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< href=chap9.#X7A03A83E87FB1189>9<span"Ambiguous ClassFusions the class"pkgGAP/trong Library</

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 <a href="chap9.html#X7F5186E28201B027">9.3-1 BN7 → B (March 2002)</a>

 <a href/panjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

 <a href="chap9.html#X85822C647B29117B">9.3-3 A_6 × L_2(8).3 → Fi_24^' (November 2002)</a>



< /< class""&bsp;nbsp<span<a href=.#7962DD4387D63675"9.- span =Heading">spanclass"impleMath"> .) →MMay2009)span<a

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< class"">< /a =chap9.tmlX7E9C203C7C4D709D>.-3 =Heading"< =S"M_11 →B( 2009)/span>/a>

 <a href="chap9.html#X85821D748716DC7E">9.7-4 L_2(

 <a href="chap9.html#X828D81487F57D612">9.7-5 L_3(3) → B (April 2009)

 <a href="chap9.html#X7B4E13337D66020F">9.7-6 L_2(17).2 → B />

 <a href="chap9.html#X8528432A84851F7B">9.7-7 L_2(49).2_3 → B (June 2006)</a>

 

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<spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
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< =ContSS"<r /<pan class="&;&;/>a href".tml#7A94F78C792122D5">.7-2 span class=Heading>span class"impleMath>.3^.^.S_4 2A12( 2011</pan>/>

 <a href="chap9.html#X7E2AF30C7E8F89F9">9.7-13 <pan>
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 <a href="chap9.html#X80051B297DF244CF">9.7-14 L_2(59) → M (May 2009) – Do not use this!</a>

 <a href="chap9.html#X8409DA2E83A41ABE">9.7-15 
 <a href="chap9.html#X78B3B1BE7A2CA4D1">9.7-16 L_2(41) → M (April 2012)</a>

</div></div>
</div>

<h3>9 Ambiguous Class Fusions in the GAP Character Table Library</h3>

<>:January11th 2004 these examples, is ambiguous in the notdetermine table strategy is to compute first all possibilities with the GAP function <code class="func">PossibleClassFusions</code> (<a href="../../../doc/ref/chap73.html#X7883271F7F26356E">Reference: PossibleClassFusions</a>), and then to use either other character

The purpose of this writeup is twofold. On the one hand, the computations are documented this way. On the other hand, the GAP 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 GAP Character Table Library <a href="chapBib.html#biBCTblLib">[Bre25]</a>.

The examples use the GAP Character Table Library, so we first load this package.

<div class="example"><pre>
<span<The examples use the <trong class=pkg"GAPCharacter Library, first load this .>
<div class=example><>
</<pan class"APprompt"&t;/pan <panclass"APinput">oadPackage("ctbllib" )a="X784492877DB04FE9"="784492877DB04FE9">/a>/>

<h4p< id=X784492877DB04FE9"=X784492877DB04FE9"<>/p>

The function <code class="code">SetOfComposedClassFusions</code> takes two java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

Note that the returned list may be a proper subset of the set of all possible class

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
span class=GAPinput"SetOfComposedClassFusions:=function(hfusg, ufush )/pan>
spanclass"">gt</pan>< classGAPinput>result= [;
< classGAPprompt>gt
< class="&;><pan class"APprompt">gt;/pan>< ="" map1 ufush/pan
<;<span="APinput" od</
 od/>
<="GAPinput"> return;/pan>
> &t;/pan><pan="APinput" returnresult</div>

<a id="X7EA839057D3AD3B4" name="X7EA839057D3AD3B4"></a>

<h4>9.2 Fusions Determined by Factorization through Intermediate

This situation clearly occurs java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<a id="X78DCEEFD85FF1EE2" name="X78DCEEFD85FF1EE2"></a>

<h5>9.2-1 Let H be the Sylow 5

<div class="example"><pre>

haracterTable "o3")
gap( "o3")
)" )
("^12:242")
gap> Length(span=GAPprompt>ap&t;span spanclass=GAPinput"hfusco3=PossibleClassFusionsh java.lang.StringIndexOutOfBoundsException: Range [111, 100) out of bounds for length 111
/><div>
</pre></div>

<As <panclass=SimpleMath"H/span> maximal in <pan class=SimpleMathCo_3</pan>,we lookat those maximal subgroups of <pan ="impleMath</pan>whose order divisible by that of < class=SimpleMath>H/pan>
<span< =">gap&; span ="":=Maxes(co3 )
[ "McL McL." "S" U(3)(^){}" M23","^:2xm11),
 2S6(),"(2" "^+:s6" "^.8, "()D12
 "2xm12", "2"xm12,22[^*^]S3 "3xpsl2,8)3" "4xs5java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
gap> maxes:= List( mx =""gapgt;/ spanclass"APinput"filt=Filtered ,x&t; Size( x ) mod Size( h ) = 0 );
gap> filt:= Filtered( maxes, x -> Size( x ) mod Size( h ) = 0 );
[ CharacterTable( "McL.2" ), CharacterTable( "HS" ),
 CharacterTable( "U3(5).3.2" ) ]
</pre></div>

According to the 

We p to < class=pkg"Atlas(ee href=chapBib.java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 728

<div class="example"><pre>
gap> max:= filt[1];;
<;<spanclass"APinput>:=PossibleClassFusions(h,max );/>
gap> gap> max:= filt[1];;
2
gap> reps:= RepresentativesFusions( h, comp, co3 );
[ [1,2,3, 4,8,8 ,9,10,11 17,17 19,22,23 ,27
 30, ,34 40 40,, 40,42]]
</pre></div>

 factoring through a maximal subgroup of type McL.2 determines the fusion from Hspanclass=GAPprompt"gap>/span>(comp/spanjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84

Alternatively, we can use the group U_3(5).3.2 as intermediate subgroup, which leads to the same result.

<>So through a maximal subgroup of <spanclass="SimpleMath"McL2/pan determinesthe fusion<spanclass"SimpleMath"H/pan to<class="SimpleMath"Co_3/uptotableautomorphisms<pjava.lang.StringIndexOutOfBoundsException: Index 229 out of bounds for length 229
gap>gap>:( h,;>/ <=": (h,);<>
gtcomp:= SetOfComposedClassFusions( maxfusco3, hfusmax );;
<java.lang.StringIndexOutOfBoundsException: Range [60, 5) out of bounds for length 125
gap ">&;/ span =G>=;/>
/>
p stored of< =""<>Finally,span class=GAPprompt"gap&t; span ="APinput"GetFusionMap(h,co3 ) in reps;

<div class="example"java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
gap> GetFusionMap( h, co3 ) in java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 0
true
</pre></div>

<p<aid"86CEA907EC4C833"name=X86BCEA907EC4C833>/>/

<>922 < class="eading">31:15 → B (March 2003)</h5>

< class=GAPprompt>gap>

<panclass"GAPprompt"gapg;/span <spanclass"APinput">h:= CharacterTable( "31:15" );;
 hfusb:= PossibleClassFusions( h, b );;
ap>/span> b:= CharacterTable( "B" );;
gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> hfusb:= PossibleClassFusions( h, b );;

2
</pre></div>

We span classGAPprompt>&t/span><pan class"GAPinput"hfusth: PossibleClassFusions(h th);;/span>

<div class="example"><pre>
class="&t<java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 98
<spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 84
>< =G"comp=( );gap> Length( comp );
2
gap/> span class="GAPinput">reps:= RepresentativesFusions( h, comp, b );
[ [ 1, 145, 146, 82, 82, 19, 82, 7, 19, 82, 82, 19, 7, 82, 19, 82, 82[[1,145146,82 82,19,,7 19,82 82,19,7 ,19 ,82
 ] ]
gaph5>.2- < class="Heading">span class=SimpleMath>SuzN3→Suz/>(eptember2002)/>/>
true
</pre></div>

<a id="X7C719F527831F35A" name="X7C719F527831F35A"></a>

SuzN3 → uz(eptember 2002)</h5>

The class fusion from the Sylow 3gap> h:= CharacterTable( "SuzN3" );

<div class="example"><preCharacterTable("Suz")
gap> gap> hfussuz:= PossibleClassFusions( h, suz );;
CharacterTable( "3^5:(3^2:SD16)" )
gap> suz:=
CharacterTable( "Suz" )
Since H is not maximal in Suz, we try to factorize the fusion through a java.lang.StringIndexOutOfBoundsException: Index 146 out of bounds for length 0
 
2
</pre></div>

Since H is not maximal in <span class="java.lang.StringIndexOutOfBoundsException: Range [0, 84) out of bounds for length 67

<div java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
= (suz java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 115
<pan""&t<>span="filt=( , -g; ( x )mod (h ) =0 ;/pan
 "2U()23"), haracterTable35:M11,
 p>Definitelyclass"</> a with span class="SimpleMath">H, because the Sylow 3 normalizer in M_11java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>

The group 3_2.U_4(3).2_3^' does not admit a fusion from >/>< "APinput">: ,max;<>

<div class="example"><pre>
gap> PossibleClassFusions( h, filt[1] );
[ ]
</

< <class=""3^M</> a <pan ="H in span ="">11/>has the span=""3^:; using span ="impleMath"32+:2(2^ A_4)</pan> would lead to the sameresult as we get below. We compute the compositions of possible class fusions.

<div class="example"><pre>
gap> max:= filt[2];;
gap> hfusmax:= PossibleClassFusions( h, max );;
gap> maxfussuz:= PossibleClassFusions( max, suz );;
gap> comp:= SetOfComposedClassFusions( maxfussuz, hfusmax );;
gap> repr:= RepresentativesFusions( h, comp, suz );
[ [ 1, 2, 2, 4, 5, 4, 5, 5, 5, 5, 5, 6, 9, 9, 14, 15, 13, 16, 16, 14,
 15, 13, 13, 13, 16, 15, 14, 16, 16, 16, 21, 21, 23, 22, 29, 29,
 29, 38, 39 ] ]
</pre></div>

So the factorization determines the fusion map up to table automorphisms. We check that this map is equal to the stored one.

<div class="example"><pre>
gap> GetFusionMap( h, suz ) in repr;
true
</pre></div>

<a id="X828879F481EF30DD" name="X828879F481EF30DD"></a>

<h5>9.2-4 F_{3+}N5 → F_{3+} (March 2002)</h5>

The class fusion from the table of the Sylow 5 normalizer H in the sporadic simple group F_{3+} into F_{3+} is ambiguous.

<div class="example"><pre>
gap> f3p:= CharacterTable( "F3+" );;
gap> h:= CharacterTable( "F3+N5" );;
gap> hfusf3p:= PossibleClassFusions( h, f3p );;
gap> Length( RepresentativesFusions( h, hfusf3p, f3p ) );
2
</pre></div>

H is not maximal in F_{3+}, so we look for tables of maximal subgroups that can 15, 13,13,13 16 ,14 16,16 16 21,21 , 22 29,29,

<div class="example"><pre>
gap> <span<div class"example"><pre>
[ CharacterTable( "Fi23" ), CharacterTable( "2.Fi22.2" ),
 CharacterTable( "(3xO8+(3):3):2" ), gap> GetFusionMap( h, suz ) in repr;
 CharacterTable( "(A4xO8+(2).3).2" ), CharacterTable( "He.2" ),
 CharacterTable( "F3+M14" ), CharacterTable( "(A5xA9):2" ) ]
gap> possfus:= List( filt, x -> PossibleClassFusions( h, x ) )</re>/div>
[ ] [ ] [ ], [ ,java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 [ [ 1, 69, 110,12,80, 121, 4 72,113 11, 11, 79, 79, 120 120,
 3The class fusionfromthe table the span class="impleMath"5/ normalizerspanclass"impleMath">H/ the groupspanclass="SimpleMathF_3}/span> into span ="SimpleMath"F_{+is 
 37, 105, 105, 139, 140< ="GAPprompt">/>spanclass=GAPinput>=CharacterTableF3 ;
 44, 167, 167, 48, 48, 171, 171, 57, 57, 180, 180, 66, 66,
 189, 189 ],
 [ 1, 69, 110, 12, 80, 121, 4, 72, 113, 11, 11, 79, 79, 120,
 120, 3, 71, 11, 79, 23< class"APprompt">>/> spanclassGAPinput>( RepresentativesFusions(h ,f3p))
 37, 37,
 44,,44,167,167, 48,48, 171,171,, 57,57 180, 180 66
 66, 189, 189 ] ], [ ], [ ], [ ] ]
</pre></div class"xample"<>

We see that from the eight possible classes of maximal subgroups in F_{3+} that might containspan class="GAPprompt">gap> filt:= Filtered( maxes, x -> Size( x ) mod java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 58

<div class="example"><pre>
&t/pan><class"APinput"possfus=Listfilt x &;PossibleClassFusions ,x ))
CharacterTable( "(A4xO8+(2).3).2" )
gap>gap> maxfusf3p:= PossibleClassFusions( max, f3p );;
gap>&;/> span=G>( comp)</>
2
gap> <span 120, 3, 71, 11, 79, 23, 91, 112, 120, 13211 79, ,112,120 132,,32,97 ,
[ [ 1, 2, 4, 12, 35, 54, 3, 3, 16, 9, 9, 11, 11, 40, 40, 2, 3,44 ,167 ,48, 48 ,171 , 180 ,
 , ,90 ,22 ,20,,43 ,50 ,823
 23, 46, 46, 47, 47, 10, 10, 9, 9, 10, 10, 11, 11, 26, 26, 28,
 28, 67,
>/div

<Finallywe map in tablejava.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 80

 span=>java.lang.StringIndexOutOfBoundsException: Range [76, 74) out of bounds for length 100
true
<<>

span class=>&;/> span""Length)/

<< =G>&t9.3 Fusions 35,36,13 40,7 22,19 20 43 ,50,50 8,8,23
Subgroups</h4>

In each of the>Finally, we the stored thetable is<center> <img src="ambigfus1.png" alt

<><a=X7F5186E28201B027"name="75186E28201B027>/9.3-1 BN7 → java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

Let H be a Sylow 7 normalizer in the sporadic simple group < id="F81AC6 name"7981579278">/a>/p>

<div class="example"><pre>
<G" B")/java.lang.StringIndexOutOfBoundsException: Range [95, 94) out of bounds for length 95
CharacterTable
gap> h:= CharacterTable( "BN7" );
CharacterTableh5.31 < ="Heading"BN7 B/pan (arch2002)/><h5
< <pan =S">/> be a <pan class=java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 327
<panclass""gapg;</pan spanclass"APinput"Length RepresentativesFusions,hfusb, )/>
2
</pre></div>

