[128X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     x:= Random( u );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 6;[127X[104X
    [4X[25Xgap>[125X [27Xsiz:= Size( Centralizer( g, x ) );[127X[104X
    [4X[28X36[128X[104X
    [4X[25Xgap>[125X [27XFiltered( [ 1 .. NrConjugacyClasses( t ) ],[127X[104X
    [4X[25X>[125X [27X             i -> SizesCentralizers( t )[i] = siz );[127X[104X
    [4X[28X[ 14 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo [22XU[122X contains [10X6C[110X elements in [22XG_2(5)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) in Filtered( sfust, map -> 14 in map );  [127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-9 [33X[0;0Y[22X5^{1+4}.2^{1+4}.A_5.4 → B[122X[101X[1X (April 2009)[133X[101X
  
  [33X[0;0YThe  sporadic  simple  group  [22XB[122X  contains  a  maximal subgroup [22XM[122X of the type
  [22X5^{1+4}.2^{1+4}.A_5.4[122X whose class fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "5^(1+4).2^(1+4).A5.4" );;[127X[104X
    [4X[25Xgap>[125X [27Xmfusb:= PossibleClassFusions( m, b );;[127X[104X
    [4X[25Xgap>[125X [27XLength( mfusb );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xrepres:= RepresentativesFusions( m, mfusb, b );; [127X[104X
    [4X[25Xgap>[125X [27XLength( repres );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   restriction  of  the  unique  irreducible  character  of  degree  [22X4371[122X
  distinguishes the two possibilities,[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xchar:= Filtered( Irr( b ), x -> x[1] = 4371 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( char );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xrest:= List( repres, map -> char[1]{ map } );;[127X[104X
    [4X[25Xgap>[125X [27Xscprs:= MatScalarProducts( m, Irr( m ), rest );;[127X[104X
    [4X[25Xgap>[125X [27Xconstit:= List( scprs,[127X[104X
    [4X[25X>[125X [27X               x -> Filtered( [1 .. Length(x) ], i -> x[i] <> 0 ) );[127X[104X
    [4X[28X[ [ 2, 27, 60, 63, 73, 74, 75, 79, 82 ], [128X[104X
    [4X[28X  [ 2, 27, 60, 63, 70, 72, 75, 79, 84 ] ][128X[104X
    [4X[25Xgap>[125X [27XList( constit, x -> List( Irr( m ){ x }, Degree ) );[127X[104X
    [4X[28X[ [ 1, 6, 384, 480, 400, 400, 500, 1000, 1200 ], [128X[104X
    [4X[28X  [ 1, 6, 384, 480, 100, 300, 500, 1000, 1600 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  database [WWT+]  contains  the  [22X3[122X-modular  reduction of the irreducible
  representation  of  degree  [22X4371[122X  and  also  a  straight  line  program  for
  restricting  this  representation  to  [22XM[122X.  We  access these data via the [5XGAP[105X
  package  [5XAtlasRep[105X (see [WPN+22]), and compute the composition factors of the
  natural module of this restriction.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasSubgroup( "B", Dimension, 4371, Ring, GF(3), 21 );;[127X[104X
    [4X[25Xgap>[125X [27Xmodule:= GModuleByMats( GeneratorsOfGroup( g ), GF(3) );;[127X[104X
    [4X[25Xgap>[125X [27Xdec:= MTX.CompositionFactors( module );;[127X[104X
    [4X[25Xgap>[125X [27XSortedList( List( dec, x -> x.dimension ) );[127X[104X
    [4X[28X[ 1, 6, 100, 384, 400, 400, 400, 480, 1000, 1200 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see that exactly one ordinary constituent does not stay irreducible upon
  restriction  to  characteristic  [22X3[122X. Thus the first of the two possible class
  fusions is the correct one.[133X
  
  
  [1X9.7-10  [33X[0;0YThe  fusion from the character table of [22X7^2:2L_2(7).2[122X[101X[1X into the table[101X
  [1Xof marks (January 2004)[133X[101X
  
  [33X[0;0YIt  can  happen that the class fusion from the ordinary character table of a
  group [22XG[122X into the table of marks of [22XG[122X is not unique up to table automorphisms
  of the character table of [22XG[122X.[133X
  
  [33X[0;0YAs  an  example,  consider  [22XG  =  7^2:2L_2(7).2[122X,  a  maximal subgroup in the
  sporadic simple group [22XHe[122X.[133X
  
  [33X[0;0Y[22XG[122X  contains  four  classes  of cyclic subgroups of order [22X7[122X. One contains the
  elements  in  the  normal  subgroup  of  type  [22X7^2[122X,  and the other three are
  preimages  of the order [22X7[122X elements in the factor group [22XL_2(7)[122X. The conjugacy
  classes  of  nonidentity elements in the latter three classes split into two
  Galois   conjugates   each,   which  are  permuted  cyclicly  by  the  table
  automorphisms  of  the  character table of [22XG[122X, but on which the stabilizer of
  one  class  acts  trivially.  This  means  that determining one of the three
  classes determines also the other two.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "7^2:2psl(2,7)" );[127X[104X
    [4X[28XCharacterTable( "7^2:2psl(2,7)" )[128X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( tbl );[127X[104X
    [4X[28XTableOfMarks( "7^2:2L2(7)" )[128X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleFusionsCharTableTom( tbl, tom );[127X[104X
    [4X[28X[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 10, 9, 16, 7, 10, 9, 8, 16 ], [128X[104X
    [4X[28X  [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 9, 8, 10, 16, 7, 8, 10, 9, 16 ], [128X[104X
    [4X[28X  [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 10, 9, 8, 16, 7, 9, 8, 10, 16 ], [128X[104X
    [4X[28X  [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ], [128X[104X
    [4X[28X  [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 10, 8, 9, 16, 7, 8, 9, 10, 16 ], [128X[104X
    [4X[28X  [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 9, 10, 8, 16, 7, 10, 8, 9, 16 ] ][128X[104X
    [4X[25Xgap>[125X [27Xreps:= RepresentativesFusions( tbl, fus, Group(()) );        [127X[104X
    [4X[28X[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ], [128X[104X
    [4X[28X  [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 10, 9, 16, 7, 10, 9, 8, 16 ] ][128X[104X
    [4X[25Xgap>[125X [27XAutomorphismsOfTable( tbl );[127X[104X
    [4X[28XGroup([ (9,14)(10,17)(11,15)(12,16)(13,18), (7,8), (10,11,12)[128X[104X
    [4X[28X  (15,16,17) ])[128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( tbl );[127X[104X
    [4X[28X[ 1, 7, 2, 4, 3, 6, 8, 8, 7, 7, 7, 7, 14, 7, 7, 7, 7, 14 ][128X[104X
    [4X[25Xgap>[125X [27Xperms1:= PermCharsTom( reps[1], tom );;[127X[104X
    [4X[25Xgap>[125X [27Xperms2:= PermCharsTom( reps[2], tom );;[127X[104X
    [4X[25Xgap>[125X [27Xperms1 = perms2;      [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XSet( perms1 ) = Set( perms2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  table  of  marks  of [22XG[122X does not distinguish the three classes of cyclic
  subgroups,  there are permutations of rows and columns that act as an [22XS_3[122X on
  them.[133X
  
