Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma

Benutzer


products/Sources/formale Sprachen/GAP/pkg/idrel/doc/ (GAP Algebra Version 4.15.1^©) Datei vom 2.9.2025 mit Größe 12 kB

Impressum chap9.html

Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/ctbllib/doc2/chap9.html

<?xml 10"="TF8"java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
<
<title>GAP (CTblLibXpls) - Chapter
<-equivc-"content=texthtml;charset=TF-8"/
<meta name="generator" content="java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0
<rel"tylesheet" =text href=manual." />
<script src="manual.js" type="text/javascript"></script>
<script type"ext/avascript>();/>
</head>
<body class="div class="ChapSects"><a href="chap9.html#X7A03A83E87FB1189">9 Ambiguous Class Fusions in class=ContSect"< =t">< class"ocss>nbsp;</>< href="hap9htmlX784492877DB04FE9. span class"eading" < class"kg></> /a>

<div class="chlinktop">Goto Chapterjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 7

< ="hlinkprevnexttop"&;a href"hap0.tml"[opofBook<a>&bsp < href".#ontents>[ontents]/>n; a =chap8.tml"[ Chapter<a&; <a ="html"[Next Chapter]/>nbsp; <div

<a href="chap9_mj.html">[MathJax on]</a>
<a id="X7A03A83E87FB1189" name="X7A03A83E87FB1189"></a>
<sa href=chap9html"9 class="Heading>Ambiguous in<strong=">GAP</trong> CharacterTable Library/>
<div class="java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 7

<div class="ContSect"> <a href="chap9/pan>
</pan>
<div class="ContSSBlock">
 <a href="chap9.html#X78DCEEFD85FF1EE2">9.2-1span

 <a href="chap9.html#X86BCEA907EC4C833">9.2-2 31:15 → B</spanSubgroups/span>/>

 <a href="chap9.html#X7C719F527831F35A">9.2-3 SuzN3 → Suz (September 2002)</a>
</pan>
 <a href="chap9.html#X828879F481EF30DD">9.2-4 F_{3+}N5 → java.lang.StringIndexOutOfBoundsException: Index 171 out of bounds for length 7

</div></div>
<div class="ContSect"> <a href="chap9.html#X7981579278F81AC6">9.3 Fusions Determined Using Commutative Diagrams Involving Smaller
ups<a
</spanspan class="br /<pan ="ocss>nbsp;nbsp;/>a href=chap9html#7962DD4387D63675"9.5< class"eading< =SimpleMath7^1+4:(3 ×2S_7 →M( 2009)/>/>
<div java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7
<class=ContSS>< ><class=nocss>nbsp&</pana".tml#7F5186E28201B027">.31<class"eading"< class"">BN7 →</> (arch 2002)/>/a

 <a href="chap9.html#X79710B137B5BB1B8">9.3-2 (A_4 × O_8^+(2)./>

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

/>

nbsp;span>ahref"hap9.tml#7F2B104686509CAA>.- span ="eading>span ="SimpleMath">3._ 3Suz>(ecember2010/>
<spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
< class"ContSS"< /<class"">nbsp&;/> =chap9h#860B6C30812DE3FC936<span="">span class"impleMath"3^._7()2→Fi_24>(ovember)<span<>

<class"ContSS"<r/>< ="ocss>nbsp;nbsp;/>a href="hap9htmlX7C3AC42F8342EE2E.-7 span class=SimpleMath"^E_6()3C →^E_6()/pan>(January 2019)/>

</div></div>
<div class="ContSect"> <a href="chap9.html#X84F966E2824F5D52
roupsspan<a>

<div class="ContSSBlock"/
<"> class"ocss>nbsp&<span<hrefc.X7F2B104686509CAA"9.- <pan class""<a>

 
</spandivclass"ontSect"spanclass=tocline"<panclass=nocss"&bsp;span<a href=chap9.tmlX85E2A6F480026C959. < class"eading"Conditions byBrauer Tables<>
</div<spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<div">nbsp<span< href"hap9.html#">95<pan class"eading>Fusions Determined Using Commutative DiagramsInvolving
Automorphic Extensions

<div class="ContSSBlock">
spanclass">;&bsp;/pan>a ="hap9htmlX7E91F8707BA93081>9.-1 <panclass"eading"<panclass"impleMath"U_38)3_ →^()</a>


</pan>

div="ContSect"><spanclass=nocss"&bsp;/>a href=chap9.html#85E2A6F480026C95"9. <panclass=Heading"> by Brauer Tables<><a>
</>
<div class="ContSSBlock">
<pan=SimpleMath>()4 →J_32(anuary 004)/pan>/>

nbsp&;/>a =chap9X7AE2962E82B4C814>.- span class=Heading"_33. Fi_24^</>(ovember )<>/>

 span>
</>
</div></div>
<ivclass=ContSect"<pan =tocline"< =nocss>&bsp;/pan< =chap9#8225D9FA80A7D20F>. < ="eading"FusionsDeterminedInformation the<span/a>

<div class="ContSSBlock">
 <a href="chap9.html#X7AE2962E82B4C814">9.span =ContSS>br >spanclass""&;nbsp<span<href".#"973 <spanclass"eading"spanclass"impleMath"M_11 / April2009<<java.lang.StringIndexOutOfBoundsException: Index 197 out of bounds for length 197

 <a href="chap9.html#X83061094871EE241">9.7-2 L_2(13).2 → Fi_24^' (September 2002)</a>
<
 <a href="chap9.html#X7E9C203C7C4D709D">9.7-3 M_11 → B (April 2009)</a>
</panjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 <a href="chap9.html#X85821D748716DC7E">9.7-4 L_2(11

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

 
 

< class=ContSS>br/<class""&&</spana href=.tmlX7EAD52AA7A28D9569.- <panclass=Heading> class"">^._()→G_25)</>(anuary )span<>

/>
<>
br/<span="ocss"&bsp;nbsp;span>a =".tml#X7B1C689C7EFD07CB">97-1< class="< =SimpleMath"3 ×U_42 _.()/pan> (March 2010)/><a>
into the table of marks (January 2004)</a>

 <>

 &spanclass""<r/< =nocss>nbsp&bsp<span< =chap9h#7A94F78C792122D597-2 < class""< class=""23423→./ September)/>a
</pan>
 <a href="chap9.html#X7E2AF30C7E8F89F9">9.7-13 127:7 → L_7(2) (January 2012)<//>

 <a href="chap9.html#X80051B297DF244CF">9.7-14 L_2(59) → M (May 2009) – Do not use this!</a>
</pan>
 <a href="chap9.html#X8409DA2E83A41ABE">9.7-15 L_2(71) → M (May 2009)</a>

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

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

<h3>9 

This a collectionofexamples showing how classfusions between tables bedetermined using the <trong class"pkg"></>system< chapBib.tml#biBGAP"[AP24]/> In each of these examples,the fusion <>ambiguous the sensethat charactertablesdo itup to automorphisms.Our java.lang.StringIndexOutOfBoundsException: Range [357, 356) out of bounds for length 796

This is a collection of examples showing how class fusions between character tables can be determined using the GAP system <a href="chapBib.html#biBGAP">[GAP24]</a>. In each of these examples, the fusion is ambiguous in the sense that the character tables do not determine it up to table automorphisms. Our strategy is to compute first all possibilities with the GAP function <code class="func">PossibleClassFusions</code> (<a href="../../../doc/ref/chap73.html#X7883271F7F26356E">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 <trong class=pkg>/>CharacterTableLibrarysowe loadthispackage./p

< class""<re
<pan ="APprompt"gapgt;/>< ="APinput">oadPackage("ctbllib" false;/>
true
</pre><p< id"X784492877DB04FE9"nameX784492877DB04FE9<a</>

<>a id=X784492877DB04FE9"name=X784492877DB04FE9"<a

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

The function <code class="code">SetOfComposedClassFusions</code> takes two list of class java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 0

Note that the returned list may be a proper subset of java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<div class="example"><pre>
<">gapgt;/pan>< class=GAPinput"SetOfComposedClassFusions:=function(hfusg, ufush )/pan>
> =GAPprompt>gt;/pan>span="" : [;/>
<span=""&;/><pan ="GAPinput" for map2 in hfusg do
<spanclass=GAPprompt>gt map1 in ufushdo
<pan class=GAPprompt&;/pan><panclass"APinput> for in do</pan>
&> class"APinput" od;/span>
span class"APprompt"&t;/> > 
<panclass"APprompt"&t;/pan><pan classG" return result;/>
> end;;
</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<a id="X7EA839057D3AD3B4" name="X7EA839057D3AD3B4"></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4>9.2 Fusions Determined by Factorization through

This situation clearly occurs only for nonmaximal subgroups. java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 0

<a id="X78DCEEFD85FF1EE2" name="java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0

<h5>9.2-1 Co_3N5 → Co_3 (September 2002)</h5>

Let H be the Sylow 5 normalizer in

<div=example>pre
( "o3")
CharacterTable "o3")
gap&CharacterTable( "5^(1+2):(24:2
haracterTable 5(+)(:) java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
< class"">ap&t;/>< class=GAPinput"hfusco3: ( ,co3 );;
gap> Length( RepresentativesFusions( h, hfusco3, co3 ) );
2
<pre<div>

As<>As < class=SimpleMath"HCo_3/, thosemaximalsubgroupsof< class"">Co_3/pan whose order isdivisible by that of spanclass="H</pan>/>

<div class="example ="xample>pre
<panclass"APprompt"gapgtgap> maxes:= List( mx, CharacterTable );;
&;/pan>< class=""filt:=Filtered(maxes,x -&java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
[ CharacterTable( "McL.2" )java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0

</pre></div>

<>Accordingto thestrong class=pkg"Atlas(ee a href=chapBib.tml#biBCCN85">[CCN+85, pp. 34 and 100]</a>), H occurs as the java.lang.StringIndexOutOfBoundsException: Index 175 out of bounds for length 0