< ="g;/ span class">fusb(,b);;/
g<
2
gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
CharacterTable( "Th" )
gap> 7 subgroup of B have the types ^3D_4(2).3 and 7^2:(3 × 2S_4), and the Sylow <
/>java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

The class fusion of U into B<pre>
<">>=java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 101
java.lang.StringIndexOutOfBoundsException: Range [30, 14) out of bounds for length 31
< ""gt< =>: java.lang.StringIndexOutOfBoundsException: Range [118, 116) out of bounds for length 133
> <[ [ 1, 31, 7, 7, 5, 28, 28, 17, 72, 72, ,
[ [ 1, 31, 7, 7, 5, 28, 28, 17 09,17 ,45,72 72,127 ,127 127 ] java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
 109, 17, 45, 45, 72, 72, 127, 127, 127, 127 ] ]
</</iv>

The : PossibleClassFusions,h)pre
< =GAPinput"( ( h ,b )<spanjava.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
gap> gt;<> < =" -> CompositionMaps x )in );
gap> Length( RepresentativesFusions( h, filt, b ) );
s ">ap&;<><class"GAPinput>GetFusionMap(h, b)infilt
</pre></div>

Finally, we compare the result with the

<div class="example"><pre>
<pan class"APprompt">gap&t;<> spanclass=GAPinput>(h, b filt/>
true
<re/

<a id="X79710B137B5BB1B8" name="X79710B137B5BB1B8"></a><pan=>span=G"t: ( "";

<h5>9.3-2 span classGAPpromptgapgt;< class=GAPinput"repr= RepresentativesFusions m,mfust,t);

The class fusion of the java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 1

<div class="example"><pre>
gap> gap> t:= CharacterTable( "F3+" );;
gap> mfust:= PossibleClassFusions( m, t );;
gap> repr:= class="GAPprompt">gap> OrdersClassRepresentatives;/pan>
gap> Length( repr );
2
</pre></div>

We span class>> 

<div class="example"><pre>
<anclass=GAPprompt"gap> span class="GAPinput">a4inm:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lass"n &SizesConjugacyClasses ) n})=12)
[ [ 1, 69, 110 ] ]
gap&</pan>u:= CharacterTable("yclic" *CharacterTable "O8()3 )
[ 4, 4 ]
<spanclass"APinput"OrdersClassRepresentatives ){[ 1. 4 }
[1 , 2,3]
</pre></div>

Let us take one such element gpan class=GAPpromptgap&t;/pan spanclass=GAPinput>sfust: PossibleClassFusionss t;/pan

<div class="example"><pre>
<= CharacterTable( "F3+3A")
CharacterTable( "(3xO8+(3):3):2" )
</prespan class&t;/pan span =GAPinput">ilt: (mfust,/>

< intersection< class""(g)= M/span>contains subgroup < class="U/span>of the type span ="SimpleMath>3×O_8().<,andthewe the class of<class=S>/ "></> < =SimpleMath"<,thenthefact thosefrom class""M < ""G>possiblewhose from<class"U/> < ="impleMathM</>into< class"SimpleMath" from<spanclass="S</pan span =S">/>.p

<div class="example"><pre>
< class=G>&t</>< =G>: ( "Cyclic,3 )*CharacterTable O+2.")<spanjava.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
CharacterTable( "gap&t <pan ="GetFusionMap );
gap> gap> ufusm:= java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0
gap>/span>fust=PossibleClassFusions(s,t ;;/span>
gap> comp:= SetOfComposedClassFusions( sfust, ufuss );;
span ="GAPprompt">gap> Length( comp );
6
< class"APprompt">gapgt;/span><spanclass="APinput">filt:= Filtered( mfust,
> The class fusion of the maximal subgroup the java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 159
gap> repr:= RepresentativesFusions( m, filt, t );;

1
gap> GetFusionMap( m, t ) in repr;
true
</pre></div>

So the class fusion from M</java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 117

<a id="X85822C647B29117B"

<h5>9.3-3 A_6 ×p> use fact direct factor spanclass=java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 268

The class fusion of the maximal subgroup M ≅ A_6 × L_2(8).3 of G = Fi_24^' is ambiguous.

< class=">
<spanclass="APprompt"gap&t<spanclass=GAPinput"m= CharacterTable( "6(8)3";;
gap> <gap>/span><class=GAPinputList( dppos[] l -; Sum SizesConjugacyClasses( m {l}))
gap> 3Apos:= Position( OrdersClassRepresentatives( t ), 3 );
gap> gap> <span"3Ainm:=List( mfust,map >Position((map 3pos )
2
</pre></div>

We will use the fact that the direct factor of the type

<div class="example"><pre>
 < =GAPinput>=(m)gap> gap> List( dppos[1], l -&java.lang.StringIndexOutOfBoundsException: Range [0, 84) out of bounds for length 33
360,1512 ]
gap> 3Apos:= Position( java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 34
4
gap> gap> comp:= SetOfComposedClassFusions( sfust, ufuss );;
[23,23 23,,34,34 34,34 java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
gap> /span>map-gt ForAny ,/span
true
</pre></div>

Since the normalizer of an element of span class="GAPprompt">gap> <spanGAPinput>repr: RepresentativesFusions( m,filt,t );/span

(Note that all classes in the 3^2:2 type group are java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 1

<divclass=example>pre>
span class=SimpleMath>32D_8 ×U_4(.22).2→Bgap> ufuss:= PossibleClassFusions( u,
<spanclass=GAPprompt"gap>/span>
gap&
< class=GAPprompt"gap&t <pan ="APinput>=Filteredmfust/
>
> <divclass="example"><pre>
gap&spanclass""gap<> <spanclass"" java.lang.StringIndexOutOfBoundsException: Range [82, 80) out of bounds for length 96
gap> <java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 29
1
gap> gap> Length( nfusb );
true
</pre></div>

<a id="/></iv>

<>34 H>classS>32D_8 ×(3.^2. → B

Let 

Accordingjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 92

The class fusion from java.lang.StringIndexOutOfBoundsException: Range [90, 89) out of bounds for length 99

<div class="example"><pre>
<class="APprompt">;>span">=CharacterTable " ;gap> n:= java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 0
CharacterTable( "S3xFi22.2" )
gap> nfusb:= PossibleClassFusions( 3[1,/
gap> Length( nfusb );
1
gap> nfusb:= nfusb[1];;&t;/> span ="APinput>rec :,quick: java.lang.StringIndexOutOfBoundsException: Range [119, 106) out of bounds for length 119
</pre></div>

<

<div class="example"><pre>
gap> fi222:= CharacterTable( "< ="gap&t< =G>=(fi222 ;<>
<=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 122
gap> s3:= CharacterTable( "S3" );;
gap> inter:= s3 * fi222n3a;;
gap><Next ="G ∩ />to<java.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 154
<pre>
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
</div>

<="gapg;< ="> inter ))<java.lang.StringIndexOutOfBoundsException: Index 128 out of bounds for length 128

<div class="example"><pre>
>
< =java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 113
gap> approx:= List( interfusg,
gap> interfusn:= PossibleClassFusions( inter, n,
< class"APinput>rec : t ));/>
</div>

The lift is unique. span class="GAPprompt">gap> Length( gfusb );

<div class="example"><pre>
gap> <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> interfusn[1] = Concatenation( List( [ 0 .. 2 ],
> i -> intermods3fusnmods3[1] + i * nccl ) );
true
</pre></div>

Next we compute the class fusions java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 0

< classexample<pre
< class=GAPprompt"gap>/span>span ="GAPinput>: t, );/java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
CharacterTable( "(3^2:D8xU4(3).2^2).2" )
gap> interfusg
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 89
2
</div>

The approximation of the class fusion fromspan class=""gap></ class=">(""

<div class="example"><pre>
< class=GAPprompt">gap></pan> nterfusb= CompositionMaps([1]);;/span
gap<pan=G>ap&;</pan>< =G>1=Normalizer(g1,s1;<>
"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 108
gap> gt; map -> PossibleClassFusions( tblg, b,
< ="APprompt"&t/><spanclass=GAPinput"> rec( fusionmap:= map ) ) ) ) );;
gapg;/span><panclass=GAPinput"Length( RepresentativesFusions(tblg,gfusb,b ) );
1
</pre></div>

<>, result that onlibrary./>

<class>pre
gap> </div>

<a id="X7962DD4387D63675" name="X7962DD4387D63675"></a>

<h5>9.3-5 7^1+4:(3 × 2.S_7) → M (May 2009)java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

The class fusion of the

< class="xample"pre>
gap> <G">:=CharacterTable(M");/>
<( tblu, m );;<>
gap> Length( RepresentativesFusions( tblu, ufusm, m ) );
< ""&;java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 128
</pre></div>

<>The Uspan> containsaSylow 7</pan>-subgroupofM</> onlymaximal of span =SimpleMath>with thisproperty are the class For that, we use representations from the Atlas of Group Representations <a href="chapBib.html#biBAGRv3">[WWT+]</a>, and access these representations via the GAP package AtlasRep (<a href="chapBib.html#biBAtlasRep">[WPN+22]</a>).

<div class="example"><pre>
gap> LoadPackage( "atlasrep", false );
true
gap> gap> gap> n1:= java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 1
<span=>&;/>< ="=( ufusm, );/java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
<=>>/ span">:=SylowSubgroup,7);/>
gap> n2:= Normalizer( g2, s2 );;
<spanclass="APprompt">ap;/span spanclass""Size(n1 (n2 )<spanjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
true
gap> ( Size( m ) / Size( s1 ) ) mod 7 <> 0;
true
</pre></div>

So let N be a <pThe class fusionof themaximal subgroup span =SimpleMath"M ≅3^.O_7(3):</pan> of span class=SimpleMath">G = Fi_24 = F_{3+}.2 is ambiguous.

<pspan class=GAPprompt"&t;/span> span class=GAPinput"t:( "3.");/span>

<div class="example">gap>Length( RepresentativesFusions( m, mfust, t ) );
gap> tbln:= CharacterTable( Image( IsomorphismPcGroup( n1 ) ) );;
gap> tbls:= CharacterTable( "7^(2+1+2):GL2(7)" );;
gap> nfusm<pre>div
We will use the fact thatheelementaryabeliannormal of < class=>^<spaninxsay in the code class=c>A/ode of . Thisfact can as follows./>
2
<class"APprompt">gt<span<spanclass=GAPinput>nsg: ClassPositionsOfNormalSubgroups( m)
<pan class="APprompt"gap;</pan>< class"GAPinput"sfusm: PossibleClassFusions );/>
gap> Sum( SizesConjugacyClasses( m ){ nsg[2] } );
ap&t
1
</></div>

Now we use the condition that the class fusions from ap> List( rest, l -> ClassNames( t, "Atlas" ){ l } );

<java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [60, 5) out of bounds for length 113
gap> ufusm:= Filtered( ufusm
> map1 -> CompositionMaps( map2, < "java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 102
> </div>

Let <span<panclass="GAPprompt">gap> Size( s ) / size_u;

<div class="example"><pre>
<G"gapg;/ java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 130
CharacterTable( "7^1+4.2A7" )
gap> cfusm:= java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> Length( RepresentativesFusions( tblc, cfusm, m ) );</java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 104
2
gap>< ="gapgt;/ java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 130
<spanclass=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 120
gap> Length( RepresentativesFusions( tblc, cfusm, m ) );
1
</pre></div>

<a id="X860B6C30812DE3FC" name="X860B6C30812DE3FC"></a>

<h5>9.3-6 3^7.O_7(3):2 → Fi_24 (November 2010)</h5>

<>The classfusion ofthe subgroup ≅(2>of ="=Fi_24 ={3+}2/pan ambiguous

<div class="example"><pre>
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 106
gap> t:= CharacterTable( "F3+.2" );;
gap> 
<class=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 117
2
</pre></div>

We will use the fact that the elementary abelian normal subgroup of order gap> ufusm:= Concatenation( List( approx, map ->

<div class="example"><pre>
&t<> < class=GAPinput">= ,s)<>
gap> Sum( java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 1
2187
<class">gt;/>< "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 73
2187
gap>> PossibleClassFusions( u, s, rec( fusionmap:= map ) ) ) );;
[>/
java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0
[ 1" 3" 3" 3"] ]
</pre></div>

The normalizer S of java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
 span class="GAPinput">s:= CharacterTable( "F3+.2N3A" );
CharacterTable( "S3xO8+(3):S3" )
gap> PowerMap(gap&t</pan> span class="APinput>= SizesCentralizersm)[2]gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
3360
</pre></div>

Using the list java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<div class="example"><pre>
<"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 104
gap> mx:= List( Maxes( o8p3 ), CharacterTable );;
 < class"APinput"filt=(mx,x& 3360 modIndex o8p3, x)=0)<spanjava.lang.StringIndexOutOfBoundsException: Index 130 out of bounds for length 130
[ CharacterTable( "3^6:L4(3)" ), java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 1
 CharacterTable( "O8+(3)M9" ) ]
gap> java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 12
[ 1120, 1120,1120 ]
</pre></div>

We compute the classexample>>

<div class=/re<div
gap> uu:= filt[p>< =X7="/<>
gap> u:= CharacterTable( "Symmetric", 3 ) * h5>9.3-7 ^2E_6(2)N3C → ^2E_6(2) (January 2019) in G is ambiguous.
span class="GAPpromptspan">Length( rep );
gap> uufusm:= PossibleClassFusions( uu, m );;
gap> gap>>
> gap;/> <pan classGAPinput( sfust -& ;/>
gap> pos3CinS:= 6;;
<span="GAPprompt">>> <span"APinput"> (u , recjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
gap> class>>> <=""> (
span="GAPprompt">gap;</><pan="GAPinput>Length(uufuss );</>
8
> <span="GAPinput">Size ) / Size( u );/span
gap> the class fusions of U in <spanclass=SimpleMathN <">N in must be equaltothe composition of the class fusions of U in S and S in G. This reduces the number of candidates forjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
8
</pre ">=java.lang.StringIndexOutOfBoundsException: Range [59, 57) out of bounds for length 134

pclass java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 273

<div class="example"><pre>
<span="">gap> <span,t;/
gap> > 
8
>

It turns out that>Next thefactthat< class="SimpleMath">g thus classSimpleMath">N is invariant underanouter < =SimpleMath">αspan, class="SimpleMath">G that an classes class> the <="> < class=">3 G<> java.lang.StringIndexOutOfBoundsException: Range [508, 507) out of bounds for length 770

<div class="example"><pre>
gap> 
<="">gap;</> <=G( filt );
4
<pan=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 116
1
</pre></div>