  [33X[0;0YNote  that  an [22XS_3[122X acts on the classes in question in the [13Xrational[113X character
  table.  So  it  is due to the irrationalities in the character table that it
  contains more information.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplay( tbl );[127X[104X
    [4X[28X7^2:2psl(2,7)[128X[104X
    [4X[28X[128X[104X
    [4X[28X      2  4  .  4  3  1  1  3  3   1   .   .   .   1   1   .   .   .[128X[104X
    [4X[28X      3  1  .  1  .  1  1  .  .   .   .   .   .   .   .   .   .   .[128X[104X
    [4X[28X      7  3  3  1  .  .  .  .  .   2   2   2   2   1   2   2   2   2[128X[104X
    [4X[28X[128X[104X
    [4X[28X        1a 7a 2a 4a 3a 6a 8a 8b  7b  7c  7d  7e 14a  7f  7g  7h  7i[128X[104X
    [4X[28X     2P 1a 7a 1a 2a 3a 3a 4a 4a  7b  7c  7d  7e  7b  7f  7g  7h  7i[128X[104X
    [4X[28X     3P 1a 7a 2a 4a 1a 2a 8b 8a  7f  7i  7g  7h 14b  7b  7d  7e  7c[128X[104X
    [4X[28X     5P 1a 7a 2a 4a 3a 6a 8b 8a  7f  7i  7g  7h 14b  7b  7d  7e  7c[128X[104X
    [4X[28X     7P 1a 1a 2a 4a 3a 6a 8a 8b  1a  1a  1a  1a  2a  1a  1a  1a  1a[128X[104X
    [4X[28X    11P 1a 7a 2a 4a 3a 6a 8b 8a  7b  7c  7d  7e 14a  7f  7g  7h  7i[128X[104X
    [4X[28X    13P 1a 7a 2a 4a 3a 6a 8b 8a  7f  7i  7g  7h 14b  7b  7d  7e  7c[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1      1  1  1  1  1  1  1  1   1   1   1   1   1   1   1   1   1[128X[104X
    [4X[28XX.2      3  3  3 -1  .  .  1  1   B   B   B   B   B  /B  /B  /B  /B[128X[104X
    [4X[28XX.3      3  3  3 -1  .  .  1  1  /B  /B  /B  /B  /B   B   B   B   B[128X[104X
    [4X[28XX.4      6  6  6  2  .  .  .  .  -1  -1  -1  -1  -1  -1  -1  -1  -1[128X[104X
    [4X[28XX.5      7  7  7 -1  1  1 -1 -1   .   .   .   .   .   .   .   .   .[128X[104X
    [4X[28XX.6      8  8  8  . -1 -1  .  .   1   1   1   1   1   1   1   1   1[128X[104X
    [4X[28XX.7      4  4 -4  .  1 -1  .  .  -B  -B  -B  -B   B -/B -/B -/B -/B[128X[104X
    [4X[28XX.8      4  4 -4  .  1 -1  .  . -/B -/B -/B -/B  /B  -B  -B  -B  -B[128X[104X
    [4X[28XX.9      6  6 -6  .  .  .  A -A  -1  -1  -1  -1   1  -1  -1  -1  -1[128X[104X
    [4X[28XX.10     6  6 -6  .  .  . -A  A  -1  -1  -1  -1   1  -1  -1  -1  -1[128X[104X
    [4X[28XX.11     8  8 -8  . -1  1  .  .   1   1   1   1  -1   1   1   1   1[128X[104X
    [4X[28XX.12    48 -1  .  .  .  .  .  .   6  -1  -1  -1   .   6  -1  -1  -1[128X[104X
    [4X[28XX.13    48 -1  .  .  .  .  .  .   C  -1  /C  /D   .  /C   C   D  -1[128X[104X
    [4X[28XX.14    48 -1  .  .  .  .  .  .   C  /C  /D  -1   .  /C   D  -1   C[128X[104X
    [4X[28XX.15    48 -1  .  .  .  .  .  .  /C   D  -1   C   .   C  -1  /C  /D[128X[104X
    [4X[28XX.16    48 -1  .  .  .  .  .  .   C  /D  -1  /C   .  /C  -1   C   D[128X[104X
    [4X[28XX.17    48 -1  .  .  .  .  .  .  /C   C   D  -1   .   C  /D  -1  /C[128X[104X
    [4X[28XX.18    48 -1  .  .  .  .  .  .  /C  -1   C   D   .   C  /C  /D  -1[128X[104X
    [4X[28X[128X[104X
    [4X[28X      2   1[128X[104X
    [4X[28X      3   .[128X[104X
    [4X[28X      7   1[128X[104X
    [4X[28X[128X[104X
    [4X[28X        14b[128X[104X
    [4X[28X     2P  7f[128X[104X
    [4X[28X     3P 14a[128X[104X
    [4X[28X     5P 14a[128X[104X
    [4X[28X     7P  2a[128X[104X
    [4X[28X    11P 14b[128X[104X
    [4X[28X    13P 14a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1       1[128X[104X
    [4X[28XX.2      /B[128X[104X
    [4X[28XX.3       B[128X[104X
    [4X[28XX.4      -1[128X[104X
    [4X[28XX.5       .[128X[104X
    [4X[28XX.6       1[128X[104X
    [4X[28XX.7      /B[128X[104X
    [4X[28XX.8       B[128X[104X
    [4X[28XX.9       1[128X[104X
    [4X[28XX.10      1[128X[104X
    [4X[28XX.11     -1[128X[104X
    [4X[28XX.12      .[128X[104X
    [4X[28XX.13      .[128X[104X
    [4X[28XX.14      .[128X[104X
    [4X[28XX.15      .[128X[104X
    [4X[28XX.16      .[128X[104X
    [4X[28XX.17      .[128X[104X
    [4X[28XX.18      .[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(8)-E(8)^3[128X[104X
    [4X[28X  = Sqrt(2) = r2[128X[104X
    [4X[28XB = E(7)+E(7)^2+E(7)^4[128X[104X
    [4X[28X  = (-1+Sqrt(-7))/2 = b7[128X[104X
    [4X[28XC = 2*E(7)+2*E(7)^2+2*E(7)^4[128X[104X
    [4X[28X  = -1+Sqrt(-7) = 2b7[128X[104X
    [4X[28XD = -3*E(7)-3*E(7)^2-2*E(7)^3-3*E(7)^4-2*E(7)^5-2*E(7)^6[128X[104X
    [4X[28X  = (5-Sqrt(-7))/2 = 2-b7[128X[104X
    [4X[25Xgap>[125X [27Xmat:= MatTom( tom );;[127X[104X
    [4X[25Xgap>[125X [27Xmataut:= MatrixAutomorphisms( mat );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( mataut, "\n" );[127X[104X
    [4X[28XGroup( [ (11,12)(23,24)(27,28)(46,47)(53,54)(56,57), [128X[104X
    [4X[28X  ( 9,10)(20,21)(31,32)(38,39), ( 8, 9)(20,22)(31,33)(38,40) ] )[128X[104X
    [4X[25Xgap>[125X [27XRepresentativesFusions( Group( () ), reps, mataut );[127X[104X
    [4X[28X[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  could  say  that thus the fusion is unique up to table automorphisms and
  automorphisms  of  the  table of marks. But since a group is associated with
  the  table  of  marks,  we  compute  the character table from the group, and
  decide which class fusion is correct.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= UnderlyingGroup( tom );;[127X[104X
    [4X[25Xgap>[125X [27Xtg:= CharacterTable( g );;[127X[104X
    [4X[25Xgap>[125X [27Xtgfustom:= FusionCharTableTom( tg, tom );;[127X[104X
    [4X[25Xgap>[125X [27Xtrans:= TransformingPermutationsCharacterTables( tg, tbl );;[127X[104X
    [4X[25Xgap>[125X [27Xtblfustom:= Permuted( tgfustom, trans.columns );;[127X[104X
    [4X[25Xgap>[125X [27Xorbits:= List( reps, map -> OrbitFusions( AutomorphismsOfTable( tbl ),[127X[104X
    [4X[25X>[125X [27X                                             map, Group( () ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XPositionProperty( orbits, orb -> tblfustom in orb );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XPositionProperty( orbits, orb -> FusionToTom( tbl ).map in orb );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo we see that the second one of the possibilities above is the right one.[133X
  