We compute the possible span class="GAPprompt">gap>> =""hfusmax PossibleClassFusions(,max;< =G">:SetOfComposedClassFusions(maxfusco3 hfusmax );/>
<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [44, 5) out of bounds for length 84
gap> <span class ,2 , 4 ,8,7 9,10 ,17,17,19, 19 22 ,27 ,
< 30,33 ,,40,40 40 java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
gt<>class"Length( );
2
gap> reps:= RepresentativesFusions( h, comp, co3 );
[ [ 1, 2, 3, 4, 8, 8, 7, 9, 10, 11, 17, 17, 19, 19, 22, 23, 27, 27,
 30, 33, 34, 40, 40, 40, 40, 42 ] ]
</pre></div>

factoring a type =">.</pan> the from span =SimpleMath></>to </pan> uniquely up to ./>

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

<div class="example"><pre>
gap> max:= filt[3];;
<> fusmax: PossibleClassFusions h,max );/span>
gap&<panclass="APprompt"gap;span class=GAPinput>hfusmax= PossibleClassFusions(h,max ;;
< java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 125
gap> gap> comp:= SetOfComposedClassFusions( maxfusco3, hfusmax );;
<class"APinput"reps2 reps/iv

<>Finally,wecomparethe result withthe map that is onthelibrary table of <pan class="impleMath"H/span></>

<div
<=>g;<<classG> ,
true
</pre></div>

<a id="X86BCEA907EC4C833" name="X86BCEA907EC4C833"></a>

<h5> =XB =">a>

h59.- <pan class="/java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 98

<div class
<pan "APprompt"gap&;/ span">: ( "B
< ="&t<> < ="java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 100
<=>&gap></span=><java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 96
2
</pre></div>

We determine the correct fusion using the factjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 107

< ">pre>
<spanjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<=""gapg;= , ;gap><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 th:= CharacterTable( "Th" );;
gap> Length(="APprompt">gap> thfusb:= PossibleClassFusions( th, b );;span class="GAPprompt"gap;/>spanclass"APinput">comp: SetOfComposedClassFusionsthfusb,hfusth );
2
<gt;gap> GetFusionMap( h, b ) in reps;
true
</pre></div>

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

<h5>2- span ">span ""SuzN3 → S 2002</span<h5>

<>

<div class ="> =>uz/>Sjava.lang.StringIndexOutOfBoundsException: Range [87, 86) out of bounds for length 104
<classGjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 99
CharacterTable( "3^5:(3^2:SD16)" )
gap> suz
CharacterTable java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 111
gap> Length( RepresentativesFusions( h, hfussuz, suz ) );
java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 1
</pre></div>

<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 169

<div class="example"><pre>
gap> <span">></> span => RepresentativesFusionsjava.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 121
<span class="GAPprompt"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ CharacterTable( "3_2.U4(3).2_3'" ), CharacterTable( "3^5:M11" ),
 CharacterTable( "3^2+4:2(2^2xa4)2" ) ]
</pre></div>

The group 3_2.U_4(3).

<div class="example"><pre>
gap> maxes:=List( Maxes( ),CharacterTable );;
[<pan class=GAPprompt>gap&t;/span : Filtered maxes x-&t Size x )Size =0)</pan>
</pre>CharacterTable( "_.43._'",( "3^5:" )

< 3^:M_11span containsgroupisomorphic<java.lang.StringIndexOutOfBoundsException: Range [94, 93) out of bounds for length 450

<div class="example"><pre>
gap> max:= filt[2];;
< class"GAPprompt"gap;<panclass=GAPinputhfusmax=PossibleClassFusions(h, );;/span>
gap> maxfussuz:= PossibleClassFusions( max, suz );;
</div
[ [ 1, 2,p>Definitely span ="impleMath"3^:_11span>containsa group isomorphicwith<class=SimpleMath><>becausetheSylowclass=SimpleMath3/ normalizer< class"impleMath">_ thestructure< class"SimpleMath"3^2SD_16><classS"^+22^×)</ tothe java.lang.StringIndexOutOfBoundsException: Range [375, 374) out of bounds for length 450
13 ,,15,,16 ,,21 ,23 ,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=java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<APinput
java.lang.StringIndexOutOfBoundsException: Range [16, 4) out of bounds for length 61
<<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12

<[] ][ ], ]

<h5>9.2-4 <span1110 ,, 121 ,72,,11,11,79, ,

p the of Sylow <class"> normalizer < ="">H</pan> in sporadicsimple {+<span< classSimpleMath"{}/>isambiguous./>

<div class="example"><pre>
<panclass""gap;3p: ( "+");/java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
gap> h:= CharacterTable( "F3+N5" );;
gap> hfusf3p:= PossibleClassFusions( h, f3p );;
<panclass=G>apgt;span< ="GAPinput"Length RepresentativesFusions , hfusf3p ;/>
2
</pre></div>

H44 44 167 167 48 48 171 171, 57 ,180, 180,66,

< class=e><re
gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 131
[ CharacterTable( "Fi23" ), CharacterTable( "2.Fi22.2" ),
 CharacterTable( "(3xO8+(3):3):2" ), CharacterTable( "O10-(2)" ),
 CharacterTable( "(A4xO8+(2).3).2" ), CharacterTable( "He.2" ),
 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<class="GAPprompt">gap> possfus""gapg;: ( , -gt (h );/>
[ [ ], [ ], [ ], [ ],
 [[1 69 110 12,80,121 4 72 ,11,11,79,79,120,,
 3, 71, 11, 79, 23, 91, 112, 120, 132, 29, 32, 97, 100, 37,
 , ,,146155 ,,,44
 44, 167, 167, 48, 48, 171, 171, 57, 57, 180, 180, 66, 66,
 189, 189 ],
 [ 1, 69, 110, 12, 80, 121, 4<spanclass"">ap&t<span < class="APinput"Length comp )</pan
 120, 3, 71, 11, 79,,23,91 112,120,132,29 32 ,100
 37, 37, 105, 105, 140, 139, 146, 145, 156, 156, 155, 155,
 ,44 167,167 48, 48,171 171,57 57,180, 180, 66
 66, 189, 189 ] ], [ ], [ ], [ ] ]
</pre></div>

We see that from the eight possible classes of maximal subgroups in F_{3+} that might contain H, only the group of type (A_4 × O_8^+(2).3).2 admits a class fusion35,36,13, 40 90,7 22,19 20,43 43,50 50,8, , 23,

<div <div>
, check whether the stored in the library is correct.
CharacterTable( "(A4xO8+(2).3).2" )
gap> hfusmax:= possfus[5];;
<="GAPprompt">gap> maxfusf3p:= PossibleClassFusions(< =G>gap></pan < class"GAPinput">GetFusionMap( h, f3p ) in repr;
gap> /div
apgt Length( comp )
2
<pan class"APprompt">apgt;/span>repr:= RepresentativesFusions( h, comp, f3p );
[ [ 1, 2, 4, 12,
 ,,, 90 ,22,,,43,43 50 ,8,8 ,
 23, 46, 46, 47, 47, 10, 10, 9, 9, 10, 10, 11, 11, 26, 26, 28,
 28, 67, 67, 68, 68 ] ]
</pre></div>



<div class="example"><pre>
<a>/>
true
</pre></div>

Note that we did not determine the class fusion from the maximal subgroup (A_4 × O_8^+(2).3).2 into <

<paX7981579278" =X7981579278F81AC6<a<pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h4>9.3 Fusions Determined Using Commutative Diagrams Involving Smaller
Subgroups</h4>

In each of the following examples

<center> <img src="ambigfus1.png" alt="setup: some subgroupsspan class="APprompt>gap> b:= CharacterTable(" )

<p=Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 97

<>9- <spanclass"> ="impleMathBN7→</>M 2002<span</>

<pLet<panclass"impleMath">7 normalizer in the sporadic simple group B. The class fusion of H into &t</>( ( h hfusb, b))gap> b:= CharacterTable( "B" );
CharacterTable( "B" )
gap> h:= CharacterTable( "BN7" );
CharacterTable( "BN7" )
spanclass=GAPprompt"gap&t;/pan>< class=GAPinput">:= PossibleClassFusions h, );;
class="gap&t; Length( RepresentativesFusions( h, hfusb, b ) );
2
</pre></div>

Let us consider a maximal subgroup of the type Th</java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 22

<>Notethat the only subgroups< class"></> is Sylow < =Sjava.lang.StringIndexOutOfBoundsException: Range [151, 150) out of bounds for length 525

<div class="example"><pre>
gap> Number( Factors( Size( b ) ), x -> x = 7 );
2
gap> th:= CharacterTable( "Th" );
CharacterTable( "Th" )
gap> Filtered( Maxes( th ), x -> Size( CharacterTable( x ) ) mod 7^2 = 0 );
[ "3D4(2).3", "7^2:(3x2S4)" ]
</pre</div>

The class fusion of Ujava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
gap> thn7:= CharacterTable( "ThN7" );
gap&CharacterTable( "7^2:(3x2S4)" )
> PossibleClassFusions( thn7spanclass=GAPprompt>gap>/span>
[ 6, 6, 7, 28, 27, 27,109,java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 , 17,45 45,,72,127,127 ,127 ]
</pre></div>

The condition that the class fusion of U into B factors through H determines the class fusion of H into B up to table automorphisms.