<>class the class"GAPstrong> ./>

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 98
true
</pre></div>

<a id="X7C3AC42F8342EE2E" name="X7C3AC42F8342EE2E"></a>

<h5>9.3-7 <java.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 4

Let G = ^2E_6(2), and <spanpThe of class"span< ="26/spanjava.lang.StringIndexOutOfBoundsException: Range [384, 118) out of bounds for length 607

<div class="example"><pre>
gap>gap&div="example"><
7
gap;> <="GAPinput">:= ( [ 1 .. NrConjugacyClasses]>
/pan><pan="GAPinput">filt:= Filtered( rep, map -> ForAny ord3beta -&t</pan>
< classgt<><class">( map, alpha)=Permuted( map, ) ) ;<>
< class"gt;/>span =>:= PossibleClassFusions( , t ;
gap>Again number candidates class
true
gap> ForAll ></>< "java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 104
true
</pre></div>

The class fusion 72,,77 78 , ,64 ,66 , ,82 5 ,

<div class="example"><pre>
gap> rep31 ,31 ,,32,31 ,32,51 ,52
</div></div

We>The class="pkg">GAP
< GAPprompt;</span="GAPinput">( n, t1;/
<span="GAPprompt">gap;</ =">:= ( s, t );
<pre>
true
<spanclass="GAPprompt">gap> pos3CinS:= 6;;
</pre></div>

</div>
CharacterTable16)2(+4:^:")
9.<= Determined
> i -> > i -> uclasses
[ 2 maximalsubgroups type Suz> lift groups the <span=SimpleMath</ </. can <"S<>not ajava.lang.StringIndexOutOfBoundsException: Range [345, 344) out of bounds for length 397
gap></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<span="">gap;<span<pan="">ForAll,x-&[1 ]= )</div>

<>Composing class of <pan =">U in N> and< classSimpleMathin<span="impleMath> must be equal to the composition of the class fusions of U in S and S in G. This reduces the number of candidates for the fusion of N span ="GAPpromptspan <span<spanclassGAPinput">PossibleClassFusions(s 3 );<>

<div class&;3:= ( 3, t);/
gap> ufusn:= PossibleClassFusions( u
<class=">gap> < class"GAPinput=SetOfComposedClassFusions ;gap>> </div
gap>>We in second <">.Suz<>. Since the in < class"">3Suz of thespan class=""/> java.lang.StringIndexOutOfBoundsException: Range [224, 223) out of bounds for length 673
2
</pre></div>

Next we use the fact java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<="example><pre
/><class="APinput"( sfust
gap><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
14
gap> autt:= AutomorphismsOfTable( t );;
gap> ord3:= Filtered( autt, x -> Order( x ) = 3 );;
< class/> <span=Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 84
2
<[ ,,9 16 16 ,,, ,33 34, 47 ,47,,
</divpre></div>

<pThecharacter of N has 26 table automorphisms of order three. We do not java.lang.StringIndexOutOfBoundsException: Index 149 out of bounds for length 0

<div class="example[[ 1, 2 3,7 8, 9,16,16,26 ,28 32,33 34,47,47,47 48,
gap> autn:= AutomorphismsOfTable( n );;
gap> <gap> <span">etFusionMap(3, 3t) in ;
26
gap> Add, ( )gap> </prediv>
>
gap> Length( filt );
2
</pre></div>

Again,the number candidates the of N> in class=SimpleMathG>is fromto Moreover are because one satifies the java.lang.StringIndexOutOfBoundsException: Range [244, 236) out of bounds for length 257

<div class="example"><pre>
gapdiv="example"><pre
[ [ 1, 7,/span> 
 10, 14, 19, 19, 19, 16, 17, 18, 21, 58, 61, 62, 67, 68, 69, 57,
 72 59 75, 76,77,78,79 80,64 65,66 60,81 82,5 6,java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
 7, 6, 7, 7, 7, 7, 6, 7, 6, 7, 7, 24, 25, 27, 26, 28, 30span class=G><> <=java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 109
 31, 32, 31, 32,
4,76,,75 ,100,101 ,4, 20,
 29, 31, 32, 36, 36, 42, 42, 39,
 71,112 112,114,115 116 ]]
</pre></div>

The class The simple group" = U_42</> contains maximal <pre>
gap
true
</pre></div>

Remark:

Note that the structure of N is 3^1+6:2^3+6:haracterTable "62_")

java.lang.StringIndexOutOfBoundsException: Range [10, 4) out of bounds for length 26
gap> Length( LinearCharacters( n ) );
2
</pre></div>

<a id="X84F966E2824F5D52" name="X84F966E2824F5D52"></a>

<h4>9.4 Fusions Determined Using Commutative Diagrams Involving Factor
Groups</h4>

<a id="X7F2B104686509CAA" name="X7F2B104686509CAA"></a>

<h5>9.4-1 3.A_7 → 3.Suz#!/bin

<>The subgroups of 11 , 9, 10, 15, 16, 17, 3,4 15 25 ,26,26]]

<div class="example"><pre>
gap
gap

gap> 2sfuss=( 2s );/java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
gap> PossibleClassFusions class>>< span=GAPinput( ->( 2/>
[java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
</pre></div>

The class fusion of 3.A_7 into 3.Suz is

<div class="example<a id="X7E91F8707BA93081" name="X7E91F8707BA93081"></a>
gap> 
6
Accordingto
 49, 50, 48, 49, 50 ],
 [ 1, 11, 12, 4, 36, 37, 13, 16, 23, 82, 83, 32, 100, 101, 44, 38,
 ,112,116,48 115,113 ]]
gap> gap> </div>

We see that the possible

<div class="example<>Only half ofthe possible class fusions from U into G are compatible with the embeddings of U into G.2 via U.2 and G, and the compatible
gap> sfust:= PossibleClassFusions( s, t );;
gap> Length( sfust );
1
gap> filt:= Filtered( 3sfus3t, map -> CompositionMaps( GetFusionMap( 3t, t ),
> map )
>
[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48,
 49, 50, 48, 49, 50 ],
 1,3 2,7,9,8,16,16 26, 28,27, 32,34, 33,,47 47,48,
 50, 49, 48, 50, 49 ] ]
</pre></div>

So the class <pan "3 subgroups .Suz is totableOnethese istheof class">3.A_7.

<div class="example"><pre>
gap> RepresentativesFusions( 3s, filt, 3t );
[ [ 1, 2, 3, 7, 8, 9, 16, 16, 26, 27, 28, 32, 33, 34, 47, 47, 47, 48,
 49 50 48, 49,50] java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
gap> GetFusionMap( 3s, 3t ) in filt;
true
</pre></div>

Also the class fusions in the other orbit belong to/re</div

<div class="example"><pre>
</pan> u= CharacterTable "32.U4();</pan>
gap> PossibleClassFusions
[ ]
gap> Length( PossibleClassFusions( 3s, u ) );
8
</pre></div classGAPprompt>Set, x; x 31 . 36 });/span>

<>a id"82FB71647D37F4FD" name="X82FB71647D37F4FD"></a>

<h5>9.4-2 S_6 → U_4[ 30,31 28, 29 30,31,[30,31,30 ,28,29]

The simple group G = U_4(2) contains a maximal subgroup U of type <spanpre</div

<div class="example"><pre>
gap> s:= CharacterTable( "S6" );
CharacterTable( "A6.2_1" )
gap> t:= CharacterTable( "U4(2)" );
CharacterTable("4(2) java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
gap> sfust:= PossibleClassFusions( s, t );
[ [ 1, 3, 6, 7, 9, 10, 3, 2, 9, 16, 15 ],
 [ 1, 3, 7, 6, 9, 10, 2, 3, 9, 15, 16 ] ]
gap> Length( RepresentativesFusions( s, sfust[ 65,70 71 ] [65, 70 71 ] [65 71,70 ] [ 65 71 java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
1
<pre<div>

In the double cover 2.G of <pre>
<span="GAPprompt">gap> 2t:= CharacterTable( "2.U4(2)" );
CharacterTable( "2.U4(2)" )
gap> 2s:= CharacterTable( "2.A6.2_1" );
CharacterTable( "2.A6.2_1" )
gap> 2sfus2t:= PossibleClassFusions( 2s, 2t );
[ [ 1, 2, 4, 11, 12, 9, 10, 15, 16, 17, 3, 4, 15, 24, 25, 26, 26 ],
 [ 1, 2, 4, 11, 12, 9, 10, 15, 16, 17, 3, 4, 15, 25, 24, 26, 26 ] ]
gap> Length( RepresentativesFusions
1
</pre>

GetFusionMap( s, t ) in;

<div class="example"><pre>
gap> <java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 12
[ 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 11 ]
2tfust= ( 2tt );;
gap> induced:= Set( 2sfus2t, x -> CompositionMaps( 2tfust
> 9..</div>

Thep> to the[CCN+85, p 115]</>) the group G = U_6(2) contains a maximal subgroup U of the type L_3(4).2_1. The class fusion of U into G is ambiguous.

<div="example"><pre>
gap> auts:= AutomorphismsOfTable( s );
Group,)78)10,11)]
gap> other:= gap> t:= CharacterTab
1,1, 2,4 4,3,3 5,6 6,8,7, 9 11 ,10 10 ]
gap>gap> Length( sfust );
>java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
[ [1,3, 6 7,9, 10, 3,2 9,16,15] ]
</pre></div>

The library table of U</span</re</div>

<div class="example"><pre>
<spanclass="GAPprompt">gap> GetFusionMap( s, t ) in induced;
true
</pre></

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4>9.5 Fusions Determined Using Commutative Diagrams Involving
Automorphic Extensions</h4>

<a id="X7E91F8707BA93081" name="X7E91F8707BA93081"></a>

<h5.- U_3(8).3_1 → ^2E_6(2) (December 2010)</h5>

According to the Atlas class=GAPprompt>/ ( s3fust3/>

<div class classjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 121
gap> s:= CharacterTable( "U3(8).3_1" );;</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
gap> sfust:= PossibleClassFusions( s, t );;
gap> Length( sfust );
24
gap> Length( RepresentativesFusions( s, sfust, t ) );
2
</pre></div>

In the automorphic extension G.2 = ^2E_6(2).2 of G, the subgroup U extends to a group U.2 of the type U_3(8).6 (again, see <a href="chapBib.html#biBCCN85">[CCN+85, p. 191]</a>). The class fusion of U.2 into G.2 is unique up to table automorphisms.

<div class="example"><pre>
gap> s2:= CharacterTable( "U3(8).6" );;
gap> t2:= CharacterTable( "2E6(2).2" );;
gap> s2fust2:= PossibleClassFusions( s2, t2 );;
gap> Length( s2fust2 );
2
gap> Length( RepresentativesFusions( s2, s2fust2, t2 ) );
1
</pre></div>

Only half of the possible class fusions from U into G are compatible with the embeddings of U into G.2 via U.2 and G, and the compatible maps form one orbit under table automorphisms.

<div class="example"><pre>
gap> sfuss2:= PossibleClassFusions( s, s2 );;
gap> comp:= SetOfComposedClassFusions( s2fust2, sfuss2 );;
gap> tfust2:= PossibleClassFusions( t, t2 );;
gap> filt:= Filtered( sfust, map -> ForAny( tfust2,
> map2 -> CompositionMaps( map2, map ) in comp ) );;
gap> Length( filt );
12
gap> Length( RepresentativesFusions( s, filt, t ) );
1
</pre></div>

Let us see which classes of U and G are involved in the disambiguation of the class fusion. The <q>good</q> fusion candidates differ from the excluded ones on the classes at the positions 31 to 36: Under all possible class fusions, two pairs of classes are mapped to the classes 81 and 82 of G; from these classes, the excluded maps fuse classes at odd positions with classes at even positions, whereas the <q>good</q> class fusions do not have this property.

<java.lang.StringIndexOutOfBoundsException: Range [11, 4) out of bounds for length 26
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 74, 74, 81, 82, 81, 82 ], [ 74, 74, 82, 81, 82, 81 ],
[,82, 74,74 81,82 ], [81, 82 81,82 74, 74 ]
 [ 82, 81, 74, 74, 82, 81 ], [ 82, 81, 82, 81, 74, 74 ] ]
<span";>"">Set( sfust ), x; x 31 . 36 ]})<spanjava.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129
[[74,74,81 82 , 81] [ 74 74,82 ,81 82 ]
 [ 81, 82, 74, 74, 82, 81 ], [ 81, 82, 82, 81, 74, 74 ],
 ,74 74,81 82], [82 81,81,82 74,74]]
</pre></div>

None of the possible class fusions from U to > map2 -> CompositionMaps( map2 ) in comp ));Consider the elements order four in ' L_3()/span>, which fuse to one class of U.3.
[ [ 28, 29, 30, 31, 30, 31 ], [ 29, 28, 31, 30, 31, 30 ],
 [ 30, 31, 28, 29, 30, 31 ], [ 30, 31, 30, 31, 28, 29 ],
[ 30 29,28,31 30 ] 31,30, 31 30,29 28] ]
</pre></div>

This suffices to exclude the <q>bad</q> fusion candidates because no further fusion of the relevant classes of G happens in G.2.

<div class="example"><pre>
gap> List( tfust2, x -> x{ [ 74, 81, 82 ] } );
[ [ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ],
 [ 65,70, 71 ] [65 70,71 , [ 65,71 70 ] 65,70 ]
 [ 65, 70, 71 ], [ 65, 70, 71 ], [ 65, 71, 70 ], [ 65, 71, 70 ] ]
</preclass"xample"<pre

<pThe holds for the of relevant of U.2 in G.2.)

<div class="example"><pre>
nput> x;x 28.31 };/
[ [ 65, 65, 70, 71 ], [ 65, 65, 71, 70 ] ]
/>

Finally, we check that a correct map is stored on the library table.

gap> GetFusionMap( s, t ) in filt;
true
</pre></div>

<a id="X81B37EF378E89E00" name="X81B37EF378E89E00"></a>

<h5>9.5-2 L_3(4).2_1 class>> <( gt{ .12 ;/

<>,p ]a, group spanclass=SimpleMath"G = U_6(2</pan> containsa subgroup U of the type L_3(4).2_1. The class fusion of U into G is ambiguous.