  
  [1X9.7-11 [33X[0;0Y[22X3 × U_4(2) → 3_1.U_4(3)[122X[101X[1X (March 2010)[133X[101X
  
  [33X[0;0YAccording  to  the  [5XAtlas[105X  (see [CCN+85,  p.  52]),  the simple group [22XU_4(3)[122X
  contains  two  classes  of  maximal  subgroups of the type [22XU_4(2)[122X. The class
  fusion of [22XU_4(2)[122X into [22XU_4(3)[122X is unique up to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu42:= CharacterTable( "U4(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xu43:= CharacterTable( "U4(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xu42fusu43:= PossibleClassFusions( u42, u43 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( u42fusu43 );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( u42, u42fusu43, u43 ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YMore  precisely,  take  the  outer  automorphism group of [22XU_4(3)[122X, which is a
  dihedral  group  of  order eight, and consider the subgroup generated by its
  central  involution  (this  automorphism is denoted by [22X2_1[122X in the [5XAtlas[105X) and
  another  involution  called  [22X2_3[122X in the [5XAtlas[105X. This subgroup is a Klein four
  group  that  induces  a  permutation group on the classes of [22XU_4(3)[122X and thus
  acts on the four possible class fusions of [22XU_4(2)[122X into [22XU_4(3)[122X. In fact, this
  action is transitive.[133X
  