<div class="example"><prepre>/
<GAPprompt>gapgt</pan spanclass"">hn7fush= ( thn7 );/>

<pan="GAPprompt">&t
1
</pre></div>

Finally, we compare the<pan class"APprompt"> ForAny(thn7fush,y&CompositionMaps(,y  comp));/java.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143

<div class="example"><pre>
<panclass="APprompt">apgt</div>

<a id="X79710B137B5BB1B8java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<< =GAPprompt&t; 

<pre><div>

<div class="example"><pre>
gap> ">gap>< class"APinput>t=CharacterTable "F3+");/
gap> mfust:= PossibleClassFusions( m, t
&t;</ spanclass=>: ( ;/
gap> </div>

We first observe that the elements of order three in the normal subgroup of type <java.lang.StringIndexOutOfBoundsException: Range [0, 89) out of bounds for length 0

<div class="example"><pre>
gap> a4inm:= Filtered( ClassPositionsOfNormalSubgroups( m ),
> n -> Sum( SizesConjugacyClasses( m ){ n } ) = 12 );
[ [ 1, 69, 110 ] ]
<span( m ){ a4inm[1] };/>
[ 1, 2, 3 ]
gap> List( repr, map -> map[110] );
[ 4, 4 ]
<="GAPprompt"gap;/> <spanclass=G>( t{[1 . 4]}</>
[ 1, 2, 2, 3 ]
</pre></div>

<<s ="gt<><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 124

<div class="example"><pre>
gap> s:= CharacterTable( "F3+N3A" );
CharacterTable( "(3xO8+(3):3):2" )
</pre></div>

The intersection > 

<div [ 1, 2, 3 ,3 ]
< =""gt;/pan>span"APinput"u (Cyclic" 3)*( "O8+2.";/>
CharacterTable( "C3xO8+(2).3" )
gap>gap></span=G>g;( t ) 1 . ];/
gap> g;/>< ="">sfust=( , );;/>
gap> comp:= java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> span class="GAPprompt">gap> s:+3 ;/>
6
gap&t;/pan> < class=GAPinput">:Filtered(mfust,/pan>
Theintersection span=SimpleMath>N_M)=S∩< of <class">3 ^+23/span>, in the following computethe fusions span "impleMath"Uinto and utilize fact that only class fusions into<panclass="SimpleMath>are compositionwith the class fusionfrom span class="SimpleMath>/span> equalsa of span=""US and into <class"impleMath><span.</>
gap> Length( repr <pan="APprompt"gapg;/span spanclass"APinput"u=CharacterTable"Cyclic" 3)*("8()3 ;/>
1
"g;/span <pan class=GAPinput>GetFusionMap( m,t )in repr

</pre></div>

So the class fusion from M into G is determined up to tableclass""gt<><"">: (,t);/>

<<classjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 84

<span=G&< "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 92

The classfusion of themaximal subgroup M ≅ A_6 × L_2(8).3 of G = Fi_24^' is ambiguous.

<example>pre
gap> t:= CharacterTable( "F3+" );;
gap> <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap> Length( RepresentativesFusions( m, mfust, t ) );
2
</pre></div>

<pWewill the that the directofthe type< class"SimpleMath">A_6 in M contains elements in the class <code class="code">3A</code> of G. This fact can be shown as follows.

<div class="example"><pre>
gap> dppos:= ClassPositionsOfDirectProductDecompositions( m );</
[ [ [ 1, 12 .. 67 ], [ 1 .. <iv=example><pre>
gap></&;/ span ">: AxL2)");<>
[ 17733424133316996808705, 4545066196775803392 ]
<=GAPpromptgtList 1,l-gt (( m) ;/
[ 360, 1512 ]
<panjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 124
4
< class="GAPinput"3= -gt ,A));
[ 23, 23, 23, 23, 34, 34, 34, 34 ]
</div>

Since the normalizer of an java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 0

(Note that all classes in span class="GAPprompt">gap>

<div class="example"><pre>
gap> u:= CharacterTable( "3^2:2" ) * CharacterTable( "L2(8).3" );
CharacterTable( "3^2:2xL2(8).3" )
gap> s:= CharacterTable( "F3+360 1512]
CharacterTable( "(3xO8+(3):3):2" )
gap> ufuss:= PossibleClassFusions( u, s );;
<java.lang.StringIndexOutOfBoundsException: Range [60, 5) out of bounds for length 119
gap></span[23 ,23,23 34 ,34,34]
gap> filt:= Filtered( mfust,
gt< class" -gt;( ufusm,
> map2 -> CompositionMapsjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
< class=""repr= ,,)<>
gap> Lengthjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
1
gap> GetFusionMap( m, t ) in repr;
true
</pre></divdiv ="<

<p("^:xL2(8.")

<>.4<H"< ="(^: ×()2^). B/>J 2007)/>

Let G be a maximal java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 34

According to <a href="chapBib.htmlspan ="gap< spanG":=(java.lang.StringIndexOutOfBoundsException: Range [103, 101) out of bounds for length 119

The span =>g;/ <classG"filt:=( ,

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26
=GAPprompt"gap>/span ="GAPinput">b:=CharacterTable( "B" );;
gap> n:= CharacterTable( "BN3A" );
CharacterTable( "S3xFi22.2" )
gap> nfusb:= PossibleClassFusions( n, b );;
<java.lang.StringIndexOutOfBoundsException: Range [60, 5) out of bounds for length 85
1
gap> <java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 12
<pre/

The computation of<59.-4 <spanclass="eading"(une 2007)/><h5

< class="xample"<prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
gap> fi222:= CharacterTable( "Fi22.2" );;
gap> fi222n3a:= CharacterTable( "S3xU4(3).(2^2)_{122}" );;
gap> s3:= CharacterTable( "S3" );;
gap> inter:= s3 * fi222n3a;;
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
gap&t>ength( intermods3fusnmods3 );
2
gap> Length( RepresentativesFusions( fi222n3a, intermods3fusnmods3, fi222 ) );
1
</pre></div>

span G>gap>b: CharacterTable( "");

<div class="example"><pre>
gap> interfusn:= CompositionMaps( java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 29
<3fusnmods3[1]
> GetFusionMap( inter, fi222n3a ) ) );;
gap> interfusn:= PossibleClassFusions( inter, n,1
span=&
gap> Length( interfusn );
1
</pre></div>

The lift is unique. Since we

<div class="example"
<panclass=GAPprompt>g;/span>gap<pan class"APprompt">gap> fi222n3a:= CharacterTable( "S3xU4(3).(2^2)_{122}" );;
> i -> intermods3fusnmods3[1] + i * nccl ) );
true
</pre></div>

<> we computetheclassfusionsfrom <spanclass=SimpleMath" ∩ Nto span class="SimpleMath">G. We get two equivalent solutions.

="
gap> tblg:= CharacterTable( "BM142
CharacterTable( "(3^2:D8xU4(3).2^2).2" )
<="GAPprompt">gap> interfusg:= PossibleClassFusions( inter, tblg );;
gap> Length( java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 1
2
gap&t/ span class=GAPinput">Length( RepresentativesFusions( inter, interfusg,tblg ) ;/span>
1
</pre></div>

The approximation of thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div="xample">pre
gap> > CompositionMaps( intermods3fusnmods3[1],
<java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for length 94
> map -> CompositionMaps( interfusb, InverseMap( map ) ) );;
gap> gfusb:= Set(<="GAPprompt>gt;/>span class=G" (fusionmap=interfusn, quick:= rue ;;/pan
<panclass="APprompt">> map -> PossibleClassFusions( tblg, b,
gap> Length( java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0
1
</pre></div>

Finally, we compare the result java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

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

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

<h5>9.3-5 7^1+4:(3 × 2.S_7) → M (May 2009)</h5>

The class fusion of the maximal subgroup U of type 7^1+4:(3 × 2.S_7) of the Monster group M into M is ambiguous.

<div class="example"><pre>
gap> tblu:= CharacterTable( "7^(1+
gap> mdiv="xample"><pre>
span class=GAPprompt>gt< class"">fusm=PossibleClassFusions( blu,m;
gap> Length( RepresentativesFusions( tblu, ufusm, m ) );
2
</pre></div>

The subgroup Uspan class="GAPprompt">gap> Length( interfusg );

<pre>
gap>oadPackage( "atlasrep, false );
true
gap> g1:= AtlasGroup( "7^(2+1+2):GL2
gap> >/ "i: (nfusb, interfusn1 <>
< class"APprompt">ap&t</pan spanclass"APinput">1: Normalizer g1 );/span
gap> g2:= AtlasGroup( "7^(1+4):(3x2.S7)" );;
gap> s2:= SylowSubgroup( g2, 7 );;
gap> n2:= Normalizer );
g;< =java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 129

</pre></divclass="gap&t;<pan => ( <java.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 120

So let <<pFinally,we compare the with theclass fusionthat isstored the table</java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96

We compute the character table of N<div ="example"<pre>

<div class="example"><pre>
gap> gap> tbls:= CharacterTable( "7^(2+1+2):GL2(7)" );;<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> nfusm:= PossibleClassFusions( tbln, m );;
gap> nfuss:= PossibleClassFusions( tbln, tbls );;
gap> sfusm:= PossibleClassFusions( tbls, m );;
gap> nfusm:= SetOfComposedClassFusions( sfusm, nfuss );;
</div>

Now we use the condition that the class fusions from div="xample"<>

< class">> <pan class="APinputm CharacterTable "" ;gap> nfusu:= span class="GAPprompt">gap> ufusm:= PossibleClassFusions
gap> ufusm:= Filtered( ufusm, map2 -> ForAny( nfusu, 
<panclass="APprompt>> map1 -> CompositionMaps( map2, map1 ) in nfusm ) );;
gap> Length( java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 12
1
</ with this propertyarethe ofspan=> and another class of subgroups, of the type 7^2+1+2:GL_2(7). Moreover, it turns out that the Sylow 7 normalizers in the subgroups in both classes have the same order, hence they are the Sylow 7 normalizers in M.