<div class="example"><pre
gap>div="example"><pre
<panjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 100
spanGAPprompt><=": ( s );;<>
<span="">gap;filt( ,mapgtIsSubset(map,[ 10.));/pan>
27
<pan class>/span> <span="GAPinput">Length( RepresentativesFusions( s, sfust, t ) );
3
<pre></div>

In the automorphic1 3,7, 11 10,,15, ,4,14,23 ,26 28 ],java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

""java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
G"gt/java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 103
gap> t3:= CharacterTable( "U6(2).3" );;
=PossibleClassFusions(s3 );;
gap> Length( s3fust3 );
2
gap> Length
1
</pre></div>

Here the argument used in Section class"xample><pre>>

<div class="example"><div
gap> <a id="X85E2A6F480026C95" name="X85E2A6F480026C95>
gap>9.6<="eading"Conditions h4
/> <span="">sfust( -&tForAny ,> map2 -> CompositionMaps
true
</pre></div>

<Consider elementsof order four in <pre>
gap> <spangap> sfuss3;
[ [ 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 10, 10 ] ]
</pre></div>

These classes of <span="GAPprompt">gap;sCharacterTable( "216).>

<div class="example"><pre>
<; map{ [ 4 .. 6]} )</pan
[ [ 10, 10, 10 ], [ 10, 10, 11 ], [ 10, 10, 12 ], [ 10, 11, 10 ],
 [ 10, 11, 11 ], [ 10, 11, 12 ], [ 10, 12, 10 ], [ 10, 12, 11 ],
 [ 10, 12, 12 ], [ 11, 10, 10 ], [ 11, 10, 11 ], [ 11, 10, 12 ],
 [ 11, 11, 10 ], [ 11, 11, 11 ], [ 11, 11, 12 ], [ 11, 12, 10 ],
 [ 11, 12, 11 ], [ 11, 12, 12 ], [ 12, 10, 10 ], [ 12, 10, 11 ],
 [ 12, 10, 12 ], [ 12, 11, 10 ], [ 12, 11, 11 ], [ 12, 11, 12 ],
 [ 12, 12, 10 ], [ 12, 12, 11 ], [ 12, 12, 12 ] ]
gap> Set( tfust3, map -> map{ [ 10 .. 12 ] } );
[ [ 10, 10, 10 ] ]
<pre

This means that the automorphism can exclude two possible fusions by the fact that their images all lie inside the normal subgroup J_3, but J_3 does not contain a subgroup of type L_2(16).4; so still one orbit of length two remains.

<div classspan="GAPprompt">gapgt;> <span="GAPinput">PossibleClassFusions( s, j3 );
gap> Filtered( sfust, map]
[ ]
<classjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 135
[ [ 1, 3, 7, 10, 11, 12, 15, 24, 4, 14, 23, 26, 27, 28 ],
 [ 1, 3, 7, 10, 12, 11, 15, 24, 4, 14, 23, 26, 28, 27 ],
 [ 1, 3, 7, 11, 10, 12, 15, 24, 4, 14, 23, 27, 26, 28 ],
 [ 1, 3, 7, 11, 12, 10, 15, 24, 4, 14, 23, 27, 28, 26 ],
 [ 1, 3, 7, 12, 10, 11, 15, 24, 4, 14, 23, 28, 26, 27 ],
 [ 1, 3, 7, 12, 11, 10, 15, 24, 4, 14, 23, 28, 27, 26 ] ]
gap> Length( RepresentativesFusions( s, filt, [1,2 3,6 14, 16,15 ,5 ,12 ,19,22,23 23 java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
1
</pre></div>

Finally, we check that a correct map is stored on the library table.

<div class="example"><pre> class= mod;/
gap> := t mod 2;;

</pre></div>

<a id="X85E2A6F480026C95" name="X85E2A6F480026C95"></a>

<h4>9.6 Conditions Imposed by Brauer Tables</h4>

The 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

<a id="X7ACC7F588213D5D5" name="X7ACC7F588213D5D5"></a>> , map>

<h5>9.6-1 L_2(16).4 → J_3.2 (January 2004)</h5>

It can happen that Brauer tables decide ambiguities of <pre>/iv

<div class="
gap> s:= CharacterTable( "L2(16).4" );;
<panclass=GAPprompt"gap&t;/span><="APinput>: CharacterTable("3.");/>
gap> fus:= PossibleClassFusions( s, t );
[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ],
 [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ],
 [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ],
 [<div class=e>pre>
gap> RepresentativesFusions( s, fus, t );
[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 5, 5, 8, 8, 13, 13 ],
 [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
</pre></div>

Using Brauer tables, we will see that just one fusion is admissible.

We can exclude two possible fusions by the fact that their images all lie inside the normal subgroup J_3, but J_3 does not contain a subgroup of type L_2(16).4; so still one orbit of length two remains.

<div class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
gap> j3:= CharacterTable( "J3" );;
gap> PossibleClassFusions
[ ]
gap> GetFusionMap( j3, t );
[ 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16,
 17, 17 ]
gap> filt:= Filtered( fus,
> x -> not IsSubset( ClassPositionsOfDerivedSubgroup( t ), x ) );
[ [ 1, 2, 3, 6, 14, 15, 16, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ],
 [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
</pre></div>

Now the remaining wrong fusion java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0

<div class="example"><pre>
gap> &;/>< ="tmod2=t 2;/pan>
gap>
 span class=G">or map filt do
> modmap:= CompositionMaps( InverseMap( GetFusionMap( tmod2, t ) ),
<panclass="APprompt"&;/ class""s ( "() <>
span="APinput"t: CharacterTable("3 ;;/span>
< =GAPprompt&;class"APinput" ( ) -gt;{modmap}) n )</pan
> AddSet( java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 49
> fi;
t">/> <pan class=">od</>
gap> admissible ,2 ,6, ,11 ,10 ,14 ,20]
[ [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
</pre></div>

The [1 2, 4,7 ,11 ,10 ,1320, ,

<div class="example"><pre>
gap> CTblLib.Test.Decompositions( s, fus, t ) = admissible;
true
</pre></div>

<</><div>

<div class="example"><pre>
gap> GetFusionMap( s, t ) in admissible;
true
</pre></div>

Remark:

In May 2015, the 19-modular java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 34

<a id="X7ACB86CB82ED49D1" name="X7ACB86CB82ED49D1"></a>

h596-< class=Heading>span ="SimpleMath">L_2(17 →S_82 (uly)/span><h5

The class E17)3+17^+()6E()7E)^10+E(17)11E)^java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

<div class=<>oreover, the subgroups those table automorphisms fusionsinvariant orbits thisthis. Indeed,the two orbits belong to essentially decompositions therestriction < class">_3./>
gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> t:= CharacterTable( "S8(2)" );;
gap> mfust:= PossibleClassFusions( m, t );;</span[1 2 4,6,7,,10 11 13 ,20, ]]
gap> Length( RepresentativesFusions( m, mfust, t ) )<class"APprompt"&t</pan> InverseMap(GetFusionMap(tmod19 t))</pan>
4
</pre></div>

The Brauer tables for M</java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 41

<classexample"<>
span ="APprompt">&;/> span class=GAPinput": Decomposition(Irr smod19 )rest nonnegative")
< =GAPprompt>&; classjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 109
1
gap> GetFusionMap( m, t ) in repr;
true
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<a id="X7DED4C437D479226" name="X7DED4C437D479226"></a>

<h5>9.6-3 <span<divclass=example"<re>

It can happen that Brauer tables impose conditions such that ambiguities arise which are not visible if one considers only ordinary tables.<

<> classfusion the ordinary character tables of <pan=SimpleMathL_2(9)andspan ="impleMath> unique up to table automorphisms./>

<div class="standardization=1,subgroupname :)"version="")
span >ens: OneAtlasGeneratingSetInfo(J3 Characteristic,19,/span
<spanclass=">g;<> span =GAPinput"t:CharacterTable "3 ;</pan>
gap> &t;/ <pan class=">:AtlasGeneratorsgens;/>
[ [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 20, 21 ],
 [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 21, 20 ],
 [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ],
 [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 21, 20 ],
 [ 1, 2, 4, 6, 7, 12, 10, J1,19 ] repname : J3G1, repnr : 35,
 [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 21, 20 ],
 ,2,4,7,6, 10 , 12,14 20,21 ]
 [ 1, 2, 4, 7, 6, 10, 11, 12, 14, 13, 21, 20 ],
 [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 20, 21 ],
 [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 21, 20 ],
 [ 2,4,7,6 12,10 ,14 ,20 21 ]
 [ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 21, 20 ] ]
gap> 
[ [ 1[ 5 7 9 11,13, 15 19]
</>/iv

<ppre><div>

<div class="example"><pre>
gap> tmod19:= t mod 19;
BrauerTable( "J3", 19 )
gap> deg110:= Filtered( Irr( tmod19 ), phi -> phi[1] = 110 );
[ Character( BrauerTable( "J3", 19 ),
 [ 110, -2, 5, 2, 2, 0, 0, 1, 0,
 -2*E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6-2*E(9)^7,
 E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6+E(9)^7,
 +E9)3++*E(9)4+2*E(9)^(9)7, -2, -2,-,0,java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
 0, E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12
 +E(17)^14,
 E(17)+E(17)^2+E(17)^4+E(17)^8+E(17)^9+E(17)^13+E(17)^15+E(17)^16
 ] ) ]
gap> 9A:= Position( OrdersClassRepresentatives( tmod19 ), 9 );
10
gap>>emark:</gap> AtlasIrrationality( "-1+2y9+&4" ) = deg110
true
</pre></div>

It turns out that four of the twelve possible class fusions are not compatible with the 19-modular tables.

<div class="example"><pre>
gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
BrauerTable( "L2(19)", 19 )
gap> compatible:= [];;
gap> for map in sfust do
> comp:= CompositionMaps( InverseMap( GetFusionMap( tmod19, t ) ),
> CompositionMaps( map, GetFusionMap( smod19, s ) ) );
< class="APprompt">>/span> rest: List(Irr(tmod19 ), phi -> phi{ comp } );
> if not fail in Decomposition( Irr( smod19 ), rest, "nonnegative" ) then
> > fi;
> od;
gap> compatible;
[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ],
 [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 21, 20 ],
 [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ],
 [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 21, 20 ],
 [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 20, 21 ],
 [ 1, 2, 4, 7, 6, 11, 12, 10, 14, 13, 21, 20 ],
 [ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 20, 21 ],
 [ 1, 2, 4, 7, 6, 12, 10, 11, 14, 13, 21, 20 ] ]
</pre></div>

Moreover, the subgroups java.lang.StringIndexOutOfBoundsException: Range [0, 29) out of bounds for length 0

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 1, 2, 4, 6, 7, 11, 12, 10,<iv ="xample"><re
 [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14, 20, 21 ] ]
gap> compatiblemod19:= List( reps, map -> CompositionMaps(
> InverseMap( GetFusionMap( tmod19, t ) ),
> CompositionMaps( map, GetFusionMap( smod19, s ) ) ) );
[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14 ],
 [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14 ] ]
gap> rest:= List( compatiblemod19, map -> Irr( tmod19 )[3]{ map } );;
gap> dec:= Decomposition( Irr( smod19 ), rest, "nonnegative" );
[ [ 0, 0, 1, 2, 1, 2, 2, 1, 0, "6K", 7A "B, "A","B,C,"A" 9,"9C","9D", "E,java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
gap> List( Irr( smod19 ), phi -> phi[1] );
[,3 5,7 9,11,13 15,17 19
</pre></div>

In order to decide which class fusion is correct, we take the matrix representation of J_3 that affords 
gap> LoadPackage( "atlasrep", false );
true
gap> prog:= AtlasProgram( "J3", "maxes", 2 );
rec( groupname := "J3", identifier := [ "J3", "J3G1-max2W1", 1 ],
 program := <straight line program>, size := 3420,
 OrdersClassRepresentatives( s );
gap> gens:= OneAtlasGeneratingSetInfo( "J3", Characteristic, 19,
> Dimension, 110 );;
gap> gens:= AtlasGenerators( gens );<pan class="GAPprompt">gap> sfust:= List( [ "1A", "2B", "3D", "3E", "4C", "4C", "6J", "7B", "8C",
rec( charactername := "110a", constituents := [ 3 ],
 contents := "core", dim := 110,
 enerators : [ lt immutablecompressed matrix 110110 overGF(19) &t;java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
 < immutable compressed matrix 110x110 over GF(19) > ],
 groupname := "J3", id := "",
 identifier := [ "J3", [ "J3G1-f19r110B0.m1", "J3G1-f19r110B0.m2" ],
 1, 19 ], repname := "J3G1-f19r110B0", repnr := 35,
 ring := GF(19), size := 50232960, standardization := 1,
 type := "matff")
gap> restgens:= ResultOfStraightLineProgram( prog.program, gens.generators );
[ < immutable compressed matrix 110x110 over GF(19) >,
 < immutable compressed [ 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 28 ,12]
gap> module:= GModuleByMats( restgens, GF( 19 ) );;
gap> facts:= SMTX.CollectedFactors( module );;
gap> Length( facts );
7
gap> List( facts, x -> x[1].dimension );
[ 5, 7, 9, 11, 13, 15, 19 ]
gap> List( facts, x -> x[2] );
[ 1, 2, 1, 2, 2, 1, 1 ]
</pre></div>

This means that there are seven pairwise nonisomorphic composition factors, the smallest one of dimension five.