  [33X[0;0YThe  automorphism  [22X2_1[122X  swaps each pair of mutually inverse classes of order
  nine,  that  is, [10X9A[110X is swapped with [10X9B[110X and [10X9C[110X is swapped with [10X9D[110X. All [22XU_4(2)[122X
  type  subgroups of [22XU_4(3)[122X are invariant under this automorphism, they extend
  to subgroups of the type [22XU_4(2).2[122X in [22XU_4(3).2_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu43_21:= CharacterTable( "U4(3).2_1" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus1:= GetFusionMap( u43, u43_21 );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 17, [128X[104X
    [4X[28X  18 ][128X[104X
    [4X[25Xgap>[125X [27Xact1:= Filtered( InverseMap( fus1 ), IsList );[127X[104X
    [4X[28X[ [ 16, 17 ], [ 18, 19 ] ][128X[104X
    [4X[25Xgap>[125X [27XCompositionMaps( ClassNames( u43, "Atlas" ), act1 );[127X[104X
    [4X[28X[ [ "9A", "9B" ], [ "9C", "9D" ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  automorphism  [22X2_3[122X swaps [10X6B[110X with [10X6C[110X, [10X9A[110X with [10X9C[110X, and [10X9B[110X with [10X9D[110X. The two
  classes of [22XU_4(2)[122X type subgroups of [22XU_4(3)[122X are swapped by this automorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu43_23:= CharacterTable( "U4(3).2_3" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus3:= GetFusionMap( u43, u43_23 );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 13, 14, [128X[104X
    [4X[28X  15 ][128X[104X
    [4X[25Xgap>[125X [27Xact3:= Filtered( InverseMap( fus3 ), IsList );[127X[104X
    [4X[28X[ [ 4, 5 ], [ 11, 12 ], [ 13, 14 ], [ 16, 18 ], [ 17, 19 ] ][128X[104X
    [4X[25Xgap>[125X [27XCompositionMaps( ClassNames( u43, "Atlas" ), act3 );[127X[104X
    [4X[28X[ [ "3B", "3C" ], [ "6B", "6C" ], [ "7A", "7B" ], [ "9A", "9C" ], [128X[104X
    [4X[28X  [ "9B", "9D" ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [5XAtlas[105X  states that the permutation character induced by the first class
  of  [22XU_4(2)[122X  type  subgroups is [10X1a+35a+90a[110X, which means that the subgroups in
  this  class  contain  [10X9A[110X  and  [10X9B[110X  elements.  Then the permutation character
  induced  by the second class of [22XU_4(2)[122X type subgroups is [10X1a+35b+90a[110X, and the
  subgroups in this class contain [10X9C[110X and [10X9D[110X elements.[133X
  
  [33X[0;0YSo  we choose appropriate fusions for the two classes of maximal [22XU_4(2)[122X type
  subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfirstfus:= First( u42fusu43, x -> IsSubset( x, [ 16, 17 ] ) );[127X[104X
    [4X[28X[ 1, 2, 2, 3, 3, 5, 4, 7, 8, 9, 10, 10, 12, 12, 11, 12, 16, 17, 20, [128X[104X
    [4X[28X  20 ][128X[104X
    [4X[25Xgap>[125X [27Xsecondfus:= First( u42fusu43, x -> IsSubset( x, [ 18, 19 ] ) );[127X[104X
    [4X[28X[ 1, 2, 2, 3, 3, 4, 5, 7, 8, 9, 10, 10, 11, 11, 12, 11, 18, 19, 20, [128X[104X
    [4X[28X  20 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  us  now  consider  the  central  extension  [22X3_1.U_4(3)[122X. Since the Schur
  multiplier of [22XU_4(2)[122X has order two, the [22XU_4(2)[122X type subgroups of [22XU_4(3)[122X lift
  to  groups  of  the  structure  [22X3  ×  U_4(2)[122X  in [22X3_1.U_4(3)[122X. There are eight
  possible  class  fusions  from  [22X3  ×  U_4(2)[122X to [22X3_1.U_4(3)[122X, in two orbits of
  length four under the action of table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X3u42:= CharacterTable( "Cyclic", 3 ) * u42;[127X[104X
    [4X[28XCharacterTable( "C3xU4(2)" )[128X[104X
    [4X[25Xgap>[125X [27X3u43:= CharacterTable( "3_1.U4(3)" );[127X[104X
    [4X[28XCharacterTable( "3_1.U4(3)" )[128X[104X
    [4X[25Xgap>[125X [27X3u42fus3u43:= PossibleClassFusions( 3u42, 3u43 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( 3u42fus3u43 );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XLength( RepresentativesFusions( 3u42, 3u42fus3u43, 3u43 ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YMore  precisely,  each of the four fusions from [22XU_4(2)[122X to [22XU_4(3)[122X has exactly
  two lifts. The four lifts of those fusions from [22XU_4(2)[122X to [22XU_4(3)[122X with [10X9A[110X and
  [10X9B[110X  in  their  image form one orbit under the action of table automorphisms.
  The  other  orbit  consists  of the lifts of those fusions with [10X9C[110X and [10X9D[110X in
  their image.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinducedmaps:= List( 3u42fus3u43, map -> CompositionMaps([127X[104X
    [4X[25X>[125X [27X       GetFusionMap( 3u43, u43 ), CompositionMaps( map,[127X[104X
    [4X[25X>[125X [27X       InverseMap( GetFusionMap( 3u42, u42 ) ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XList( inducedmaps, map -> Position( u42fusu43, map ) );[127X[104X
    [4X[28X[ 1, 1, 2, 2, 4, 4, 3, 3 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis solves the ambiguity: Fusions from each of the two orbits occur, and we
  can assign them to the two classes of subgroups by the choice of the fusions
  from [22XU_4(2)[122X to [22XU_4(3)[122X.[133X
  
  [33X[0;0YThe reason for the asymmetry is that the automorphism [22X2_3[122X of [22XU_4(3)[122X does not
  lift to [22X3_1.U_4(3)[122X. Note that each of the classes [10X9A[110X, [10X9B[110X of [22XU_4(3)[122X has three
  preimages  in  [22X3_1.U_4(3)[122X,  whereas  each of the classes [10X9C[110X, [10X9D[110X has only one
  preimage.[133X
  