Let C be the centralizer in U of the normal subgroup of order 7; note that C is the <code class="code">7B</code> centralizer on M. We

<div class="example"><pre>
gap> tblc:= CharacterTable( "MC7B" ); 
CharacterTable("71+.A")
gap> cfusm:= PossibleClassFusions( tblc, m );; 
gap> Length( RepresentativesFusions( tblc, cfusm, m ) );
2
gap> cfusu:= PossibleClassFusions( tblc, tblu );;
cfusm: SetOfComposedClassFusionscfusu;
gap> Length( RepresentativesFusions( tblc, cfusm, mspan class=GAPprompt"gap; 
1
</pre></div ">><> < class="GAPinput> )=Size ;/>

<a id="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 4

<h5>9.3-6 subgroup<class=>≅7O_7:/ <=java.lang.StringIndexOutOfBoundsException: Range [120, 119) out of bounds for length 166

<div class="example"><pre>
<
<spanclass="gapg<>< =": CharacterTableF+2 ;<>
<span"">java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129
2
<</>

<t the abelian subgroup orderspan=SimpleMath"37/> =SimpleMath>M</> contains < ="x<> ,inclass < "ode"3</>of< =></pan fact be shown asfollows./p

<div class="example"><pre>
<pan =G"gap&;/> span =GAPinput"nsg=ClassPositionsOfNormalSubgroups ;/
[ [ < =G>>/ span=>: ( tbls, m;gapclass"">&;/ span=> )/>
2187
<pre>
[ [ 1, 4, 5, 6 ] ]
<class>ap;/java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 121
[ [ "1A", "3A", "3B", "3C" ] ]
</pre></java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0

The normalizer S span class="GAPprompt">gap> nfusu:= PossibleClassFusions( tbln, tblu );;

<div class="example"><pre>
<panclass="GAPprompt">gap> s:= CharacterTable( "F3+.2N3A" );
CharacterTable( "S3xO8+(3):S3" )
PowerMap( m, 2 )[4];
4
gap> size_u:= 2 * java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 12
53059069440
< =java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 88
3360
</pre></div>

Using the list of maximal subgroups of &t tblc:= CharacterTable( "MC7B" ); 

<div class="example"><pre>
gap> o8p3:= CharacterTable( "O8+(3)" );;
gap> mx:= List( Maxes( o8p3 ), CharacterTable );;
<panclass=GAPprompt"gap&<spanclass="GAPinput">filt:= Filtered( mx, x -> 3360 mod Index( o8p3, x ) = 0 );
[gap> cfusm:= SetOfComposedClassFusions( ufusm, cfusu );;
 CharacterTable( "O8+(3)M9" ) ]
gap
[ 1120, 1120, 1120 ]
</pre></div>

We compute the possible class fusionsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="example"><pre>
gap> is ambiguous./p>
gap> m:= CharacterTable( "3^7.O7(3):2" );;
gap> uufusm:= PossibleClassFusions( uu, m );;
gap> Length( uufusm );

gap> approx:= List( uufusm<pan class="APprompt">gap> Length( RepresentativesFusions( m, mfust, t ) );
> InverseMap( GetFusionMap( uu, u ) ) ) );;
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 115
> PossibleClassFusions( u, m, rec( fusionmap:= map ) ) ) );;
<span java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
=GAPpromptgap&;/spanspan ""uufuss: PossibleClassFusions(uu s );/span>
gap> Length( uufuss );
8
gap> approx:= List( uufuss, map -> CompositionMaps( map,
<pan class="APprompt"gap&;/pan>span class="APinput">3^7;
gap><2187
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 126
gap> Length( ufuss );
8
</pre</div>

Now we compute the possible class fusions from S into G, and the compositions of [["A","3A","B,"C java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

<div class="example"><pre>
<=""gapgt<span<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 102
< ="">>/>< class="GAPinput">comp:= SetOfComposedClassFusions( sfust, ufuss );;
gap> Length( comp );
< class=""gapg;/pan<G"size_u: 2*( m )[2 ;/>
</pre></div53059069440

It turns out that only one orbit of the possible class fusions java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 4

<div class="example
gap> filt:= Filtered( mfust, map2 -><pan class="GAPprompt">gap> o8p3:= CharacterTable( "O8+(3)" );;
<spanclassG>gapgt;</pan span=G": Filtered( -gt;3360mod ( o8p3, x ;/>
gap> Length( filt );
4
gap> Length( RepresentativesFusions( m, filt, t ) );
1
</pre></div>

The class fusion stored in the pre
/>

<p< id"7C3AC42F8342EE2E" name=X7C3AC42F8342EE2E"<a></pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<></h5>

Let G = ^2E_6(2), and g ∈ G in the conjugacy class <code class="code">3C</code>. Using a permutation representation of G, Frank Lübeck has computed a representation and the character table of the maximal subgroup N = N_G(⟨ g ⟩) of G.

<div class="example"><pre>
gap> t:= CharacterTable( "2E6(2)" );;
gap> pos3CinG:= Position( ClassNames( t ), "3c" );
7
gap> n:= CharacterTable( "2E6(2)N3C" );;
gap> nclasses:= SizesConjugacyClasses( n );;
gap> pos3CinN:= Filtered( [ 1 .. NrConjugacyClasses( n ) ],
> i -> nclasses[i] = 2 );
[ 2 ]
gap> nfust:= PossibleClassFusions( n, t );;
gap> ForAll( nfust, x -> x[ pos3CinN[1] ] = pos3CinG );
true
gap> Size( n ) = 2 * SizesCentralizers( t )[ pos3CinG ];
true
gap> ForAll( Irr( n ), x -> IsInt( x[ pos3CinN[1] ] ) );
true
</pre></div>

The class fusion of N<pre>
gap> rep:= RepresentativesFusions( n, nfust, t );;
<">gap> </div>

We use the fact that g is contained in a subgroup S ≅ Fi_22java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<div class="example"><pre>
gap> s:= CharacterTable( "Fi22" );;
gap> sfust:= PossibleClassFusions( s, t );;</spanjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
<class=GAPprompt>ForAll, xgt; x[6] = pos3CinG)</div>



<div class="example"><pre>
gap> u:= CharacterTable( Maxes( s )[11] );
java.lang.StringIndexOutOfBoundsException: Range [21, 14) out of bounds for length 41

> i -> uclasses[i] = 2 );
[ 2 ]
gap;
4
<></div>

<pComposing ="> and java.lang.StringIndexOutOfBoundsException: Range [213, 212) out of bounds for length 532

<div class="example"><pre>
gap> ufusn:= PossibleClassFusions( u, n );;
gap> comp:= SetOfComposedClassFusions( sfust, ufuss );;
gap> good:= Filtered( nfust, map2 -> ForAny( ufusn,
<spanclass="GAPprompt>> map1 -> CompositionMaps( map2, map1 ) in comp ) );;
gap> Now we compute the possible fusions fromS into G, and the compositions of these maps with the possible class fusions from U into S.
1728
gap> goodrep:= RepresentativesFusions( n, good, t );;sfust:= PossibleClassFusions( s t );
<span="">gap>>< class"( goodrep );
s class="">gap;( );</div

and <span=" automorphism span class"</>, say of orderthreeof <span.Note such automorphism acts nontrivially on the conjugacy of <span="SimpleMath">G</span, for example because class fusion ofspan class"impleMath">GintospanclassSimpleMath"G.3 =⟨ , α ⟩ shows the existenceof orbits of length three, and that the permutation action of α on the classes of G is given by the unique subgroup of order three in the group of table automorphisms of G.

<div class="xample"<pre
span <pan class"APinput">Length
gap> < class="APprompt">gap> Length( RepresentativesFusions( m, filt, t ) );
14
gap>p>he class fusion stored in <strong="pkg>GAP CharacterTable Libraryis one of them
gap> gap> gap> GetFusionMap( m, t ) in filt;
2
gap> gap> pos3CinG ^ alpha = pos3CinG;
true
</pre></div>

 character table <span="SimpleMath>N</> has <pan class=SimpleMath>26 table automorphisms of order three. We do not know which of them (or perhaps the identity permutation) is induced by the restriction α_N of α to N, but the embedding ι: N → G satisfies α ∘ ι = ι ∘ α_N, and we can check each fusion candidate for the existence of a candidate for α_N such that this relation holds.

< class<pre>
gap
<span=GAPprompt">gap> ord3:= Filtered( autn, x -> Order( x ) = 3 );;
<>> ">:= SizesConjugacyClasses( n );;
26
gap;</spanspan classpos3CinNFiltered( n ) ]gapgt;</>< class;(, g;/pan
span="GAPprompt">>gap&span="GAPprompt>gap&;</div>

<p, the ofcandidates for the fusion of<span="SimpleMath">N> in <span="SimpleMath"G>is from to.Moreovercandidate also first we checked/

<div class="java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 4
<spanclass="GAPprompt">gap;/span>inter:= Intersection( good, filt );
[ [ 1, 7, 5, 6, 7, 2, 3, 4, 27, 30, 24, 32, 25, 26, 9, 11, 12, 13,
 10, 14, 19, 19,
 , 59, 75,76, 77, 78,79, 80,64,65,66,60, 81 82, 5,6
 7, 6, 7, 7, 7, 7, 6, 7, 6, 7, 7, 24, 25, 27, 26, 28, 30, 29,
 31,32 31,32,32,32,32, 31,32,31, ,52,52, 52, 52,
 74, 76, 77, 77, 75, 74, 76, 74, 75, 99, 100, 101, 102, 4, 20,
 29,31 ,36 36,4239 40, 41,49,49,49, 49, ,49,
 71, 112, 112, 114, 115, 116 ] ]
</pre>

 Character Table is candidate>

<div class="<div class="example"<pre>
<panclass="GAPprompt">gap>> 
true classGAPprompt>span> <spanclass"APinput"sfustPossibleClassFusions;/span
</></div

Note that the structure of <pre>
gap></span<iv="example">pre>
2
</re

<a( "3^(1+6):^(+)322 )

<h49. span class"Heading">Fusions UsingCommutativeDiagramsInvolvingFactor
Groups</h4>

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

<h5>9.4-1 3.A_7 → 3.Suz (December 2010)</h5>

Themaximal of ="impleMath">A_7> in sporadic Suzuki <spanclassSimpleMath</spanlift to of type class"">3.A_7span> inspan class="SimpleMath">3.Suz</pan>.This beseen from the fact that span class="SimpleMath>3.uzdoes admit a class fusion from A_7.