<div class="example"><pre>
gap> GetFusionMap( s, t ) = java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 0
true
<div=<>

<<>Remark:em/>

In May 2015, the 19<a id="X8225D9FA80A7D20F"[ 12096 192 216 18 32,24,7 ,12 48,48 6,8,12 12 java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

<h4>9.7 Fusions Determined by Information about the Groups</h4>

In 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 additionally9 15 ,10

The group representations are taken from the Atlas of Group Representations <a p>Candidates for M are those subgroups of G that contain"3</< "></> < "ode><java.lang.StringIndexOutOfBoundsException: Range [283, 280) out of bounds for length 322

<div class="example"><pre>
gap> LoadPackage( "atlasrep", false );
true
</pre></div>

<a id="X7AE2962E82B4C814"span class="GAPprompt">gap> for name in Maxes( t ) do

<h5>9.7-1 U_3(3).2 → Fi_24span class=G"&;/span>< class=>mfust:= GetFusionMap(m t ) 

aximal subgroup<class="impleMath"H/pan of type <""U_332</spanand <panclass=SimpleMath">H,one a lot ofdifferent possibilities (ndadditionally this takes quite some time). We use the description of H as the normalizer in G of a U_3(3) type subgroup containing elements in the classes <code class="code">2B</code>, <code class="code">3D</code>, <code class="code">3E</code>, <code class="code">4C</code>, <code class="code">4C</code>, <code class="code">6J</code>, <code class="code">7B</code>, <code class="code">8C</code>, and <code class="code">12M</code> (see <a hrefspan ="GAPprompt>g;/> classGAPinput"fi;/>

<div class="example"><pre>
gap> t:= CharacterTable( "F3+" );
CharacterTable( "
<div=example>prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
CharacterTable( "U3(3).2"[1 ,3,7 8,11,9 ,23 ,28 ,43 50 ]
gap> <span class="[ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50
[ "1A", "2A", "2B", "3A", "3B", "3C", "3D", "3E", "4A", "4B", "4C",
 "5A", "6A", "6B", "6C", "6D", "6E", "6F", "6G", "6H", "6I", "6J",
 "6K", "7A", "7B", "8A", "8B", "8C", "9A", "9B", "9C", "9D", "9E",
 "9F", "10A", "10B", "11A", "12A", "12B", "12C", "12D", "12E",
 12" "G "H,"I,"12J","12K"12,"12" "3A,java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
 "14A", "14B", "15A", "15[ 1, 3 3 7, 811,11 22 , 28 50 50, ]]
 "18C", "18D", "18E", "18F", "18G", "18H", "20A", "20B", "21A",
 "21B", "21C", "21D", "22A", "23A", "23B", "24A", "24B", "24C",
 "24D", "24E", "24F", "24G", "26A", "27A", "27B", "27C", "28A",
 "29A", "29B", "30A", "30B", "33A", "33B", "35A", "36A", "36B",
 "36C", "36D", "39A", "39B", "39C", "39D", "42A", "42B", "42C",
 "45A", "45B", "60A" ]

[ 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 2, 4, 6, 8, 12, 12 ]
gap> sfust:= List( [ "1A", "2B", "3D", "3E", "4C", "4C", "6J", "7B", "8C",
> "12M" ], x -> Position( tnames, x ) );
[ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50 ]
gap> sfust:= PossibleClassFusions( s, t, rec( fusionmap:= sfust ) );
[ [ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50, 3, 9, 23, 28, 43, 43 ],
 [ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50, 3, 11, 23, 28, 50, 50 ] ]
span">&tOrdersClassRepresentatives;/pan
[ 1, 2, 3, 3, 4, 4, 6, 7, 8, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>

So we still

Our idea is to take a subgroup U of H that contains such<pan class="APprompt"gap&t<java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 98

< <spanclass"impleMath"U ⟨g ⟩)</>where<panclass"SimpleMath"g/span java.lang.StringIndexOutOfBoundsException: Range [112, 111) out of bounds for length 307

<div class="example"><pre>

java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
gap> SizesCentralizers( s );
216 ,,32,24,7 ,12 48,48,6,8,12 java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
gap> u:= CharacterTable( Maxes( s )[2] );;
gap> ufuss:= GetFusionMap( u, s );
[ 1, 2, 11, 3, 4, 5, 12, 7, 13, 9, 9, 15, 16, 10 ]
</pre></div>

p> for<pan ="SimpleMath">M are those subgroups of G that contain elements in the class <code class="code">3D</code> of G whose centralizer is the full <code class="code">3D</code> centralizer in G.

<div class="example"><pre>
gap> 3Dcentralizer:= SizesCentralizers( t )[7];
153055008
gap> cand:= [];; 
gap> for name in Maxes( t ) do
> m:= CharacterTable( name );
> mfust:= GetFusionMap( m, t ); 
> if ForAny( [ 1 .. Length( mfust ) ], 
> i -> mfust[i] = 7 and SizesCentralizers( <div ejava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
> then
> Addspan ="APprompt>ap>/span> span class="GAPinput">SizesCentralizers( t )[ pos ];
> fi;
> od;
gap> <spanspan ="GAPprompt>gap&;<spanclass="APinput ,pos)</pan>
[ CharacterTable( "3^7.O7(3)" ),
 CharacterTable( "3^2.3^4.3^8.(A5x2A4).2" ) ]
</pre></div>

< span=">/java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 396

<div class="example"><pre>
gap> possufust:= List( sfust, x -> CompositionMaps( x, ufuss ) );
[ [ 1, 3, 3, 7, 8, 11, 9, 22, 23, 28, 28, 43, 43, 50 ],
 [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ]
gap> m:= cand[1];;
 ClassOrbit m, pos )=[pos] then/>
gap> < class="GAPinput"> Add( cand, m );
242
gap> comp:= List( ufusm, x -> CompositionMaps( GetFusionMap( m, t ), x ) );;
gap> Intersection( possufust, java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 74
[ [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ]
gap> m:= cand[2];;
gap> ufusm:= PossibleClassFusions( u, m );;
gap> Length( ufusm )java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
256
gap> comp:= List( ufusm, x -> CompositionMaps( GetFusionMap( m, t ), x ) );; 
</div>

Finally, we check that the correct fusion is stored in the GAP Character Table Library.

<div class="example"><pre>
<panclass=GAPprompt>gapgt<span<class""GetFusionMap(s t)= sfust2]</div>

<a id="X83061094871EE241" name="X83061094871EE241"></a>

<h5>9.7-2 L_2(13).2 → Fi_24^' (September 2002)</h5>

<java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<div class="example"><pre>
gap> t:= CharacterTable( "F3+" );;
gap> <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> fusdivclass="xample"><pre>
gapspan class="GAPprompt">gap> GetFusionMap( u, t ) = filt[2];
gap> Length/pre>/div>
3
</pre></div>

In <a href="chapBib.html#biBLW91">[LW91, p. 155]</a>, it is stated that U^'>9.- < class"Heading"<class=SimpleMath"M_11 →(pril 2009)</h5>

<div class="example"><pre>
gap> filtjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 1, 3, 7, 22, 25, 25, 25, 51, 3, 9, 43, 43, 53, 53, 53 ],
 [ 1, 3, 7, 22, 25, 25, 25, 51, 3, 11, 50, 50, 53, 53, 53 ] ]
gap> ClassNames( t ){ [ 43, 50 ] };
[1
</pre></div>

<<p1A, 2" 2"] "3A" 3B ]

The order 12 elements in question lie inside subgroups of type 13 : 12 in U["6C","E","6H" 6I "J" ]

<div class="example"><pre>
gap> pos:= Position( OrdersClassRepresentatives>Accordingto a =".html#iBWil93a>[il93a,Thm.121<a>,M contains no <code class="code">5A</code> elements of B. By the proof of <a href="chapBib.html#biBWil99">[Wil99, Prop. 4.1]</a>, the involutions in any S_5 type subgroup U java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
51
gap> gap> ClassOrbit( t, pos );
[ 51 ]
<=GAPprompt>ap>gap> for name in Maxes( t ) do
> m:= CharacterTable( name );
> pos:= Position( OrdersClassRepresentatives( m ), 13 );
> if pos <> fail and 
> SizesCentralizers( m )[ pos ] = 234 
> and ClassOrbit( m, pos ) = [ pos ] then
=GAPprompt>>/java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 88
> fi;
> od;
gap> cand;
[ CharacterTable( "(3^2:2xG2(3)).2" ) ]
gap> s:= cand[1];;
gap> sfust:= PossibleClassFusions( s, t );;
</pre></div>

As no 13:12 type subgroup is contained in the derived subgroup of (3^2:2 ×[[ 1,5,7,17,19,29,45,45,54 ]

<div class="example"><pre>
gap> der:= ClassPositionsOfDerivedSubgroup( s );;
(Using the information the _10/span> type subgroups of /span are alsocontainedin span class"SimpleMath">typesubgroups wouldnotjava.lang.StringIndexOutOfBoundsException: Range [211, 210) out of bounds for length 359
gap> sfust:= PossibleClassFusions( s, t );;
gap> imgs:= Set( Flat( List( sfust, x -> x{ outer } ) ) );

 83, 87, 98 ]
gap> t.12f in imgs;
false
gap> t.12m in imgs;
true
</pre></div>

So L_2(13).2 ∖ L_2(13) does not contain <code class="code">12F</code> elements of G, i. e., we have determined the class fusion of U in G.

Finally, we check whether the correct16

<div class="example"><pre>
< ="GAPprompt">&t>
true
</pre></div>

<a id="X7E9C203C7C4D709D" name="X7E9C203C7C4D709D"></a>

<h5>9.7-3 M_11 → B (April 2009)</h5>

The sporadic simple group B contains a maximal subgroup M of the type M_11 12"12S

<class"example">pre
gap> b:= CharacterTable( "B" );;
gap> m11:= CharacterTable( "M11" );;
gap> m11fusb:= PossibleClassFusions( m11, b );;
gap> Length( m11fusb );
31
gap> CompositionMaps( ClassNames( b, "ATLAS" ), Parametrized( m11fusb ) );
[ "1A", [ "2B", "2D" ], [ "3A", <pan=GAPprompt"gap&<spanclass=GAPinput">m:= CharacterTable( "M11" );;
 [span ="GAPprompt"gap&t: CharacterTable("L2(11)");</
 ["C,"6" "H,"6I "J"]
 [ "8B", "8E", "8G", "8J", "8K", "8L", "8M", "8N" ],
 [ "8B", "8E", "8G", "8J", "8K", "8L", "8M", "8N" ], "11A", "11A" ]
</pre></div>

According to <a href="chapBib.html#biBWil93a">[Wil93a, Thm. 12.1]</a>, M contains no <code class="code">5A</code> elements of B. By the proof of <a href="chapBib.html#biBWil99">[Wil99, Prop. 4.1]</a>, ,, ,7 , ,,5 , ]

<> weusethe embeddingof<span="SimpleMath"UB/ span class=SimpleMath"M/span>and<panclass=SimpleMath>Thspanfordeterminingclassfusionof<spanclass=SimpleMath"></> <class=SimpleMath">B. fusion the embedding ofU via Th is uniquely determined.

<div class="example"><pre>
gap> th:= CharacterTable( "Th" );;
gap> s5:= CharacterTable( "S5" );;
<h5.-<pan ="<java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 99
[ [ 1, 2, 4, 8, 2, 7, 11 ] ]
<span="gap>/span> span class="GAPinputthfusb: PossibleClassFusions( th, b );;
gap> s5fusb:= Set( thfusb, x -> CompositionMaps( x, s5fusth[1] ) );
[ [ 1, 5, 7, 19, 5, 17, 29 ] ]
</pre></div>

<<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 100

<div class="example"><pre>
gap> s5fusm11:= PossibleClassFusions( s5, m11 );<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
[ [ 1, 2, 3, 5, 2, 4, 6 ] ]
gap> m11fusbp> < href"chapBib.html#iBWil99">[Wil99, Section 9]</a>, T contains a subgroup U of the type 3^2:2S_4 that is contained also in a maximal subgroup M of the type 3^2.3^3.3^6.(S_4 × 2S_4). So we
ionMaps( map, s5fusm111 =s5fusb[1]);/panjava.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143
[ [ 1, 5, 7, 17, 19, 29, 45, 45, 54, 54 ] ]
gap> CompositionMaps( ClassNames( b, "ATLAS" ), m11fusb[1] );
[ "1A", "2D", "3B", "4J", "5B", "6J", "8N", "8N", "11A", "11A" ]
</pre></div>

(Using the information that the M_10 type subgroups of M are also contained in Th type java.lang.StringIndexOutOfBoundsException: Index 193 out of bounds for length 132

<a id=span class="GAPprompt">> map2 -> CompositionMaps( map, map2 ) in ufusb ) );;

<h5>9.7-4 L_2(11):2 →[[ 1 5, 6,7 12,27, 41 41,75 75,75, 75]

< 1,,7, 7 12 28,41,41 75,75,75 75 ,

<div class="example"><pre>
gap> b:= CharacterTable( "B" );;
gap> l:= CharacterTable( "L2(11).2" );;
gap> lfusb:= PossibleClassFusions( l, b );;
gap> Length( lfusb );
16
<span class="java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
[ "1A", [ "2B", "2D" ], [ "3A", "3B" ], [ "5A", "5B" ],
 [ "5A", "5B" ], [ "6C", "6H", "6I", "6J" ], "11A", [ "2C", "2D" ],
 [ "4D", "4E", "4F", "4G", "4H", "4J" ], [ "10C", "10E",/pre</iv>
 [ "10C", "10E", "10F" ],
 [ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12We check that this map is stored on the library table.
 "12R", "12S" ],
 [ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12P", "12Q",
 "12R", "12S" ] ]
</pre></div>

According to <a href="chapBib.html#biBWil93a">[Wil93a, Thm. 12.1]</a>, L contains no <code class=java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 12

In the same way as we proceeded in Section <a href="chap9.html#X7E9C203C7C4D709D">9.7-3</a>,172→</(arch2004)/</5

<div class="example"><pre>
gap> m:= CharacterTable( "M11" );;
gap> u:= CharacterTable( "L2(11)" );;
gap></java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 96
gap> mfusb:= GetFusionMap( m, b );;
gap> ufusb:= Set( ufusm, x -> CompositionMaps( mfusb, x ) );
[ [ 1, 5, 7, 19, 19, 29, 54, 54 ] ]
gap> ufusl:= PossibleClassFusions( u, l );
[ [ 1, 2, 3, 4, 5, 6, 7, 7 ], [ 1, 2, 3, 5, 4, 6, 7, 7 ] ]
gap> lfusb:= Filtered( lfusb, 
> map2 -> ForAny( ufusl, 
> map1 -> CompositionMaps( map2, map1 ) in ufusb ) );
[ [ 1, 5, 7, 19, 19, 29, 54, 5
</>

<a id="X828D81487F57D612" name="X828D81487F57D612"></a>

<h5>9.7-5 L_3(3) → B (April 2009)</h5>

<sporadic B/>contains subgroup Tspan> the <spanclass""L_33)whose fusionjava.lang.StringIndexOutOfBoundsException: Range [198, 197) out of bounds for length 212

<div class="example"><pre>
gap> b:= CharacterTable( "B" );;
gap> t:= CharacterTable( "L3(3)" );;
gap> tfusb:= PossibleClassFusions( t, b );;
gap> Length( tfusb );
36
</pre></div>

<pre>
gap> m:= CharacterTable( "3^2.3^3.3^6.(S4x2S4)" );;
gap> g:= PSL(3,3);;
<pan"GAPprompt>g;</>< =GAPinput">x=MaximalSubgroupClassReps ;gap> u:= First( mx, x -> Size( x ) = 432 );;
<pan class=GAPprompt"gt;/span>u:= CharacterTable( u );;
gap> ufusm:= PossibleClassFusions( u, m );;
gap> ufust:= PossibleClassFusions( u, t );;
gap> mfusb:= GetFusionMap( m, b );;
gap> ufusb:= Set( ufusm, map -> CompositionMaps( mfusb, map ) );;
gap> tfusb:= Filtered( tfusb, map -> ForAny( ufust,
> map2 -> CompositionMaps( map, map2 ) in ufusb ) );;
gap> tfusb;
[ [ 1, 5, 6, 7, 12, 27, 41, 41, <pan class"APprompt"gapgt</><"APinput">;/>
 [ 1, 5, 7, 6, 12, 28, 41, 41, 75, 75, 75, 75 ],
 [ 1, 5, 7, 7, 12, 28, 41, 41, 75, 75, 75, 75 ],
 [ 1, 5, 7, 7, 12, 29, 41, 41, 75, 75, 75, 75 ],
 [ 1, 5, 7, 7, 17, 29, 45, 45, 75, 75, 75, 75 ] ]
</pre></div>

< wethatspan java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 282

<div class="example"><pre>
gap> ClassNames( b, "ATLAS" ){ [ 12, 17 ] };
[ 4E,"J ]
</pre></div>

We check that this map is stored on the library table.