  [33X[0;0YIn  fact  the  two classes of [22X3 × U_4(2)[122X type subgroups of [22X3_1.U_4(3)[122X behave
  differently.  For  example,  inducing  the  irreducible  characters of a [22X3 ×
  U_4(2)[122X  type  subgroup in the first class of maximal subgroups of [22X3_1.U_4(3)[122X
  yields  no  irreducible character, whereas the two irreducible characters of
  degree [22X630[122X are obtained by inducing the irreducible characters of a subgroup
  in the second class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrep:= RepresentativesFusions( 3u42, 3u42fus3u43, 3u43 );[127X[104X
    [4X[28X[ [ 1, 4, 4, 7, 7, 10, 13, 15, 18, 21, 24, 24, 27, 27, 30, 27, 48, [128X[104X
    [4X[28X      49, 50, 50, 2, 5, 5, 8, 8, 11, 13, 16, 19, 22, 25, 25, 28, 28, [128X[104X
    [4X[28X      31, 28, 48, 49, 51, 51, 3, 6, 6, 9, 9, 12, 13, 17, 20, 23, 26, [128X[104X
    [4X[28X      26, 29, 29, 32, 29, 48, 49, 52, 52 ], [128X[104X
    [4X[28X  [ 1, 4, 4, 8, 9, 13, 10, 15, 18, 21, 25, 26, 31, 32, 27, 30, 46, [128X[104X
    [4X[28X      44, 51, 52, 2, 5, 5, 9, 7, 13, 11, 16, 19, 22, 26, 24, 32, 30, [128X[104X
    [4X[28X      28, 31, 47, 42, 52, 50, 3, 6, 6, 7, 8, 13, 12, 17, 20, 23, 24, [128X[104X
    [4X[28X      25, 30, 31, 29, 32, 45, 43, 50, 51 ] ][128X[104X
    [4X[25Xgap>[125X [27Xirr:= Irr( 3u42 );;[127X[104X
    [4X[25Xgap>[125X [27Xind:= InducedClassFunctionsByFusionMap( 3u42, 3u43, irr, rep[1] );;[127X[104X
    [4X[25Xgap>[125X [27XIntersection( ind, Irr( 3u43 ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "3_1.U4(3)" ),[128X[104X
    [4X[28X  [ 630, 630*E(3)^2, 630*E(3), 6, 6*E(3)^2, 6*E(3), 9, 9*E(3)^2, [128X[104X
    [4X[28X      9*E(3), -9, -9*E(3)^2, -9*E(3), 0, 0, 2, 2*E(3)^2, 2*E(3), -2, [128X[104X
    [4X[28X      -2*E(3)^2, -2*E(3), 0, 0, 0, -3, -3*E(3)^2, -3*E(3), 3, [128X[104X
    [4X[28X      3*E(3)^2, 3*E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, -1, -E(3)^2, -E(3) ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "3_1.U4(3)" ),[128X[104X
    [4X[28X  [ 630, 630*E(3), 630*E(3)^2, 6, 6*E(3), 6*E(3)^2, 9, 9*E(3), [128X[104X
    [4X[28X      9*E(3)^2, -9, -9*E(3), -9*E(3)^2, 0, 0, 2, 2*E(3), 2*E(3)^2, [128X[104X
    [4X[28X      -2, -2*E(3), -2*E(3)^2, 0, 0, 0, -3, -3*E(3), -3*E(3)^2, 3, [128X[104X
    [4X[28X      3*E(3), 3*E(3)^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, -1, -E(3), -E(3)^2 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xind:= InducedClassFunctionsByFusionMap( 3u42, 3u43, irr, rep[2] );;[127X[104X
    [4X[25Xgap>[125X [27XIntersection( ind, Irr( 3u43 ) );[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  [22X6_1.U_4(3)[122X  and  [22X12_1.U_4(3)[122X, one gets the same phenomenon: We have two
  orbits  of  class  fusions,  one corresponding to each of the two classes of
  subgroups  of  the  type  [22X3  ×  4  Y 2.U_4(2)[122X. We get [22X10[122X irreducible induced
  characters  from  a  subgroup  in the second class (four faithful ones, four
  with  kernel  of order two, and the two abovementioned degree [22X630[122X characters
  with  kernel  of order four) and no irreducible character from a subgroup in
  the first class.[133X
  
  
  [1X9.7-12 [33X[0;0Y[22X2.3^4.2^3.S_4 → 2.A12[122X[101X[1X (September 2011)[133X[101X
  
  [33X[0;0YThe double cover [22XG[122X of the alternating group [22XA_12[122X contains a maximal subgroup
  [22XM[122X of the type [22X2.3^4.2^3.S_4[122X whose class fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2a12:= CharacterTable( "2.A12" );;[127X[104X
    [4X[25Xgap>[125X [27Xmtbl:= CharacterTable( "2.3^4.2^3.S4" );;[127X[104X
    [4X[25Xgap>[125X [27Xmtblfus2a12:= PossibleClassFusions( mtbl, 2a12 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( mtblfus2a12 );[127X[104X
    [4X[28X32[128X[104X
    [4X[25Xgap>[125X [27Xrepres:= RepresentativesFusions( mtbl, mtblfus2a12, 2a12 );; [127X[104X
    [4X[25Xgap>[125X [27XLength( repres );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe decide the question which of the essentially different two possible class
  fusion is the right one, by explicitly constructing [22XM[122X as a subgroup of [22XG[122X.[133X
  
  [33X[0;0YFor that, let [22Xπ[122X denote the natural epimorphism from [22XG[122X to [22XA_12[122X, and note that
  [22Xπ(M)[122X  can  be  described as the intersection of a [22XS_3 ℘ S_4[122X type subgroup of
  [22XS_12[122X  with  [22XA_12[122X.  Further  note that the generators for [22XG[122X and [22XA_12[122X provided
  by [WWT+]  are  compatible in the sense that [22Xπ[122X can be defined by mapping the
  generators of [22XG[122X to those of [22XA_12[122X.[133X
  