<div class="example"><pre>
gap> classGAPprompt>/span> <pan classGAPinput( ufuss, x -> x[ pos3CinU1]] pos3CinS;/span>
gap> <spantrue
gapspan="GAPprompt">gap> Size( n ) / Size( u );
<spanp the class fusionsof< class=SimpleMath=SimpleMathN/>in gap> ="GAPinput(,3t/span
[ ]
</pre></div>

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

<div class="example"><pre>
class="GAPprompt">gapgtspan>
gap> Length( 3sfus3t );
6
gap> RepresentativesFusions( 3comp: ( sfust, ufuss );/>
[ 1 ,3, ,8 9, 16, 16, 26,27 28,32 33,34, 47,47 47 48,
 49, 50, 48, 49, 50 ],
 [ 1, 11, 12, 4, 36,37,13 16, 23, 82,83 32, 100,101 44,38
 41, 48, 112 116,, 48,115, 113 ]]
<1728
[ 1, 2, 3 ]
</>

A_7 type subgroups of Suz contain the centre of 3.Suz, we know that the class fusion of these preimages belong to the first orbit. This can be formalized by checking the commutativity of the diagram of fusions between 3.A_7, 3.Suz, and their factors 
<span="GAPprompt">gapgtgap> > map )
> = CompositionMaps( sfust[1span="GAPprompt">gap></pan class"APinput">Length( ord3 );
[ 1, 2, 3 7, 8,9,16,16, 26 27 28, 32,33, 34, 47,47 47,48
 ,50, 48, 49, 50 ]
 [ 1, 3, 2, 7, 9, 8, 16, 16, 26, 28, 27, 32, 34, 33, 47, 47, 47, 48,
 50, 49, 48, 50, 49 ] ]
</java.lang.StringIndexOutOfBoundsException: Range [12, 5) out of bounds for length 12

So the class fusion of maximal <spanp> table"java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 607

<div class="example"><pre>
 s 3 filt;/
true ( ord3))/>
<></

Also the class fusionsp of for fusion reducedfrom four two. , we lucky only candidate also firstcriterion we have checked.

< class>
span="">gap<spanspan">:= CharacterTable( 32.43));/java.lang.StringIndexOutOfBoundsException: Range [104, 103) out of bounds for length 104
gap> gap;Length( PossibleClassFusions( 3s, u ) );
8
</pre></div>

<a id="X82FB71647D37F4FD,76,74 , 99 100,101,102 4, 20,

<h5>9.4-2 U of type S_6. The class fusion from U to gap> <span class="GAPinput"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
( ".1 java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
gap><pre>
java.lang.StringIndexOutOfBoundsException: Range [139, 27) out of bounds for length 397
[[, 3,6 7, 9, 10, 3, 2,9 16,15 ,java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
 [ 1,java.lang.StringIndexOutOfBoundsException: Range [10, 11) out of bounds for length 10
<span/convert -i c Alnuth
1
</pre></div>

In the double cover 2.G of G, U lifts to the double cover 2.U of U (which is unique up to isomorphism). Also the class fusion from 2.U to 2.G is unique up to table automorphisms.

<div class="example"><pre>
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 3 ,15,25,24,26, ]java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
gap> Length( RepresentativesFusions( 2s, 2java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 0
1
</pre></div>

However, the two possible fusions from 2.U to 2.G are lifts of the same class fusion from U to G.

<div="xample"><pre
<">></pan> 
[ 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 11 ]
gap> 2tfust:= GetFusionMap( 2t, t );;
<span="GAPprompt"gap;/span>induced:= Set2sfus2t,xgt; CompositionMaps2tfust,> CompositionMaps([ ]
[ [ 1, 3, 7, 6, 9, 10, 2, 3, 9, 15, 16 ] ]
</pre></div>

The point is that the outer automorphism of S_6 that makes the two fusions from U to G equivalent does not lift to 2.U, and that we have silently assumed a fixed factor fusion from <pre>
gap> other:= OnTuples( 2sfuss, GeneratorsOfGroup( auts )[1] );
[ 1, 1, 2, 4, 4, 3, 3, 5, 6, 6, 8, 7, 9, 11, 11, 10, 10 ]
gap> Set( 2sfus2t, x -> CompositionMaps( 2tfust,
> CompositionMaps( x, InverseMap( other ) ) ) );
[ [ 1, 3, 6, 7, 9, 10, 3, 2, 9, 16, 15 ] ]
</pre></div>

The library table of U stores the class fusion to G that is compatible with the stored factor fusion from 2.U to U.

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

<a id="X7CFBC41B818A318C" name="X7CFBC41B818A318C"></a>

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

java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 62

<h5>9.5-1 </> ( 3sfus3t );</spanjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87

p the Atlas (see <a href="chapBib.html#biBCCN85">[CCN+85, p. 191]</a>), the group G = ^2E_6(2) contains a maximal subgroup U of the type U_3(8).3_1. The class fusion of U into G is ambiguous.

<div class="example"><pre>
gap> s:= CharacterTable( "U3(8).3_1" );;
gap> t:= CharacterTable( "2E6(2)" );;
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).241,48,112,116, ,115,113 ]]
< class="GAPinput">s2fust2:= PossibleClassFusions( s2, t2);
gap> Length[1 2,3]
2
gap> Length( RepresentativesFusions( s2, s2fust2, t2 ) );
1
</pre></div>

< java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 382

<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.

<div class="example"><pre>
gap> 3.A_7> type of class"3Suz determined up table automorphisms. One of these fusions stored on table of <span="SimpleMathjava.lang.StringIndexOutOfBoundsException: Range [254, 252) out of bounds for length 258
 [ 82, 81, 74, 74, 82, 81 ], [ 82, 81, 82, 81, 74, 74 ] ]
gap, 5050,48,49 ]
[ [ 74, 74, 81, 82, 82, 81 ], [ 74, 74, 82, 81, 81, 82 ],
 [ 81, 82, 74, 74, 82, 81 ], [ 81, 82, 82, 81, 74, 74 ],
 [ 82, 81, 74, 74, 81, 82 ], [ 82, 81, 81, 82, 74, 74 ] ]
</>>

None of="">gap;/:("_.(3) ;</java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104

<div class="example"><pre>
<span="">gap;<> class( sfuss2 ->{ [ 31. 36] ;span
[ p<=X82java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
 [ 30 ,28, ,30 ], ,31, ,31 28 ,
 [ 31, 30, 29, 28, 31, 30 ], [ 31, 30, 31, 30, 29, 28 ] ]
</>>

<pre>
</div>

(The same/><

<div java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span classjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 102
[ [ 65, 65, 70, 71 ], =java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 103
</pre></div>

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

<div class="example<></div
<span">>span> </div>

<p="GAPprompt">gap/> class2: GetFusionMap, t;;/

<95-2 span="Heading"><span="SimpleMath">L_3(4).2_1 → U_6(2) (December 2010)</h5>

<>According ">Atlas</> ( <a href="chapBib#biBCCN85 ./)java.lang.StringIndexOutOfBoundsException: Range [137, 136) out of bounds for length 396

<div classjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
gap[1,,2,,4 ,5,,6 , ,,11 ,10]
<span"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 85
27
>

In the automorphic extension gap> s3:=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap>95- spanHjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 113
gap; <spanclass="GAPinput">Length );</pan
2
<span="GAPprompt">gap> Length( RepresentativesFusions( s3, s3fust3, t3 ) );
1
</pre></div>

Herespan GAPprompt;spandiv class="example"><pre>

<div class="example"><pre>
gap [81 82,74 ,81,82], [81 ,81 ,74 74]
gap> comp class="GAPprompt>gap>{ [ 31. 36 ;/>
gap> tfust3:= PossibleClassFusions( t, t3 74 74 ,,82 ][ ,74 ,81 ,82]
gap> sfust = Filtered( sfust, map -> [82,81 ,74 , ,[, ,74 java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
< map ) ;/>
true
</pre></div>

<of =">U</>. are such inside< class="">^ ≅(<, java.lang.StringIndexOutOfBoundsException: Range [170, 169) out of bounds for length 226

<div class="example"><pre>
gap> sfuss3;
[ [ 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 10, 10 ] ]
</pre></div>

These 65 , 71 ,[,70 71][65,,70][65, 71, 70 ]

<div =">pre>
gap> Set( sfust, map -> map{ [ 4 .. 6 ] } );( same fusionthe classesSjava.lang.StringIndexOutOfBoundsException: Range [85, 84) out of bounds for length 141
[ [ 10, 10, 10 ], [ 10, 10, 11 ], [ 10, 10, 12 ], [ 10, "List( s2fust2, x -> x{ [ 28 . 31 ] })
 [ 10, 11, 11 ], [ 10, 11, 12 ], [ 10, 12, 10 ], [ 10, 12, 11 ]pre></div
 [ 10, 12, 12 ], [ 11, 10, 10 ], [ 11, 10, 11 ], [ 11, 10, 12 ],
 [ 11, 11, 10 ], [ 11, 11, 11 ], [<div class="example"><pre>
 [ 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 ] ]
<span="GAPprompt">gap;</spanSet tfust3, map -> map [ 10. 12 ]} );/pan>
[ [ 10, 10, 10 ] ]
</pre>According to <strong="pkg">Atlas/strong> (see <a href="chapBib.html#biBCCN85">CCN+85 p.115</>)the < =>G2)/ maximaljava.lang.StringIndexOutOfBoundsException: Range [202, 201) out of bounds for length 396

This means that the automorphism java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class>
<span<pan class="GAPprompt">gap> t:= CharacterTable( "U6(2)" );;
[gap;<>sfust=PossibleClassFusions,t/span
> gap;<< classjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 117
 [ 1, 3, 7, 10, 12, 11, 15,<<div
 [ 1,3,7 ,,12,15,24 4 14 , 27 ,28 ]
 [ 1, 3, 7, 11, 12, 10, 15, 24, 4, 14, 23, 27, 28, 26 ],
 [ 1, 3, 7, 12, 10, 11, 15<div class="xample><pre>
 [ 1, 3, 7, 12, 11, 10, 15,gap&; s3:= CharacterTable( "L3(4).6" );;
gap> Length( RepresentativesFusions( s="">gap; s3fust3:= , t3;
1
</pre></div>

Finally, we check that a correct1

<div=""<


<"></a> Imposedby Brauer Tables</h4>

The examples in this section show that symmetries can begap;</pan classGAPinputsfust = Filtered(sfust,mapg; ForAny( tfust3

<a id="X7ACC7F588213D5D5" name

<h5>9.6-1 <span<> the ="impleMath">U. There are three such classes inside U^' ≅ L_3(4), which fuse to one class of U.3.