<div class="example"><pre>
gap> GetFusionMap( t, b ) = tfusb[5];
true
</pre></div>

<a id="X7B4E13337D66020F" name="X7B4E13337D66020F"></a>

<>976 span="eading>span =SimpleMath"L_2)2→

The sporadic simple group B contains a maximal subgroup U of the type > x -> List( x, y -> y^7 ) );

<div class="example"><pre>
gap> b:= CharacterTable( "B" );;
gap> u:= CharacterTable( "L2(17).2" );;
gap> ufusb:= PossibleClassFusions( u, b );
[ [ 1, 5, 7, 15, 42, 42, 47, 47, 47, 91, 4, 30, 89, 89, 89, 89, 97,
 97, 97 ],
 [ 1, 5, 7, 15, 44, 44, 46, 46, 46, 91, 5, 29, 90, 90, 90, 90, 96,
 96, 96 ],
 , 90, 90, 90, 95
 95, 95 ] ]
</pre></div>

According to <a href="chapBib.html#biBWil99">[Wil99, Prop. 11.1]</a>, U contains elements in the classes <code class="code">8M</code> and <code class="code">9A</> < ="SimpleMathB via the <class="7> in< =SimpleMath"B/>/p

<div class="example"><pre>
gap> names:= ClassNames( b, "ATLAS" );;
gap> pos:= List( [ "8M", "9A" ], x -> Position( names, x ) );
[ 44, 46 ]
gap> ufusb:= Filtered( ufusb, map -> IsSubset( map, pos ) );
[[1,5,7 15,44,44 ,46 ,91,,29,90,90 90 ,java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
 ,96 ]]
</pre></div>

We check that this

<div class="example"><pre>
gap> GetFusionMap( u, b ) = ufusb[1];
true
</pre></div>

<a id="X8528432A84851F7B" name="X8528432A84851F7B"></a>

<h5>9.7-7 L_2(49).2_3 =>> class=GAPinput( , {[]}<

The sporadic simple group B contains a class of maximal subgroups of the type L_2(49java.lang.StringIndexOutOfBoundsException: Index 144 out of bounds for length 30

<div class="example"><pre>
gap> u:= CharacterTable( "L2(49).2_3" );;
gap> b:= CharacterTable( "B" );;
gap> ufusb:= PossibleClassFusions( u, b );;
<classGgt<>spanclass"">(( ,,b))<spanjava.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
2
gap> ufusb;
[ [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128,
 128, 128, 15, 71, 71, 89, 89, 89, 89 ],
128,
 128, 128, 17, 72, 72, 89, 89, 89, 89 ] ]
</pre></div>

We show that the fusion is determined by the embeddings of the Sylow 7 normalizer N, say, of U into U and into the Sylow 7 normalizer of B. (Note that the fusion of the latter group into B has been determined in Section <a href="chap9.html#X7F5186E28201B027">9.3-1</a>.)

Forspan class="GAPprompt"gapgt/ span class="APinput"NrConjugacyClasses( u ); NrConjugacyClasses( act );

<div class="example"><pre>
gap> g:= SL( 2, 49 );;
gap> gens:= GeneratorsOfGroup( g );;
<">f:= GF(49);;/>
gap> mats:= List( gens, x -> IdentityMat( 4, f ) );;
gap> for i in [ 1 .. Length( gens ) ] do
> mats[i]{ [ 1, 2 ] }{ [ 1, 2 ] }:= gens[i];
> mats[i]{ [ 3, 4 ] }{ [ 3, 4 ] }:= List( gens[i],
> x -> List( x, y -> y^7 ) );
> od;
gap> fieldaut:= PermutationMat( (1,3)(2,4), 4, f );;
gap> gap> diagaut[1][1]:= Z(49);;
gap> diagaut[3][3]:= Z(49)^7;;
gap> g:= Group( Concatenation( mats, [ fieldaut * diagaut ] ) );;
gap> v:= [ 1, 0, 0, 0 ] * Z(7)^0;;
gap> orb:= Orbit( g, v, OnLines );;
gap> act:= Action( g, orb, OnLines );;
gap> n:= Normalizer( act, SylowSubgroup( act, 7 ) );;
gap> ntbl:= CharacterTable( n );;
<pre>/div

Now we compute the possible class fusions of N into B, via the Sylow 7 normalizer in B.

<div class="example"><pre>
gap> bn7:= CharacterTable( "BN7" );;
gap> nfusbn7:= PossibleClassFusions( ntbl, bn7 );;
gap> Length( RepresentativesFusions( ntbl, nfusbn7, bn7 ) )
3
gap> nfusb:= SetOfComposedClassFusions( PossibleClassFusions( bn7, b ),
> nfusbn7 );;
gap> Length( RepresentativesFusions( ntbl, nfusb, b ) );
5
</pre></div>

Although there are several possibilities, this information is enough to exclude one of the possible fusions of U<

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 110
gap> Length( nfusu );
4
gap> filt:= Filtered( ufusb,
> x -> ForAny( nfusu, y -> CompositionMaps( x, y ) in nfusb ) );
[ [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128,
 128, 128, 17, 72, 72, 89, 89, 89, 89 ] ]
<class=GAPprompt>&;/pan span class=GAPinput"ClassNames(b, "ATLAS"){filt[1] };
[ "1A", "2D", "3B", "4H", "5B", "6I", "7A", "8K", "8K", "12Q", "24L",
 "24L", "25A", "25A", "25A", "&t/span 
 "16G", "16G", "16G", "16G" ]
</pre></div>

So the class fusion of U into B can be described by the property that the elements of order four inside and outside the simple subgroup L_2(49) are not conjugate in B.<gap> OrdersClassRepresentatives( s );

We check that the correct map is stored on the library table.

<div class="example"><pre>
gap> GetFusionMap( u, b ) in filt;
true
</pre></div>

Let us confirm that the two groups of the types L_2(49).2_1 and L_2(49).2_2 cannot occur as subgroups of B. First we show that L_2(49).2_1 is isomorphic with PGL(2,49), an extension of L_2(49) by a diagonal automorphism, and L_2(49).2_2 is an extension by a field automorphism.

<div class="example"><pre>
gap> NrConjugacyClasses( u ); NrConjugacyClasses( act );
24
24
gap> u:= CharacterTable( "L2(49).2_1" );;
java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 6
gap> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> act:= Action( g, orb, OnLines );;
gap> java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 26
51
51
gap> u:= CharacterTable( "L2(49).2_2" );;
< classGAPprompt">& span class=GAPinput">g:= Group( Concatenation( mats, [ fieldaut ] ) );;
gap> orb:= Orbit( g, v, OnLines );;
gap> act:= Action( g, orb, OnLines );;
gap> NrConjugacyClasses( u ); NrConjugacyClasses( act<5>9.7- <panclass="impleMath">5^{1+4}.2^{1+4}.A_5.4 → B (April 2009)</h5>
27
27
</pre></div>

The group L_2(49).2_1 can be excluded because no class fusion into B</spandiv ="xample"><>

<div class="example"><pre>
gap> PossibleClassFusions( CharacterTable( "L2(49).2_1" ), b );
[ ]
</pre></div>

For L_2(49).2_2, it is not that easy. We would get several possiblespan ="gap&t;/> <pre>
gap> n:= Normalizer( act, SylowSubgroup( act, 7 ) );;
gap> Length( PossibleClassFusions( CharacterTable( n ), bn7 ) );
0
</pre></div>

idX7EAD52AA7A28D956 ="X7EAD52AA7A28D956"<ap>

<h5>9.7-8 2^3.L_3(2) → G_2(5) (January 2004)</h5>

The Chevalley group G = G_2(5) contains a maximal subgroup U of the type 2^3.L_3(2) whose class fusion is ambiguous.&t/>< class=GAPinput>: ( "2^.3());
gap> sfust:= PossibleClassFusions( s, t );;
gap> RepresentativesFusions( s, sfust, t );
[ [1, 2, 2,5,6 ,13,16 17,15,15 ]
 [ 1, 2, 2, 5, 6, 4, 14, 16, 17, 15, 15 ] ]
gap> java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 26
[ 1, 2, 2, 4, 4, 3, 6, 8, 8, 7, 7 ]
</pre></div>

So the question is whether U contains elements in the class <code class="code">6B</code> or <code class="code"><pan class=GAPprompt>gapgt/pan><spanclass="APinput">SortedList List( dec x-g;xdimension) ;/span>

<div class="example"><pre>
gap> g:= AtlasGroup( <>aid"85C48EEB7B711C09" name=X85C48EEB7B711C09"<a>/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<up( "G2(5)", 7 );ap&t;span> span="APinput"Size( )
1344
gap></>It happen thatthe fusion from theordinary character table a < =SimpleMathG/>into ofmarks of< =SimpleMath</java.lang.StringIndexOutOfBoundsException: Index 283 out of bounds for length 283
>p>s example consider amaximalsubgroup thesporadicsimplegroupspan class=S"He.
> until Order( x ) = 6;
< class"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 103
36
gap> Filtered( [ 1 .. NrConjugacyClasses
> i -> SizesCentralizers( t )[i] = siz );
[ 14 ]
</pre></CharacterTable( "7^:2(27)")

So U contains <code class="code">6C</code> elements in G_2(5).

<div class="example"><pre>
gap> GetFusionMap( s, t ) in Filtered( sfust, map -> 14 in map ); 
rue
</pre></ [1,6, 2,4,3 5,13 13 7,10,, 8 16, , 9,8 10,16 ],

>=X79617107849A6CEA"="79617107849A6CEA>/a

<h5>9.7-9 5^{1+4}.2^{1+4}.A_5.4 → B (April 2009)</h5>

The sporadic simple group 7,8,10 9, ,7 10, 9,8 ]]

<div class="example"><pre>
gap> b:= CharacterTable( "B" );;
gap> m:= CharacterTable( "5^(1+4).2^(1+4).A5.4" );;
gap> mfusb:= PossibleClassFusions( m, b );;
<="GAPinput">Length(mfusb )</pan>
4
><span<panclass=GAPinput>epres: RepresentativesFusions(mmfusb b);</div>

The restriction of the unique irreducible java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 0

""<>
gap> char:= Filtered( Irr( b ), x -> x[1] = 4371 );;
gap> Length( char );
1
gap> rest:= List( repres, map -> char[1]{ map } );;
gap> scprs:= MatScalarProducts( m, Irr( m ), rest );;
gap> constit:= List( scprs,
> x -> Filtered( [1 .. Length(x) ], i -> x[i] <> 0 ) );
[ [ 2, 27, 60, 63, 73, 74, 75, 79, 82 ],
 [ 2, 27, 60, 63, 70, 72, 75, 79, 84 ] ]
gap> List( constit, x -> List( Irr( m ){ x }, Degree ) );
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [ 1, 6, 384, 480, 100, 300, 500, 1000, 1600 ] ]
</pre></div>

The database <a href="chapBib.html#biBAGRv3">[WWT+]</a> contains the 3-modular reduction of the irreducible representation of degree 4371 and also a straight line program for restricting this representation to M. We access these data via the GAP package AtlasRep (see <a href="chapBib.html#biBAtlasRep">[WPN+22]</a>), and compute the composition factors of the natural module of this restriction.

<div class="example"><pre>
gap> g:= AtlasSubgroup( "B", Dimension, 4371, Ring, GF(3), 21 );;
gap> module:= GModuleByMats( GeneratorsOfGroup( g ), GF(3) );;
gap> dec:= MTX.CompositionFactors( module );;
gap> SortedList( List( dec, x -> x.dimension ) );
[ 1, 6, 100, 384, 400, 400, 400, 480, 1000, 1200 ]
</pre></div>

<>We exactly oneordinaryconstituent notstayirreducible uponrestriction to characteristic <spanclass="SimpleMath">3</span. Thusthefirstofthe twopossibleclassfusions isthe one./>

<a id="X85C48EEB7B711C09" name="X85C48EEB7B711C09"></a>

<h5>9.7-10 The fusion from the character table of 7^2:2L_2(7).2
into the table of marks (January 2004)</h5>

It can happen that the class fusion from the ordinary character table of a group G into the table of marks of G is not unique up to table automorphisms of the character table of G.

As an example, consider G = 7^2:2L_2(7).2, a maximal subgroup in the sporadic simple group He.