  [33X[0;0YWe  need [22Xπ[122X only for computing one preimage of each given element. Therefore,
  we  represent  [22Xπ[122X  implicitly  by two epimorphisms from a free group to [22XG[122X and
  [22XA_12[122X,  respectively,  in  order  to  avoid  that  [5XGAP[105Xprecomputes  a  lot  of
  unnecessary  information for [22XG[122X. This way, computing a preimage of an element
  of  [22XA_12[122X  under [22Xπ[122X is cheap. However, computing the preimage of a subgroup of
  [22XA_12[122X  would  be  very  expensive.  So we construct the subgroup of [22XG[122X that is
  generated  by  preimages  of  a set of generators of [22Xπ(M)[122X; later we see that
  this subgroup is in fact equal to [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= AtlasGroup( "A12" );[127X[104X
    [4X[28XGroup([ (1,2,3), (2,3,4,5,6,7,8,9,10,11,12) ])[128X[104X
    [4X[25Xgap>[125X [27X2g:= AtlasGroup( "2.A12" );[127X[104X
    [4X[28X<matrix group of size 479001600 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xf:= FreeGroup( 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xpi1:= GroupHomomorphismByImagesNC( f, 2g, GeneratorsOfGroup( f ),[127X[104X
    [4X[25X>[125X [27X             GeneratorsOfGroup( 2g ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpi2:= GroupHomomorphismByImagesNC( f, g, GeneratorsOfGroup( f ),[127X[104X
    [4X[25X>[125X [27X             GeneratorsOfGroup( g ) );;[127X[104X
    [4X[25Xgap>[125X [27Xw:= WreathProduct( SymmetricGroup( 3 ), SymmetricGroup(4) );[127X[104X
    [4X[28X<permutation group of size 31104 with 10 generators>[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints( w );[127X[104X
    [4X[28X12[128X[104X
    [4X[25Xgap>[125X [27Xs:= Intersection( w, g );  Size( s );[127X[104X
    [4X[28X<permutation group with 8 generators>[128X[104X
    [4X[28X15552[128X[104X
    [4X[25Xgap>[125X [27Xm:= SubgroupNC( 2g, List( SmallGeneratingSet( s ),[127X[104X
    [4X[25X>[125X [27X           x -> ImagesRepresentative( pi1,[127X[104X
    [4X[25X>[125X [27X                  PreImagesRepresentative( pi2, x ) ) ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  compute  the  character  table  of  [22XM[122X, using a faithful permutation
  representation of [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiso:= IsomorphismPermGroup( m );;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( Image( iso ) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( t );[127X[104X
    [4X[28X31104[128X[104X
    [4X[25Xgap>[125X [27Xtrans:= TransformingPermutationsCharacterTables( mtbl, t );;[127X[104X
    [4X[25Xgap>[125X [27XIsRecord( trans );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow let us see where the two fusion candidates differ.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpara:= Parametrized( repres );[127X[104X
    [4X[28X[ 1, 2, 6, 10, 8, 12, 7, 11, 9, 13, 5, 5, 17, 17, 17, 17, 3, 4, 24, [128X[104X
    [4X[28X  22, 27, 25, 12, 10, 13, 11, 28, 29, 35, 37, 39, 36, 38, 40, 5, 23, [128X[104X
    [4X[28X  28, 29, 26, 14, 14, 16, 16, 33, 34, [ 33, 34 ], [ 33, 34 ], 49, 49, [128X[104X
    [4X[28X  48, 48 ][128X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( para, IsList );[127X[104X
    [4X[28X[ 46, 47 ][128X[104X
    [4X[25Xgap>[125X [27XList( repres, map -> map{ [ 44 .. 47 ] } );[127X[104X
    [4X[28X[ [ 33, 34, 33, 34 ], [ 33, 34, 34, 33 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  the  question  is whether the elements in class 44 are conjugate in [22XG[122X to
  the  elements  in class 46 or in class 47. In order to answer this question,
  we  compute  preimages  of  the relevant class representatives in the matrix
  group [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpositions:= OnTuples( [ 44 .. 47 ], trans.columns );;[127X[104X
    [4X[25Xgap>[125X [27Xclassreps:= List( ConjugacyClasses( t ){ positions },[127X[104X
    [4X[25X>[125X [27X       c -> PreImagesRepresentative( iso, Representative( c ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xtraces:= List( classreps, TraceMat );;[127X[104X
    [4X[25Xgap>[125X [27XList( traces, x -> Position( traces, x ) );[127X[104X
    [4X[28X[ 1, 2, 2, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  are  lucky,  already the traces of the elements allow us to decide which
  pairs  of  elements  are  [22XG[122X-conjugate; there is no need for an explicit (and
  expensive) conjugacy test in the matrix group [22XG[122X.[133X
  
  [33X[0;0YFinally, we check whether the stored fusion is correct.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgood:= First( repres,[127X[104X
    [4X[25X>[125X [27X                 map -> map{ [ 44 .. 47 ] } = [ 33, 34, 34, 33 ] );;[127X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( mtbl, 2a12 ) = good;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-13 [33X[0;0Y[22X127:7 → L_7(2)[122X[101X[1X (January 2012)[133X[101X
  