<>It that tables ambiguities of class fusions between the corresponding ordinary tables. An easy example is the class fusion of L_2(16).4 into J_3.2. The1, 2,3,, 4,4 5,, 2,46 ,8 8 ]

<div class
< class> span=G:= CharacterTable "().4" );;</pan
gap></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> fus:= span class="GAPprompt">gap> Set( sfust, map -> ]} ;/>
[ [ 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 ],
 [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
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 <></div>

<Wejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 305

<div class="example"><pre>
<span class="GAPprompt"
< class&tgap> gap> filt:= Filtered( sfust, map -> IsSubset( map, [ 10 .. 12 ] ) );
[ 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 ],
 [, ,3 ,14, 16 ,2 ,7 ,19 19,22, 22,23,23 ]]
</pre></div>

Now the remaining wrong fusion is excluded by the fact that the table automorphism of J_3.2 that swaps the two classes of element order 17 –which swaps two of the possible class fusions– does not live in the <1

<div class="example"><pre
ap; <span="GAPinput">smod2: s 2;
<span=">> gap> for map in filt do
> modmap:= CompositionMaps( InverseMap( GetFusionMap( tmod2, t ) ),
> CompositionMaps( map, GetFusionMap( smod2, s ) ) );</java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 217 out of bounds for length 114
> List( Irr( tmod2
< AddSet( admissible );>> od;
gap> admissible;
[ [ 1, 2, 3, 6, 14, 16, 15, 2, 5, 7, 12, 19, 19, 22, 22, 23, 23 ] ]
<</>

The test of

<div class="example"><pre>
gap&<pan =GAPprompt>gapg;span span class"APinput">:CharacterTable J32 ;;span>
true
</pre></div>

We see that p-modular tables alone determine the class fusion uniquely; in fact the primes 2 and 
gap> GetFusionMap( s, t ) in admissible;
true
</pre></div>

Remark:

In May 2015, the 19-modular character table of J_3 has been corrected, by swapping the two classes of element java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

<h5>9.6-2 L_2(17) → S_8(2) (July 2004)</h5>

The class fusion of the maximal subgroup M ≅ L_2(17) of G = S_8(2) is ambiguous.

<div class="example"><pre>
gap> m:= CharacterTable( "L2(17)" );;
gap> t:= CharacterTable( "S8(2)" );;
gap> mfust:= PossibleClassFusions( m, t );;
gap> Length( RepresentativesFusions( m, mfust, t ) );
4
</pre></div>

The Brauer tables for M and G determine the class fusion up to table automorphisms.

<div class="example"><pre>
gap> filt:= CTblLib.Test.Decompositions( m, mfust, t );;
gap> repr:= RepresentativesFusions( m, filt, t );;
gap> Length( repr );
1
gap> GetFusionMap( m, t ) in repr;
true
</pre></div>

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

<h5>9.6-3 <java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0

It can happen that<pan class"GAPprompt">apgttmod2: tmod2;;/

<spanclass=GAPprompt">/>< class"APinput>or map in do/span

<div class="example"><pre>
<pan =""gap>>/span>= CharacterTable("3");;/span
gap> sfustspan class"">gt/>List Irr(tmod2 ) chi -gt;chi ,"onnegative")then/pan>
[ [ 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, 11t>gt<span<pan "APinput" ;</pan
 [1 2,4 6, 7 11,12 ,13 14,21 ,
 [ 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, 10, 11, 12, 14, 13, 20, 21 ],

 ,2, 4 ,6 11,12 ,14 , 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 ] ]
gap> fusreps:= RepresentativesFusions( s, sfust, t );
[ [ 1, 2, 4, 6, 7, 10, 11, 12, 13, 14, 20, 21 ] ]
<pre<>

The Galois automorphism that permutes the three classes of element order <java.lang.StringIndexOutOfBoundsException: Range [0, 81) out of bounds for length 26

<div class="example"><pre>
gap> tmod19:= t mod 19;
BrauerTable( "J3", 19 )
gap> deg110:= Filtered( Irr( tmod19 ), phi -> phi[1] = 110 );</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ Characterjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [ 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,
 E(9)^2+E(9)^3+2*E(9)^4+2*E(9)^5+E(9)^6+E(<>9.6- span class"eading"<classSimpleMath>L_217)→()/ J 2004<span<>
 0,E(17)3+()5+17^+17^+(1710+)+(1712
 +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> deg110[1][ 9A ];
-2*E(9)^2+E(9)^3-E(9)^4-E(9)^5+E(9)^6-2*E(9)^7
gap> AtlasIrrationality( "-1+2y9+&4" ) = deg110[1][ 9A ];
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>
gap> smod19:= s mod 19;
BrauerTable( "L2(19)", 19 )
gap> compatible:= [];;
gap> for map in sfust do
> comp:= CompositionMaps( InverseMap( GetFusionMap( tmod19, t ) ),
> CompositionMaps( map, GetFusionMap( smod19, s ) ) );
> rest:= List( Irr( tmod19 ), phi -> phi{ comp } );
> if not fail in Decomposition( Irr( smod19 ), rest, "nonnegative" ) then
> Add( compatible, map );
> 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>

<>oreover, the subgroupsof those table automorphisms of the ordinary tables that leave the set of compatiblefusions maketwo on set Indeed,the two orbits belong to essentially different of of <panclass="SimpleMathφ/./>

<div class="example"><pre>
gap> reps:= RepresentativesFusions( s, compatible, t );
[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14, 20, 21 ],
 1,2,4,6,7,12 ,11,13,14 20,21 ]
gap> compatiblemod19:= List( reps, map -> CompositionMaps(
<pan ="APprompt"gt;/pan> =GAPinput> InverseMap(GetFusionMap , t ,/pan
> CompositionMaps
[ [ 1, 2, 4, 6, 7, 11, 12, 10, 13, 14 ],
 [ 1, 2, 4, 6, 7, 12, 10, 11, 13, 14 ] ]

<class=GAPpromptgapgt;/pan> < class=GAPinput"dec=Decomposition((smod19 ) ,"nonnegative ;/>
[ [ 0, 0, 1, 2, 1, 2, 2, <spanclass""gapgt<>spanclass="GAPinput">repr:= RepresentativesFusions( m, filt, t );;
<pan class="GAPprompt">gap> List( Irr( smod19 ), phi -> phi[1] );
[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 ]
</pre></div>

In order to decide which java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0

< class=example"<re>
gap&java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
rec( groupname := "J3", identifier := [ <>he class betweenthe ordinary character tables of < class"">L_2(9) < class"impleMath"J_3 isuniquetotable, size := 3420,
 : 1 subgroupname= "L2(19, version : 1 )
<class="GAPprompt">gap;/span><spanclass"">ens: OneAtlasGeneratingSetInfo("",Characteristic,19,<>
> =GAPpromptgap&tt: ( J");/pan>
<spanclass""gapgt<pan=GAPinputgens= ( ),
 < immutable compressed matrix 110x110 over GF(19) > ],
 groupname := "J3", id := "",
 identifier := [ "J3", [ "J3G1-f19r110B0.m1", "J3G1-f19r110B0.m2" ],
 1,19 ]repname ="-f19r110B0" repnr = 35,
 ring := GF(19), size := 50232960, standardization := 1,
 type := "matff" )
gap> ,
 < immutable compressed matrix 110x110 over GF(19) > ]
gap> module:= GModuleByMats( restgens, GF( 19 ) );;
gap> [1,2 ,7 ,12 ,11 ,13 ,21 ]
gap> Length( facts );
7
<"APprompt">gap> List( facts &t []dimension ;/
,,,11 , ,
gap></spanpre>/iv>
[ 1, 2, 1, 2, 2, 1, 1 ]
</<div

This E(9)^2(9)329^+*9^+E(9)^6+E9^, 2 2,1 0

<div class="example"><pre>
gap> GetFusionMap( s, t ) = reps[1];
true
</pre></div>

<pem>/>

In May 2015, the 19-modular character table of J_3 has been corrected, by swapping the two classes of element order 17. This affects the above computations only in one place, java.lang.StringIndexOutOfBoundsException: Range [0, 279) out of bounds for length 12

<a id="X8225D9FA80A7D20F" name="X8225D9FA80A7D20F"></a>

<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, <pan ">gt;/>< =" = java.lang.StringIndexOutOfBoundsException: Range [89, 88) out of bounds for length 123

The group representations "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 74