G contains four classes of cyclic subgroups of order 7. One contains the elements in the normal subgroup X.8 4 4 . 1 - . -B /B -B -B/ -B -B-B

<div class="example"><pre>
gap> tbl:= CharacterTable( "7^2:2psl(2,7)" );
CharacterTable( "7^2:2psl(2,7)" )
gap> tom:= TableOfMarks( tbl );
TableOfMarks( "7^2:2L2(7)" )
<span classX12 48 - . . . . . 6 -- 1. 6- 1 -
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 10, 9, 16, 7, 10, 9, 8, 16 ],1348 -1 ... C-1/C / . / D 1
 [ 1, 6, 2, X.14 48-1..... . C/C D- ./ D 1C
 [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 10, 9, 8, 16, 7, 9, 8, 10, 16 ],
 [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ],
 [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 10, 8, 9, 16, 7, 8, 9, 10, 16 ],
 [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 9, 10, 8, 16, 7, 10, 8, 9, 16 ] ]
<span1
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ],
 [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 10, 9, 16, 7, 10, 9, 8, 16 ] ]
gap> AutomorphismsOfTable( tbl );
Group([ (9,14)(10,17)(11,15)(12,16)(13,18), (7,8), (10,11,12)
 (15,16,17) ])
gap> OrdersClassRepresentatives( tbl );
[ 1, 7, 2, 4, 3, 6, 8, 8, 7, 7, 7, 7, 14, 7, 7, 7, 7, 14 ]
gap> perms1:= PermCharsTom( reps[1], tom );;
gap> perms2:= PermCharsTom( reps[2], tom );;
gap> perms1 = perms2; 
false
gap> Set( perms1 ) = Set( perms2 );
true
</pre></div>

The table of marks of G does not distinguish the three classes of cyclic subgroups, there java.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 11

X.4 1

<div class="example"><pre>
gap> Display( tbl );
7^2:2psl(2,7)

 2 4 . 4 3 1 1 3 3 1 . . . 1 1 . . .
 3 1 . 1 . 1 1 . . . . . . . . . . .
 7 3 3 1 . . . . . 2 2 2 2 1 2 2 2 2

 1a 7a 2a 4a 3a 6a 8a 8b 7b 7
 2P 1a 7a 1a 2a 3a 3a 4a 4a 7b 7c 7d 7e 7b 7f 7g 7h 7i
 3P 1a 7a 2a 4a 1a 2a 8b 8a 7f 7i 7g 7h 14b 7b 7d 7e 7c
 5P 1a 7a 2a 4a 3a 6a 8b 8a 7f 7i 7g 7h 14b 7b 7d 7e 7c
 7P 1a 1a 2a 4a 3a 6a 8a 8b 1a 1a 1a 1a 2a 1a 1a 1a 1a
 11P 1a 7a 2a 4a 3a 6a 8b 8a 7b 7c 7d 7e 14a 7f 7g 7h 7i
 13P 1a 7a 2a 4a 3a 6a 8b 8a 7f 7i 7g 7h 14b 7b 7d 7e 7c

X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 3 3 3 -1 . . 1 1 B B B B B /B /B /B /B
33 3 3 -.. 1 1 BB /B B/ BB java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
X46 6 6 2 . .. .1 1- - - - 1 - 1
X.5 7 7 7 -1 1 1 -1 -1 . . . . . . . . .
X.6 8 8 8 . -1 -1 . . 1 1 1 1 1 1 1 1 1
X.7 4 4 -4 . 1 -1 . . -B -B -B -B B -/B -/B -/B -/B
X.8 4 4 -4 . 1 -1 . . -/B -/B -/B -/B /B -B -B -B -B
X.9 6 6 -6 . . . A -A -1 -1 -1 -1 1 -1 -1 -1 -1
X.10 6 6 -6 . . . -A A -1 -1 -1 -1 1 -1 -1 -1 -1
X.11 8 8 -8 . -1 1 . . 1 1 1 1 -1 1 1 1 1
X.12 48 -1 . . . -- 1 .6-- -
X.13 48 -1 . . . . . . C -1 /C /D . /C C D -1
X.14 48 -1 . . . . . . C /C /D -1 . /C D -1 C
X.15 48 -1 . . . . . . /C D -1 C . C -1 /C /D
X.16 48 -1 . . . . . . C /D -1 /C . /C -1 C D
X.17 48 -1 . . . . . . /C C D -1 . C /D -1 /C
X.18 48 -1 . . . . . . /C -1 C D . C /C /D -1

 2 1
 3 .
 7 1


 2P 7f
 3P 14a
 5P 14a
 7P 2a
 11P 14b
 13P 14a

X.1 1
X.2 /B
X.3 B
X.4 -1
X.5 .
X.6 1
X.7 /B
X.8 B
X.9 1
X. 1
X.11 -1
X.12 .
X.13 .
X.14 .
X.15 .
X.16 .
X.17 .
X.18 .

A = E(8)-E(8)^3
 = Sqrt(2) = r2
=E(7+E(7)2E)^4
 = (-1+Sqrt(-7))/2 = b7
C = 2*E(7)+2*E(7)^2+2*E(7)^4
 = -1+Sqrt(-7) = 2b7
D=3*(-3*()22E(7^-*E(7)4-*()52E)^
 = (5-Sqrt(-7))/2 = 2-b7
gap> mat:= MatTom( tom );;
gap> mataut:= MatrixAutomorphisms( mat );;
gap> Print( mataut, "\n" );
Group( [ (11,12)(23,24)(27,28)(46,47)(53,54)(56,57),
 ( 9,10)(20,21)(31,32)(38,39), ( 8, 9)(20,22)(31,33)(38,40) ] )
gap> RepresentativesFusions( Group( () ), reps, mataut );
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ] ]
</pre></div>

We could say that thus the fusion is span "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 115

<div class="example"><pre>
gap>p <strongjava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 602
gap> tg:= CharacterTable( g );;
gap> tgfustom:= FusionCharTableTom( tg, tom );;
gap> trans:= TransformingPermutationsCharacterTables( tg, tbl );;
gap> tblfustom:= Permuted( tgfustom, trans.columns );;
gap> orbits:= List( reps, map -> OrbitFusions( AutomorphismsOfTable( tbl ),
<() ) ) );;
gap> PositionProperty( orbits, orb -> tblfustom in orb );
2
gap> PositionProperty( orbits, orb -> FusionToTom( tbl ).map<iv ">pre>
2
</pre></div>

So we see that the second one of the possibilities above is the right one.

<a id="X7B1C689C7EFD07CB" name="X7B1C689C7EFD07CB"></a>

<h5>9.7-11 3 × U_4(2) → 3_1.U_4(3) (March 2010)</h5

According to the Atlas (see <a href="chapBib.html#biBCCN85">[CCN+85, p. 52]</a>), the simple group U_4(3) contains two classes of maximal subgroups of the type U_4(2). The class fusion of U_4(2) into U_4(3) is unique up to table automorphisms.

<div class="example"><pre>
gap> u42:= CharacterTable( "U4(2)" );;
gap> u43:= CharacterTable( "U4(3)" );;
gap> u42fusu43:= PossibleClassFusions( u42, u43 );;span ="APprompt"gap>
gap> Length( u42fusu43 );
4
gap> Length( RepresentativesFusions( u42, u42fusu43, u43 ) );
1
</pre></div>

Atlas. This subgroup is a Klein four group that induces a permutation ="U_4(< thus on four <class="2)/paninto<java.lang.StringIndexOutOfBoundsException: Range [649, 648) out of bounds for length 722

The automorphism 2_1 swaps each pair of mutually inverse classes of order nine, that is, <code class="code">9A</code> is swapped with <code class="code">9B</code> and <code class="code">9C</code> is swapped with <code class="code">9D</code>. All U_4(2) type subgroups of U_4(3) are invariant under this automorphism, they extend to subgroups of the type U_4(2).2 in U_4(3).2_1.

<div class="example"><pre>
gap> u43_21:= CharacterTable( "U4(3).2_1" );;
gap> fus1:= GetFusionMap( u43, u43_21 );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 17,
 18 ]
gap> act1:= Filtered( InverseMap( fus1 ), IsList );
[ [ 16, 17 ], [ 18, 19 ] ]
gap> CompositionMaps( ClassNames( u43, "Atlas" ), act1 );
[ [ "9A", "9B" ], [ "9C", "9D" ] ]
</pre></div>

The automorphism 2_3 swaps <code class="code">6B</code> with <code class="code">6C</code>, <code class="code">9A</code> with <code class="code">9C</code>, and <code class="code">9B</code> with <code class="code">9D</code>. The two classes of U_4(2) type subgroups of U_4(3) are swapped by this automorphism.

<div class="example"><pre>
gap> u43_23:= CharacterTable( "U4(3).2_3" )26 , ,32,29,, ,52 ,
gap> fus3:= GetFusionMap( u43, u43_23 );
[ 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 13, 14,
 15 ]
gap> act3:= Filtered( InverseMap( fus3 ), IsList );
[ [ 4, 5 ], [ 11, 12 ], [ 13, 14 ], [ 16, 18 ], [ 17, 19 ] ]
gap>gap> Intersection( ind, Irr( 3u43 ) );
[ [ "3B", "3C" ], [ "6B", "6C" ], [ "7A", "7B" ], [ "9A", "9C" ],
 [ "9B", "9D" ] ]
</pre></div>

The Atlas states that the permutation character 3E)^ 3E) 0 0 ,,0 ,,0 ,0 ,0, ,0

So we choose appropriate fusions for the two classes of maximal U_4(2) type subgroups.

<div class="example"><pre>
>: -gt;(x, 16 ;/java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
[ 1, 2, 2, 3, 3, 5, 4, 7, 8, 9, 10, 10, 12, 12, 11, 12, 16, 17, 20,
 20 ]
gap> secondfus:= First( u42fusu43, x -> IsSubset( x, [ 18, 19 ] ) );
 ,2 ,3,,,7 ,9 ,10 ,11 ,11, ,19, ,
 20 ]
</pre></div>

<> java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 602

<div class="example"><pre>
gap> 3u42:= CharacterTable( "Cyclic", 3 ) * u42;
CharacterTable( "C3xU4(2)" )
<class>&<spanu43: CharacterTable 3_U4() )
CharacterTable( "3_1.U4(3)" )
gap> 3u42fus3u43:= PossibleClassFusions
gap> Length( 3u42fus3u43 );
8
 Length( RepresentativesFusions( 3u42, 3u42fus3u43, 3u43 ) );
2
</pre></div>

More precisely, each of the four fusions from U_4(2) to U_4(<pan class"">ap&t</pan>< =GAPinput>repres: RepresentativesFusions ,mtblfus2a12 2a12); /

<div class="example"><pre>
gap> inducedmaps:= List( 3u42fus3u43, map <pre><div>
> GetFusionMap( 3u43, u43 ), CompositionMaps( map,
< class=GAPprompt>&t;< class=GAPinput> InverseMap GetFusionMap(3u42,u42 ) )));
gap> List( inducedmaps, map -> Position( u42fusu43, map ) );
[ 1,1,2 , 4 4,3,3]
</pre></div>

This solvestheambiguity:Fusions from each ofthe orbits occur,andwe canassignthem the twoclasses of subgroupsby the choiceof the fusionsfrom<panclass="SimpleMath>U_4(2)/span>to U_4(3).

The reason for the asymmetry is that the automorphism 2_3 of U_4(3)

In fact the two classes of 3 × U_4(2) type subgroups of 3_1.U_4(3) behave differently. For example, inducing the irreducible characters of a 3 × U_4(2) type subgroup in the first class ofGroup([ (,2,) (,,,,,,,,,,)]

<div class="example"><pre>
gap> rep:= RepresentativesFusions( 3u42, 3u42fus3u43, 3u43 );
[ [ 1, 4, 4, 7, 7, 10, 13, 15, 18, 21, 24, 24, 27, 27, 30, 27, 48,
 49, 50, 50, 2, 5, 5, 8, 8, 11, 13, 16, 19, 22, 25, 25, 28, 28,
 31, 28, 48, 49, 51, 51, 3, 6, 6, 9, 9, 12, 13, 17, 20, 23, 26,
 26, 29, 29, 32, 29, 48, 49, 52, 52 ],
 [ 1, 4, 4, 8, 9, 13, 10, 15, 18, 21, 25, 26, 31, 32, 27, 30, 46,
 44, 51, 52, 2, 5, 5, 9, 7, 13, 11, 16, 19, 22, 26, 24, 32, 30,
 28, 31, 47, 42, 52, 50, 3, 6, 6, 7, 8, 13, 12, 17, 20, 23, 24,
 25, 30, 31, 29, 32, 45, 43, 50, 51 ] ]
gap> irr:= Irr( 3u42 );;
gap> ind:= InducedClassFunctionsByFusionMap( 3u42, 3u43, irr, rep[1] );;
gap> Intersection( ind, Irr( 3u43 ) );
[ Character( CharacterTable( "3_1.U4(3)" ),
 [ 630, 630*E(3)^2, 630*E(3), 6, 6*E(3)^2, 6*E(3), 9, 9*E(3)^2,
 9*E(3), -9, -9*E(3)^2, -9*E(3), 0, 0, 2, 2*E(3)^2, 2*E(3), -2,
 -2*E(3)^2, -2*E(3), 0, 0, 0, -3, -3*E(3)^2, -3*E(3), 3,
 3*E(3)^2, 3*E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, -1, -E(3)^2, -E(3) ] ),
 Character( CharacterTable( "3_1.U4(3)" ),
 [ 630, 630*E(3), 630*E(3)^2, 6, 6*E(3), 6*E(3)^2, 9, 9*E(3),
 9*E(3)^2, -9, -9*E(3), -9*E(3)^2, 0, 0, 2, 2*E(3), 2*E(3)^2,
 -2, -2*E(3), -2*E(3)^2, 0, 0, 0, -3, -3*E(3), -3*E(3)^2, 3,
 3*E(3), 3*E(3)^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, -1, -E(3), -E(3)^2 ] ) ]
gap> ind:= InducedClassFunctionsByFusionMap( 3u42, 3u43, irr, rep[2] );;
gap> Intersection( ind, Irr( 3u43 ) );
[ ]
</pre></div>

For 6_1.U_4(3) and 12_1.U_4(3), 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 3 × 4 Y 2.U_4(2). We get 10 irreducible induced characters from a subgroup in the second class (four faithful ones, four with kernel of order two, and the two abovementioned degree 630 characters with kernel of order four) and no irreducible character from a subgroup in the first class.

<a id="X7A94F78C792122D5" name="X7A94F78C792122D5"></a>

<h5>9.7-12 2.3^4.2^3.S_4 → 2.A12 (September 2011)</h5>

The double cover G of the alternating group A_12 contains a maximal subgroup M of the type 2.3^4.2^3.S_4 whose class fusion is ambiguous.

<div class="example"><pre>
gap> 2a12:= CharacterTable( "2.A12" );;
gap> mtbl:= CharacterTable( "2.3^4.2^3.S4" );;
gap> mtblfus2a12:= PossibleClassFusions( mtbl, 2a12 );;
gap> Length( mtblfus2a12 );
32
gap> repres:= RepresentativesFusions( mtbl, mtblfus2a12, 2a12 );; 
gap> Length( repres );
2
</pre></div>

We decide the question which of the essentially different two possible class fusion is the right one, by explicitly constructing M as a subgroup of G.

For that, let π denote the natural epimorphism from G to A_12, and note that π(M) can be described as the intersection of a S_3 ℘ S_4 type subgroup of S_12 with A_12. Further note that the generators for G and A_12 provided by <a href="chapBib.html#biBAGRv3">[WWT+]</a> are compatible in the sense that π can be defined by mapping the generators of G to those of A_12.

We need π only for computing one preimage of each given element. Therefore, we represent π implicitly by two epimorphisms from a free group to G and A_12, respectively, in order to avoid that GAPprecomputes a lot of unnecessary information for G. This way, computing a preimage of an element of A_12 under π is cheap. However, computing the preimage of a subgroup of A_12 would be very expensive. So we construct the subgroup of G that is generated by preimages of a set of generators of π(M); later we see that this subgroup is in fact equal to M.