  [33X[0;0YThe  simple group [22XG = L_7(2)[122X contains a maximal subgroup [22XM[122X of the type [22X127:7[122X
  (the  normalizer  of an extension field type subgroup GL[22X(1,2^7)[122X) whose class
  fusion is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "L7(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "127:7" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( s, fus, t );[127X[104X
    [4X[28X[ [ 1, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, [128X[104X
    [4X[28X      112, 113, 114, 115, 117, 116, 76, 76, 77, 76, 77, 77 ], [128X[104X
    [4X[28X  [ 1, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, [128X[104X
    [4X[28X      112, 113, 114, 115, 117, 116, 83, 83, 83, 83, 83, 83 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe two fusion candidates differ only for elements of order [22X7[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdiff:= Filtered( [ 1 .. Length( repr[1] ) ],[127X[104X
    [4X[25X>[125X [27X                    i -> repr[1][i] <> repr[2][i] );[127X[104X
    [4X[28X[ 20, 21, 22, 23, 24, 25 ][128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( s ){ diff };[127X[104X
    [4X[28X[ 7, 7, 7, 7, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27XList( repr, l -> l{ diff } );[127X[104X
    [4X[28X[ [ 76, 76, 77, 76, 77, 77 ], [ 83, 83, 83, 83, 83, 83 ] ][128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( t ){ [ 76, 77, 83 ] };[127X[104X
    [4X[28X[ 3528, 3528, 49 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  can decide which candidate is the correct one if we know the centralizer
  order  in  G of the elements of order [22X7[122X in [22XM[122X. So we compute this centralizer
  order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= Image( IsomorphismPermGroup( GL(7,2) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( g ); until Order(x) = 127;[127X[104X
    [4X[25Xgap>[125X [27Xn:= Normalizer( g, SubgroupNC( g, [ x ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( n ) / 127;[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27Xrepeat x:= Random( n ); until Order( x ) = 7;[127X[104X
    [4X[25Xgap>[125X [27Xc:= Centralizer( g, x );;[127X[104X
    [4X[25Xgap>[125X [27XSize( c );[127X[104X
    [4X[28X49[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see that the second candidate is the fusion from [22XM[122X into [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) = repr[2];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-14 [33X[0;0Y[22XL_2(59) → M[122X[101X[1X (May 2009) – Do not use this![133X[101X
  
  [33X[0;0YIt  had  been  claimed  that  the sporadic simple Monster group [22XM[122X contains a
  maximal  subgroup [22XG[122X of the type [22XL_2(59)[122X, see [HW04]. Later it was shown that
  [22XM[122X does [13Xnot[113X contain such subgroups, see [DLPP24].[133X
  
  [33X[0;0YHere  we  show  that  the  character tables of [22XG[122X and [22XM[122X do not provide enough
  information  to  determine a unique class fusion. (In particular, the tables
  do  not  help  to show that [22XG[122X is not a subgroup of [22XM[122X.) The class fusion of [22XG[122X
  into [22XM[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L2(59)" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( s, fus, t );[127X[104X
    [4X[28X[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 98, 52, 32, 52, 14, 12, 98, 52, 32, 5, 98, 12, 98, 52, 3 ], [128X[104X
    [4X[28X  [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 100, 50, 30, 50, 15, 11, 100, 50, 30, 4, 100, 11, 100, 50, [128X[104X
    [4X[28X      3 ], [128X[104X
    [4X[28X  [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 101, 51, 30, 51, 14, 11, 101, 51, 30, 5, 101, 11, 101, 51, [128X[104X
    [4X[28X      3 ], [128X[104X
    [4X[28X  [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 102, 53, 32, 53, 18, 12, 102, 53, 32, 6, 102, 12, 102, 53, [128X[104X
    [4X[28X      3 ], [128X[104X
    [4X[28X  [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52, [128X[104X
    [4X[28X      3 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe candidates differ on the classes of element order [22X30[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord:= OrdersClassRepresentatives( s );;[127X[104X
    [4X[25Xgap>[125X [27Xord30:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 30 );[127X[104X
    [4X[28X[ 18, 24, 28, 30 ][128X[104X
    [4X[25Xgap>[125X [27XList( repr, x -> x{ ord30 } );[127X[104X
    [4X[28X[ [ 98, 98, 98, 98 ], [ 100, 100, 100, 100 ], [ 101, 101, 101, 101 ], [128X[104X
    [4X[28X  [ 102, 102, 102, 102 ], [ 104, 104, 104, 104 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording  to [HW04], any subgroup [22XG[122X of [22XM[122X must contain elements in the class
  [10X30G[110X  of [22XM[122X. This determines the class fusion up to Galois automorphisms. (But
  note that we do not store the fusion, since [22XG[122X is not a subgroup of [22XM[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpos:= Position( ClassNames( t, "Atlas" ), "30G" );;[127X[104X
    [4X[25Xgap>[125X [27Xgood:= Filtered( fus, map -> pos in map );[127X[104X
    [4X[28X[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52, [128X[104X
    [4X[28X      3 ], [128X[104X
    [4X[28X  [ 1, 153, 152, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52, [128X[104X
    [4X[28X      3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( s, good, t );[127X[104X
    [4X[28X[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, [128X[104X
    [4X[28X      97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52, [128X[104X
    [4X[28X      3 ] ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) = fail;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-15 [33X[0;0Y[22XL_2(71) → M[122X[101X[1X (May 2009)[133X[101X
  