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

<a id="X7AE2962E82B4C814" name="X7AE2962E82B4C814"></a>

<h5>9.7-1 U_3(3).2 → Fi_24^' (November 2002)</h5>

The group G = Fi_24^' contains a maximal subgroup

<class"><>
gap> t:= CharacterTable( "F3+" );
CharacterTable( "F3+" )
gap> s:= CharacterTable( "U3(3).2" );
CharacterTable( "U3(3).2" )
gap> tnames:= ClassNames( t, "ATLAS" );
[ "1A", "2A", "2B", "3A", "3B", "3C", "3D", "3java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 41
 "5A", "6A", "6B", "6C", "6D", "6E", "6F", "6G", "6H", "6I", "6J",
 6K"", """8 8" "8","A" "B" 9 9D ""
 "9F", "10A", "10B", "11A", "12A", "12B", "12C", "12D", "12E",
 "12F", "12G", "12H", "12I", "12J", "12K", "12L", "12M", "13[1 ,5 ,9,11 ,15 ,19 ]
 "14A", "14B", "15A", "15B", "15C", "16A", "17A", "18A", "18B",
 "18C", "18D", "18E", "18F", "18G", "18H" "0" 20"21"
 "21B", "21C", "21D", "22
 "24D"<="<
 "29A", "29B", "30A", "30B", "33A", "33B", "35A", "36A", "36B",
 "36C", "36D", "39A", "39B", "39C", "39D", "42A", "42B", "42C",
 "45A", "45B", "60A" ]
="GAPprompt"gap/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 101
[ 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 2, 4, 6, 8, 12, 12 ]
<"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 138
> "12M" ], x -> Position( tnames, x ) );
[ 1, 3, 7, 8, 11, 11, 22, 25, 28, =[&t; x 19)&,
gap> sfust:= PossibleClassFusions( s, t, rec( fusionmap:= sfust ) );
[ [ 1, 3, 7, 8, 11, 11, 22, 25, 28, 50, 3, 9, 23, 28, 43, 43 ],
 [ :="
gap> OrdersClassRepresentatives( s );
, 4, 6, 8,12 12 java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
</pre></div>

So we still have two possibilities, which differ on the outer classes of element order 4 and 12.

Our idea is to take a subgroup We take U = N_H(⟨ g ⟩)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class"example">pre>
<>emRemark:/>
["U3(3)", "3(+)S" L3)" 2(+).3",42 ]
gap> SizesCentralizers( s );
[ 12096,192,216,, 96,32,24 ,8 ,48 ,6,8 ,12]
gap> u:= CharacterTable( java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 0
gap> ufuss:= GetFusionMap
[ 1, 2, 11, 3, 4, 5, 12, 7, 13, 9, ,15,16 10 ]
</pre></div>

< elements in the class <code class="code">3Dcode> of <panclass="impleMath"G/span whose centralizeristhefull <codeclass=code">D</code> centralizer in G.

<div
gap> 3Dcentralizer:= SizesCentralizers( t )[7];
153055008
gap> cand:= [];; 
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 94
> m:= CharacterTable( name );
> if ForAny( [ 1 .. Length( mfust ) ], 
> i -> mfust[i] = 7 and SizesCentralizers( m )[i] = 3Dcentralizer ) 
> then
> The group G = Fi_24^' contains a m span "></> span class=SimpleMath>().2/>. the of<pan ="G> < "">/ getslot essentially a java.lang.StringIndexOutOfBoundsException: Range [360, 359) out of bounds for length 906
<spanclass"">&t
> od;
gap> cand;
[ CharacterTable( "3^7.O7(3)" ),
 CharacterTable( "3^2.3^4.3^8.(A5x2A4).2" ) ]
</pre></div>

For these two groups M, we show that the possible class fusions from U to G via M factorize through H only if the second possible class fusion from H to G is chosen.

< class"example">>
gap> possufust:= List( sfust, x -> CompositionMaps( x, ufuss ) );
 ,3 3 ,8 11 ,22 ,28 28,43 ,50 ,java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 ] ]
gap> m:= cand[1];;
gap> ufusm:= PossibleClassFusions( u, m );;
gap> Length( ufusm );
242
gap> comp:= java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 64
gap> Intersection( "F, 12", 12" 12" 12J 12, "L" 12M,"3"
[1 ,,7,8, 11 ,,23, 28, 28,50,50, 50 ] ]
gap> m:= cand[2];;
gap> ufusm:= PossibleClassFusions( u, m );;
gap> Length( ufusm ); 
256
gap> comp:= List( ufusm, x -> CompositionMaps( GetFusionMap( m, t ), x ) );; 
gap> class="GAPprompt">gap> OrdersClassRepresentatives
[ [ 1, 3, 3, 7, 8, 11, 11, 22, 23, 28, 28, 50, 50, 50 ] ]
</pre></div>

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

<div class="example"><pre>
gap> GetFusionMap( s, t ) = sfust[2];
true
</pre></div>

<a id=

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

The class fusion of maximal subgroups U of type L_2(13).2 in G = Fi_24^' is ambiguous.

<div class="example"><pre>
<">&t; t:= CharacterTable( "F3+" );;
gap> u:= CharacterTable( "L2(13).2" );;
gap><>We take =N_H( span < =SimpleMath>is an element in the first class of order three elements of H; this is a maximal subgroup of H, of order 216.
gap> repr:= RepresentativesFusions( u, fus, t );;
<"">g;/><span=GAPinput>(repr)</
3
</pre></div>

In <a [ 12096, 192,216,18,96 32,24 ,8 ,48,48,6 8 ,12]

<div class="example"><pre>
gap> filt:= Filtered( repr, x -> t.2b in x and t.3d in x and t.7b in x );
[ [ 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 ] ]Candidates <classjava.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 322
gap> ClassNames( t ){ [ 43, 50 ] };
[ "12f", "12m" ]
</pre></div>

So we have to decide whether U contains elements in the class <code class="code">12F</code> or in <code class="code">12M</code> of G.

The order 12 elements in question lie inside subgroups of type 13 : 12 in U. These subgroups are clearly contained in the Sylow 13 normalizers of G, which are contained in maximal subgroups of type (3^2:2 × G_2(3)).2 in G; the class fusion of the latter java.lang.StringIndexOutOfBoundsException: Index 485 out of bounds for length 94

<class="xample"><pre>
gap> pos:= Position( OrdersClassRepresentatives( t ), 13 );
51
<class"">gt<span<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 99
234
<classGAPprompt"gt;</ G">ClassOrbit(t ;/panjava.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
[ 51 ]
gap> cand:= [];; 
<>Forthese twogroups < class=SimpleMath">M, we show that the possible class fusions from U to G via M factorize through H only if the second possible class fusion from H to G is chosen.
> m:= CharacterTable( name );
> pos:= Position( OrdersClassRepresentatives( m ), 13 );
> if pos <> fail and 
> SizesCentralizers( m )[ pos ] = 234 
<classG>gt > 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 × G_2(3)).2, we look at java.lang.StringIndexOutOfBoundsException: Range [0, 167) out of bounds for length 3

<div,3 3 ,,11,11,22, 23, 28, ,50 50 ] ]
gap> der:= ClassPositionsOfDerivedSubgroup( s );;
gap> outer:= Difference( [ 1 .. NrConjugacyClasses( s ) ], der );;
gap> sfust:= PossibleClassFusions( s, t );;
gap> imgs:= Set( Flat( List( sfust, x -> x{ outer } ) ) );
[ 2, 3, 10, 11, 15, 17, 18, 19, 21, 22, 26, 44< ="&;<> span class=GAPinput>(, )= [2]<>
 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="

Finally, we check whether the correct fusion is stored in the GAP Character Table java.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 98

< "java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 100
true
<<div

<a id="X7E9C203C7C4D709D" java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 12

h59.- span=>span =>→Bspan Ajava.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 97

The sporadic simple group B contains a maximal

<div class="example"><pre>
gap> b:= CharacterTable( "B" );;
gap> m11:= CharacterTable( "M11" );;
gap> m11fusb:= PossibleClassFusions( m11, b );;
gap> Length( m11fusb );
1
gap> CompositionMaps( ClassNames( b, "ATLAS" ), Parametrized( m11fusb ) );
"1A", ["B,"D ,["3,"",
 [ "4B", "4E", "4G", "4H", "4J" ], [ "5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 6 6" 6H,"6","J" ,
 [ "8B", "8E", "8G", "8J", "8K", "8L", "8M", "8N" ],
 [ "8B", "8E", "8G", "8J", "8java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>

 span="java.lang.StringIndexOutOfBoundsException: Range [99, 97) out of bounds for length 1046

Now we use the embedding of U into B via M and Th for determining the class fusion of M into B. The class fusion of the embedding of U via Th is uniquely determined.

<div class="example"><pre>
gap> th:= CharacterTable( "Th" );;
gap> s5:= CharacterTable( "S5" );;
gap> s5fusth:= PossibleClassFusions( s5, thspan class"APprompt">&;</> class=": [; 
[ [ 1, 2, 4, 8, 2, 7, 11 ] ]
gap> thfusb:= PossibleClassFusions( th, b );;
gap> s5fusb:= Set( thfusb, x -> CompositionMaps( x, s5fusth[1] ) );
[ [ 1, 5, 7, 19, 5, 17, 29 ] ]
</pre></div>

Also the class fusion of U into M is unique, and this determines the class fusion of M</ Add( cand, m );

<div class="example"><pre>
gap> s5fusm11:= PossibleClassFusions( s5, m11 );
[ [ 1, 2, 3, 5, 2, 4, 6 ] ]
gap> m11fusb:= Filtered( m11fusb,
> map -> CompositionMaps( map
[ 1 ,7,17 19 29 45,45 ,54]java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
gap> CompositionMaps( ClassNames( b, "ATLAS" ), m11fusb[1] );
[ "1A", "2D", "3B", "4J", "5B", "6J
</pre></div>

< that"">10<spantype of span =""M<>are < ="Th/> not have helped us, since these subgroups do not contain elements of order 6, and two possibilities would have remained.)