<div class="example"><pre>
gap> g:= AtlasGroup( "A12" );
Group([ (1,2,3), (2,3,4,5,6,7,8,9,10,11,12) ])
gap> 2g:= AtlasGroup( "2.A12" );
<matrix group of size 479001600 with 2 generators>
gap> f:= FreeGroup( 2 );;
gap> pi1:= GroupHomomorphismByImagesNC( f, 2g, GeneratorsOfGroup( f ),
> GeneratorsOfGroup( 2g ) );;
gap> pi2:= GroupHomomorphismByImagesNC( f, g, GeneratorsOfGroup( f ),
> GeneratorsOfGroup( g ) );;
gap> w:= WreathProduct( SymmetricGroup( 3 ), SymmetricGroup(4) );
<permutation group of size 31104 with 10 generators>
gap> NrMovedPoints( w );
12
gap> s:= Intersection( w, g ); Size( s );
<permutation group with 8 generators>
15552
<spanclass=GAPprompt"gap>gap> w:= WreathProduct( SymmetricGroup( 3 ), SymmetricGroup(4) );
> PreImagesRepresentative( pi2, x ) ) ) );;
</pre></div>

Now we compute the character table of M, using a faithful permutation representation of M.

<div class="example"><pre>
gap> iso:= IsomorphismPermGroup( m );;
gap> t:= CharacterTable( Image( iso ) );;pre>/div>
gap> Size( t );
31104
gap> trans:= TransformingPermutationsCharacterTables( mtbl, t );;
gap> IsRecord( trans );
true
</pre></div>

Now let us see where the two fusion candidates differ.

<div class="example"><pre>
gap> para:= Parametrized( repres );
[ 1, 2, 6, 10, 8, 12, 7, 11, 9, 13, 5, 5, 17, 17, 17, 17
 22,
 ], [ 33, 34 ], 49, 49,
 48, 48 ]
gap> PositionsProperty( para, IsList );
[ 46, 47 ]
gap> List( repres, map -> map{ [ 44 .. 47 ] } );
[ [ 33, 34, 33, 34 ], [ 33, 34, 34, ,22 27 25 ,1013 11 ,29 ,37 ,38,,5,,
</pre></div>

class=>>/>"APinput>(, ;/java.lang.StringIndexOutOfBoundsException: Range [103, 102) out of bounds for length 103

<div class="example"><pre<So in44 <=></ java.lang.StringIndexOutOfBoundsException: Range [122, 121) out of bounds for length 301
gap> classreps:= List( ConjugacyClasses( t ){ positions <pan class""gap&t</> span class="GAPinput"positions=OnTuples[44. 47] java.lang.StringIndexOutOfBoundsException: Range [122, 103) out of bounds for length 122
> c -> PreImagesRepresentative( iso, Representative( < "><><" - PreImagesRepresentativeiso java.lang.StringIndexOutOfBoundsException: Range [119, 118) out of bounds for length 136
gap> traces:= List( classreps, TraceMat );;
gap> List( traces, x -> Position( traces, x ) );
[ 1, 2, 2, 1 ]
</pre></div>

We are lucky, already the traces of the elements allow us to decide which pairs of elements are G-conjugate; there is no need for an explicit (and expensive) conjugacy test in the matrix group G.</

Finally, we check whether the stored fusion is correct.

<div class="example"><pre>
gap> good:= First( repres,
> map -> map{ [ 44 .. 47 ] } = [ 33, 34, 34, 33 ] );;
gap> GetFusionMap( mtbl, 2a12 ) = good;
true
</pre></div>

<a id="X7E2AF30C7E8F89F9" name="X7E2AF30C7E8F89F9"></a>

<h5>9.7-13 127:7 → L_7(2) (January 2012)</h5>

The simple group G = L_7(2)>: L72";<

<div class="example"><pre>
gap> t:= CharacterTable( "L7(2)" );;
gap> s:= CharacterTable( "java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 62
gap> fus:= PossibleClassFusions( s, t );;
gap> repr:= RepresentativesFusions( s, fus, t );
[ [ 1, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
 112, 113, 114, 115, 117, 116, 76, 76, 77, 76, 77, 77 ],
 [ 1, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
 112, 113, 114, 115 java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 127
</pre></div>

The two fusion candidates differ only for elements of order 7.

<div class="example">ga/span=GAPinput>(t ){ ,77 ;/
gap> diff:= Filtered( [ 1 .. Length( repr[1] ) ],
> i -> repr[1][i] <> repr[2][i] );
[ 20, 21, 22, 23, 24, 25 ]
gap> OrdersClassRepresentatives( s ){ diff };
[ 7, 7, 7, 7, 7, 7 ]
gap> List( repr, l -> l{ diff } );
[ [ 76, 76, 77, 76, 77, 77 ], [ 83, 83, 83, 83, 83, 83 ] ]
gap> SizesCentralizers( t ){ [ 76, 77, 83 ] };
[ 3528, 3528, 49 ]
</pre></div>

We can decide which candidate is the correct one if we know the centralizer order in G of the elements of order 7 in M. So we compute this centralizer order.

<div class="example"><pre>
gap> g:= Image( IsomorphismPermGroup( GL(7,2) ) );;
gap> repeat x:= Random( g ); until Order(x) = 127;
<panclass"APprompt">gap> n:= Normalizer( g, SubgroupNC( g, [ x ] ) );;
gap> <java.lang.StringIndexOutOfBoundsException: Range [55, 44) out of bounds for length 85
7
til Order x);/
gap> 9.-4 <pan class="eading>spanclass"">_()→M/> ( 2009 – not!/><java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 118
<class"APprompt>gap></pan>
49
</pre></div>

We see thatthesecondcandidateis fusion from< class=SimpleMath><spaninto.

gap> GetFusionMap( s, t ) = repr[2];
true
</pre></div>

<p[ ,152,97 ,97,,97, , ,97 ,97,97 ,97,

<h5>9.7-14 L_2(59) → M (May 2009) – Do not use this!</h5>

It had been claimed that the sporadic simple Monster group M contains a maximal subgroup G of the type L_2(59), see <a href="chapBib.html#biBHW04">[HW04]</a>. Later it was shown that M does Here we show that the character tables of G and M do not provide enough information to determine a unique class fusion. (In particular, the tables do not help to show that G is not a subgroup of M.) The class fusion of G into <

<div class="example"><pre>
gap> t:= CharacterTable( "M" );;
gap> fus:= PossibleClassFusions( s, t );;
gap> repr:= RepresentativesFusions( s, fus, t );
[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 97, 98, 52, 32, 52, 14, 12, 98, 52, 32, 5, 98, 12, 98, 52, 3 ],
 [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 50 ,50 ,11,100 50,30,4,100,11 100, ,java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 3 ],
 [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 97, 101, 51, 30, 51, 14, 11, 101, 51, 30, 5, 101, 11, 101, 51,
 3 ],
 [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 97, 102, 53, 32, 53, 18, 12, 102, 53, 32, 6, 102, 12, 102, 53,
 3 ],
 [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
 3 ] ]
<prediv

The candidates differ on the classes of element order 30.

<div class="example"><pre>
< class="GAPinput">ord:= OrdersClassRepresentatives( s );;
gap> ord30:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 30 );
[ 18, 24, 28, 30 ]
gap> List( repr, x -> x{ ord30 } );
[ [ 98, 98, 98, 98 ], [ 100, 100, 100, 100 ], [ 101, 101, 101, 101 ],
 [ 102, 102, 102, 102 ], [ 104, 104, 104, 104 ] ]
</pre></div>

According to <a href="chapBib.html#biBHW04">[HW04]</a>, any subgroup<pan classGAPpromptgt/ =">= M");/java.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 96

<div class="example"><pre>
gap> pos:= Position( ClassNames7,114 ,,60 ,27 , ,
gap> good:= Filtered( fus, map -> pos in map );
[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
 3 ],
 ,153 ,97 ,97 ,97 ,97 ,97 ,97,,97
 97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
 3 ] ]
gap> repr:= RepresentativesFusions( s, good, t );
[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 97, 104, 52, 33, 52, 17, 12, 104, 52, 33, 5, 104, 12, 104, 52,
 3 ] ]
gap> GetFusionMap( s, t ) = fail;
java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4
</pre></div>

<a id="X8409DA2E83A41ABE" name="X8409DA2E83A41ABE"></a>

<h5>9.7-15 L_2(71) → M (May 2009)</h5>

"/ofspan=java.lang.StringIndexOutOfBoundsException: Range [169, 168) out of bounds for length 343

<div class="example"><pre>
gap> t:= CharacterTable( "M" );;
gap> s:= CharacterTable( "L2(71)" );;
gap> fus:= PossibleClassFusions( s, t );;
gap> repr:= RepresentativesFusions( s, fus, t );
[ [ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112,
 112, 112, 11, 19, 112, 112, 114, 60, 36, 27, 114, 17, 114, 27,
 7, 60, 114, 5, 114, 60, 36, 27, 114, 3 ],
 [ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112,
 112, 112, 11, 19,apgt;/ <spanclass"APinput">pos1:= Position( ClassNames( t, "Atlas" ), "7B" );;
 7, 61, 115, 5, 115, 61, 36, 28, 115, 3 ],
 [ 1, 169, 170, 112, 112, 112, 112, 19, 112, 11, 112, 112, 19, 112,
 112,,11 19 ,117,61 ,,117 ,117 ,
 9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ],
 [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20,[[1,169 170, 113 ,113,113 20 113 12 ,113,,113,
 113, 113, 12, 20, 113, 113, 114, 60, 36, 27, 114, 17, 114, 27,
 7, 60, 114, 5, 114, 60, 36, 27, 114, 3 ],
 [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
 113, 113, 12, 20, 113, 113, 115, 61, 36, 28, 115, 17, 115, 28,
 7, 61, 115, 5, 115, 61, 36, 28, 115, 3 ],
 [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
 113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28,
 9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ]
</pre></div>

The candidates differ on the classes java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 4

<div class="example"><pre>
gap> ord:= OrdersClassRepresentatives( s )<h5>97-16(pril 2012)</h5>
gap> ord36:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 36 );
[ 21, 25, 27, 31<>The simple =Sjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 343
gap> List( repr, x -> x{ ord36 } );
[ [ 114, 114, 114, 114, 114, 114 ], [ 115, 115, 115, 115, 115, 115 ],
 [117 ,117 ,, 114 ,,,,114]java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
 [ 115, 115, 115, 115, 115, 115 ], [ 117, 117, 117, 117, 117, 117 ] ]
</pre></div>

<">G contains the java.lang.StringIndexOutOfBoundsException: Range [147, 146) out of bounds for length 300

<div class="example"><pre>
gap> pos1:= Position( ClassNames( t[1,127 64,,64,, ,30 ,64 ,,72 19
gap> pos2:= Position( ClassNames( t, "Atlas" ), "36D" );;
gap> pos:= [ pos1, pos2 ];;
gap> good:= Filtered( fus, map -> IsSubset( map, pos ) );
[ [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
 113, 113, 12, 20, 113, 113, 117, 61, 43, 28, [ 1, 127, 127,66, 33,66 ,7 33 66,12,66 3,73 73,20
 9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ],
 [ 1, 170, 169, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
 113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28,
 9, 61, 117, 5, [ 1, 127127, 127, 67, 0, 67,11,10, 30 67, 11, 67,3,72 ,19
gap> repr:= RepresentativesFusions( s, good, t );
[ [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 113,
 113, 113, 12, 20, 113, 113, 117, 61, 43, 28, 117, 17, 117, 28,
 , ,117 ,117 ,43, ,117,3]]
< class">> 
true
</pre></div>

me"">/>p

<h5>9 127 127,69 ,69 12 9 ,69,12 69 3 73 73 ,java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

The sporadic simple Monster group M contains a maximal subgroup G of the type L_2(41), see <a href="chapBib.html#biBNW12">[NW13]</a>. The class fusion of G into M is ambiguous.

<div class="example"><pre>
gap> t:= CharacterTable( "M" );;
< class>ap&;<panclass=G>:= CharacterTable( "(1) );/>
gap> fus:= PossibleClassFusions( s, t );;
gap> repr:= RepresentativesFusions( s, fus, t );
[ [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 23 ]
 70, 70, 19, 4, 70, 19, 70 ],
 [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 72, 72, 19,
 72, 72, 19, 6, 72, 19, 72 ],
 [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 73, 73, 20,
 73, 73, 20, 5, 73, 20, 73 ],
 [1, 127 127, 66,33,66 12,7,33 ,12 ,3 ,72 19
 72, 72, 19, 6, 72, 19, 72 ],
 [ 1, 127, 127, 66, 33, 66, 12, 7, 33, 66, 12, 66, 3, 73, 73, 20,
 73 73 , 5 73 ,73]
 127, 67,30,67, 11,10, 30 67,11 67, 3, 72 72 19,
 72, 72, 19, 6, 72, 19, 72 ],
 [ 1, 127, 127, 67, 30, 67, 11, 10, 30, 67, 11, 67, 3, 73, 73, 20,
 73, 73, 20, 5, 73, 20, 73 ],
 [1, 127,127,68,32 68,,12, 10,32, , 12,68 3,72,72 19, ,
 72, 72, 19, 6, 72, 19, 72 ],
 [ 1, 127, 127, 68, 32, 68, 12, 10, 32, 68, 12, 68, 3, 73, 73, 20,
 73, 73, 20, 5, 73, 20, 73 ],
 [ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 72, 72, 19,
 72, 72, 19, 6, 72, 19, 72 ],
 [ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 73, 73, 20,
 73, 73, 20, 5, 73, 20, 73 ] ]
<>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

The candidates differ on the classes of the element orders 3–8.

<div class="example"><pre>
gap> ambig:= Parametrized( repr );;
gap> ambigpos:= PositionsProperty( ambig, IsList );
[ 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22,
 23 ]
gap> Set( OrdersClassRepresentatives( t ){ ambigpos } );
[ 3, 4, 5, 6, 7, 8 ]
</pre></div>

According to <a href="chapBib.html#biBNW12">[NW13, Theorem 3]</a>, G contains elements in the classes <code class="code">3B</code> and <code class="code">4C</code> of M. This determines the class fusion uniquely.

<div class="example"><pre>
gap> pos1:= Position( ClassNames( t, "Atlas" ), "3B" );;
gap> pos2:= Position( ClassNames( t, "Atlas" ), "4C" );;
gap> pos:= [ pos1, pos2 ];;
gap> good:= Filtered( fus, map -> IsSubset( map, pos ) );
[ [ 1, 127, 127, 69, 33, 69, 12, 9, 33, 69, 12, 69, 3, 73, 73, 20,
 73, 73, 20, 5, 73, 20, 73 ] ]
gap> GetFusionMap( s, t ) = good[1];
true
</pre></div>

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