  [33X[0;0YThe  sporadic  simple  Monster  group [22XM[122X contains a maximal subgroup [22XG[122X of the
  type [22XL_2(71)[122X, see [HW08]. The class fusion of [22XG[122X into [22XM[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L2(71)" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( s, fus, t );[127X[104X
    [4X[28X[ [ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112, [128X[104X
    [4X[28X      112, 112, 11, 19, 112, 112, 114, 60, 36, 27, 114, 17, 114, 27, [128X[104X
    [4X[28X      7, 60, 114, 5, 114, 60, 36, 27, 114, 3 ], [128X[104X
    [4X[28X  [ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112, [128X[104X
    [4X[28X      112, 112, 11, 19, 112, 112, 115, 61, 36, 28, 115, 17, 115, 28, [128X[104X
    [4X[28X      7, 61, 115, 5, 115, 61, 36, 28, 115, 3 ], [128X[104X
    [4X[28X  [ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112, [128X[104X
    [4X[28X      112, 112, 11, 19, 112, 112, 117, 61, 43, 28, 117, 17, 117, 28, [128X[104X
    [4X[28X      9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ], [128X[104X
    [4X[28X  [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113, [128X[104X
    [4X[28X      113, 113, 12, 20, 113, 113, 114, 60, 36, 27, 114, 17, 114, 27, [128X[104X
    [4X[28X      7, 60, 114, 5, 114, 60, 36, 27, 114, 3 ], [128X[104X
    [4X[28X  [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113, [128X[104X
    [4X[28X      113, 113, 12, 20, 113, 113, 115, 61, 36, 28, 115, 17, 115, 28, [128X[104X
    [4X[28X      7, 61, 115, 5, 115, 61, 36, 28, 115, 3 ], [128X[104X
    [4X[28X  [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113, [128X[104X
    [4X[28X      113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28, [128X[104X
    [4X[28X      9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe candidates differ on the classes of the element orders [22X7[122X and [22X36[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord:= OrdersClassRepresentatives( s );;[127X[104X
    [4X[25Xgap>[125X [27Xord36:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 36 );[127X[104X
    [4X[28X[ 21, 25, 27, 31, 33, 37 ][128X[104X
    [4X[25Xgap>[125X [27XList( repr, x -> x{ ord36 } );[127X[104X
    [4X[28X[ [ 114, 114, 114, 114, 114, 114 ], [ 115, 115, 115, 115, 115, 115 ], [128X[104X
    [4X[28X  [ 117, 117, 117, 117, 117, 117 ], [ 114, 114, 114, 114, 114, 114 ], [128X[104X
    [4X[28X  [ 115, 115, 115, 115, 115, 115 ], [ 117, 117, 117, 117, 117, 117 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording  to [NW02, Table 3], [22XG[122X contains elements in the classes [10X7B[110X and [10X36D[110X
  of [22XM[122X. This determines the class fusion up to Galois automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpos1:= Position( ClassNames( t, "Atlas" ), "7B" );;[127X[104X
    [4X[25Xgap>[125X [27Xpos2:= Position( ClassNames( t, "Atlas" ), "36D" );;[127X[104X
    [4X[25Xgap>[125X [27Xpos:= [ pos1, pos2 ];;[127X[104X
    [4X[25Xgap>[125X [27Xgood:= Filtered( fus, map -> IsSubset( map, pos ) );[127X[104X
    [4X[28X[ [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113, [128X[104X
    [4X[28X      113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28, [128X[104X
    [4X[28X      9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ], [128X[104X
    [4X[28X  [ 1, 170, 169, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113, [128X[104X
    [4X[28X      113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28, [128X[104X
    [4X[28X      9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( s, good, t );[127X[104X
    [4X[28X[ [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113, [128X[104X
    [4X[28X      113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28, [128X[104X
    [4X[28X      9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) = repr[1];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X9.7-16 [33X[0;0Y[22XL_2(41) → M[122X[101X[1X (April 2012)[133X[101X
  
  [33X[0;0YThe  sporadic  simple  Monster  group [22XM[122X contains a maximal subgroup [22XG[122X of the
  type [22XL_2(41)[122X, see [NW13]. The class fusion of [22XG[122X into [22XM[122X is ambiguous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "L2(41)" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( s, t );;[127X[104X
    [4X[25Xgap>[125X [27Xrepr:= RepresentativesFusions( s, fus, t );[127X[104X
    [4X[28X[ [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 70, 70, 19, [128X[104X
    [4X[28X      70, 70, 19, 4, 70, 19, 70 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 72, 72, 19, [128X[104X
    [4X[28X      72, 72, 19, 6, 72, 19, 72 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 73, 73, 20, [128X[104X
    [4X[28X      73, 73, 20, 5, 73, 20, 73 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 66, 33, 66, 12, 7, 33, 66, 12, 66, 3, 72, 72, 19, [128X[104X
    [4X[28X      72, 72, 19, 6, 72, 19, 72 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 66, 33, 66, 12, 7, 33, 66, 12, 66, 3, 73, 73, 20, [128X[104X
    [4X[28X      73, 73, 20, 5, 73, 20, 73 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 67, 30, 67, 11, 10, 30, 67, 11, 67, 3, 72, 72, 19, [128X[104X
    [4X[28X      72, 72, 19, 6, 72, 19, 72 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 67, 30, 67, 11, 10, 30, 67, 11, 67, 3, 73, 73, 20, [128X[104X
    [4X[28X      73, 73, 20, 5, 73, 20, 73 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 68, 32, 68, 12, 10, 32, 68, 12, 68, 3, 72, 72, 19, [128X[104X
    [4X[28X      72, 72, 19, 6, 72, 19, 72 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 68, 32, 68, 12, 10, 32, 68, 12, 68, 3, 73, 73, 20, [128X[104X
    [4X[28X      73, 73, 20, 5, 73, 20, 73 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 72, 72, 19, [128X[104X
    [4X[28X      72, 72, 19, 6, 72, 19, 72 ], [128X[104X
    [4X[28X  [ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 73, 73, 20, [128X[104X
    [4X[28X      73, 73, 20, 5, 73, 20, 73 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe candidates differ on the classes of the element orders [22X3[122X–[22X8[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xambig:= Parametrized( repr );;[127X[104X
    [4X[25Xgap>[125X [27Xambigpos:= PositionsProperty( ambig, IsList );[127X[104X
    [4X[28X[ 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, [128X[104X
    [4X[28X  23 ][128X[104X
    [4X[25Xgap>[125X [27XSet( OrdersClassRepresentatives( t ){ ambigpos } );[127X[104X
    [4X[28X[ 3, 4, 5, 6, 7, 8 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAccording to [NW13, Theorem 3], [22XG[122X contains elements in the classes [10X3B[110X and [10X4C[110X
  of [22XM[122X. This determines the class fusion uniquely.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpos1:= Position( ClassNames( t, "Atlas" ), "3B" );;[127X[104X
    [4X[25Xgap>[125X [27Xpos2:= Position( ClassNames( t, "Atlas" ), "4C" );;[127X[104X
    [4X[25Xgap>[125X [27Xpos:= [ pos1, pos2 ];;[127X[104X
    [4X[25Xgap>[125X [27Xgood:= Filtered( fus, map -> IsSubset( map, pos ) );[127X[104X
    [4X[28X[ [ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 73, 73, 20, [128X[104X
    [4X[28X      73, 73, 20, 5, 73, 20, 73 ] ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s, t ) = good[1];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