<a id="X85821D748716DC7E" name="X85821D748716DC7E"></a>

<h5>9.7-4 L_2(11):2 → B (April 2009

The sporadic simple group B contains a maximal subgroup L of the type L_2(11):2 whose class fusion is ambiguous.

<div class="example"><pre>
gap> b:= CharacterTable( "B" );;
gap> l:= CharacterTable( "L2(11).2" );;
gap> lfusb:= PossibleClassFusions( l, b );;
gap>
[ "1A", [ "2B", "2D" ], [ "3A", "3B" ], [ "5A", "5B" ],
 [ "5A", "5B" ], [ "6C", "6H", "6I", "6J"spanclassGAPpromptgapgt; GetFusionMap( u, t ) = filt[2];</spanjava.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
 [ "4D", "4E", "4F", "4G", "4H", "4J" ], [ "10C", "10E", "10F" ],
 [ "10C", "10E", "10F" ],
 [ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12P", "12Q",
 "12R", "12S" ],
 [ "12E", "12F", "12H", "12I", "12J", "12L", "12N", "12P", "12Q",
 "R, "12" ]]
</pre></div>

According to <a href="chapBib.html#biBWil93a">[Wil93a, Thm. 12.1]</a>, L contains no <code class="code">5A</code> elements of Bdiv class=example<>

In the same way as we proceeded in Section <a href="chap9.html#X7E9C203C7C4D709D">9.7-3</a>, we use the embedding of <pre>
< class">gt;/>< "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 98
<class"g;/>< =">= L2";
 ["" 6E,6" 6I"," ,java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
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 -&pNow we use < class"SimpleMath"U<>into<class"impleMath">B</">Bvia< ="</>and <pan ="Th</> the fusion ""></pan> into <pan ="impleMath<> Theclass fusionof embedding of java.lang.StringIndexOutOfBoundsException: Range [340, 339) out of bounds for length 428
[ [ 1, 5, 7, 19, 19, 29, 54, 5, 15, 53, 53, 73, 73 ] ]
</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

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

<>9.- <pan class=Heading>span class="SimpleMath">L_3(3) → B (April 2009)</h5>

The sporadic simple group B contains a maximal subgroup class=GAPprompt>< <class">=java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 109

<div class="example"><pre>
gap> b:= CharacterTable( "B" );;
<pan class="GAPprompt">gap> t:= CharacterTable( "L3(3)" );;
gap> tfusb:= PossibleClassFusions( t, b );;
gap></java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 26
36
</pre></div>

<Accordingto<href=#java.lang.StringIndexOutOfBoundsException: Range [48, 46) out of bounds for length 696

<div class="example"><pre>
gap> m:= CharacterTable( "3^2.3^3.3^6.(S4x2S4)" );;
gap> g:= PSL(3,3);;
gap> mx:= MaximalSubgroupClassReps( > map -> Composit[])=1];/>
gap> u:= First( mx, x -> Size( x ) = 432 );;
gap> 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,
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 127
gap> tfusb;
[ 1,5, 6 ,12 , ,41 ,75 ,75 ,
 [ 1, 5, 7, 6, 12, 28, 41, 41, 75, 75, 75, 75 ],
 [1,5 , ,, 28 ,,75 , ,java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
 [ 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>

Now we use that T does not contain <code class="code">4E</code> elements of B (again see <a href="chapBib.html#biBWil99">[Wil99, Section 9]</a>). Thus the last of the five java.lang.StringIndexOutOfBoundsException: Index 249 out of bounds for length 107

<div class="example"><pre>
gap> ClassNames( b, "ATLAS" ){ [ 12, 17 ] };
[ "4E", "4J" ]
<pre>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12



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

<a id

<h5>9.7-6 L_2(17)). B/pan> (arch 2004/5>

The sporadic simple group B contains a maximal subgroup U of the type L_2(17).2</java.lang.StringIndexOutOfBoundsException: Index 176 out of bounds for length 0

<div class="example"><pre>
<pan class="GAPprompt">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 ],
 [ 1, 5, 7, 15, 44, 44, 47, 47, 47, 91, 5, 29, 90, 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<pre><div>

<div class="example"><pre>
The sporadic simplegroup<class=">B</pan>contains amaximal <class="impleMath</>of typespan ="SimpleMath>(3)/span> classfusion is ambiguous.
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, 46, 46, 91, 5, 29, 90, 90, 90, 90, 96,
 96, 96 ] ]
</pre></div>

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

<divclass"xample"><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<pan class=""gap&tyle='color:red'>span span class"">x: MaximalSubgroupClassReps(g);/>

The sporadic simple group gap&< =java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 94

<div class="example"><pre>
gap> u:= CharacterTable( "L2(49).2_3" );;
gap> b:= CharacterTable( "B" );;
gap> ufusb:= PossibleClassFusions( u, b );;
gap> Length( RepresentativesFusions( u, ufusb, b ) );
2
< ="">&;
[ [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 128,
 128, 128, 15, 71, 71, 89, 89, 89, 89 ],
 [ 1, 5, 7, 15, 19, 28, 31, 42, 42, 71, 125, 125, 128, 128, 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 Now use <class="SimpleMath">T does not contain <code class="code">4E</code> elements of B (again see <a href="chapBib.html#biBWil99">[Wil99, Section 9]</a>). Thus the last of the five candidates is the correct class fusion.

For that, we compute the character table of N from a representation of U. Note that U [ "E",""java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14

<div class="example"><pre>
gap> g:= SL( 2, 49 );;
gap> gens:= GeneratorsOfGroup( g );;
gap> f:= GF(49);;
gap> gap> for i in [ 1 .. Length( gens ) ] do
(17. B/span>(March 2004)<h5
> mats[i]{ [ 3
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 144
> od;
gap> fieldaut:= PermutationMat( (1,3)(2,4), 4, f );;
gap> diagaut:= IdentityMat( 4, f );;
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> gap> ntbl:= CharacterTable( n );;
</pre></div>

<>Now computethepossibleclass fusions <class"SimpleMath"N/pan into<spanclass"">B/,via Sylownormalizer span class=>

<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[[1,5 ,15 44,,46 ,46 ,5 29 90 ,90,90, 96

Although there are several possibilities, this information is enough to exclude one of the possible fusions of <pre>
gap> nfusu:= PossibleClassFusions( ntbl, u );;
gap> Length( nfusu );
4
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
> 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 ] ]
 ClassNames( b "ATLAS") filt1 ;
[ "1A", "2D", "3B", "4H", "5B", "6I", "7A", "8K", "8K", "12Q", "24L",
 "24L", "25A", "25A", "25A", "25A", "25A", "4J", "12R", "12R",
 "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

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. Firstspan ="APprompt">gap>/span>< class=GAPinput>ength RepresentativesFusionsu ufusb b ))/span>

<div class="example"><pre>
gap> NrConjugacyClasses( u ); NrConjugacyClasses( act );
24
24
gap> u:= CharacterTable( "L2(49).2_1" );;
gap> g:= Group( Concatenation( mats, [ diagaut ] ) );;
gap> gap> act:= Action( g, orb, OnLines );;
gap> Size(act );
117600
<="&;<=G>java.lang.StringIndexOutOfBoundsException: Range [81, 80) out of bounds for length 121
51
51
gap> gap> gap> g:= Group( Concatenation( mats, [ fieldaut ] ) );;
gap> orb:= Orbit( g, v, OnLines );;
gap> act:= Action( g, orb, OnLines );;
gap> NrConjugacyClasses( u ); NrConjugacyClasses( act );
27
27
</pre></div>

The group L_2(49).2_1 can be excluded because no class fusion into B is possible.

<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 possible class fusions into B. However, the Sylow/>div>

<div class="example"><pre>
gap> n:= Normalizer( act, SylowSubgroup( act, 7 ) );;
gap> /java.lang.StringIndexOutOfBoundsException: Range [124, 123) out of bounds for length 124
0
</pre></div>

<a id="X7EAD52AA7A28D956" name="X7EAD52AA7A28D956"></a>

<

The Chevalley group G = G_2(5) contains a maximal subgroup U of thespan class="GAPprompt">gap> nfusu:= PossibleClassFusions( ntbl, u );;

<div class="example"><pre>
gap> t:= CharacterTable( "G2(5)" );;
gap> s:= CharacterTable( "2^3.L3(2)" );;
gap> <class=> {java.lang.StringIndexOutOfBoundsException: Range [93, 92) out of bounds for length 105
G"gapg;<> span=> , </java.lang.StringIndexOutOfBoundsException: Range [107, 106) out of bounds for length 107
[ [ 1, 2, 2, 5, 6, 4, 13, 16, 17, 15, 15 ],
 [ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 101
[ 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="java.lang.StringIndexOutOfBoundsException: Index 142 out of bounds for length 0

<div class="example"><pre>
gap> g:= AtlasGroup( "G2(5)" )java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> u:= AtlasSubgroup( "G2(5)", 7 );;
gap> Size( u );
1344
gap> repeat
> x:= Random( u );
> until Order( x ) = 6;
gap> siz:= Size( Centralizer( g, x ) );
36
gap> Filtered( [ 1 .. NrConjugacyClasses( t ) ],
> i -> SizesCentralizers( t )[i] = siz );
[ 14 ]
</pre></div>

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 ); 
true
</pre></div>

<a id="X79617107849A6CEA" name="X79617107849A6CEA"></a></p<pan ="gap&t;/<=java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 119

<597-<span=< classSjava.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 114

The sporadic simple group B contains a maximal subgroup M of the type 5^{1+4}.2^{1+4}.A_5.4 whose class fusion is ambiguous.

<classe>pre
gap> b:= CharacterTable( "B" );;
gap> m:= CharacterTable( "5^(1+4).2^(1+4).A5.4" );;
gap> mfusb:= PossibleClassFusions( m, b );;
gap> Length( mfusb );
4
gap> repres:= RepresentativesFusions( m, mfusb, b );; 
<class=GAPprompt>g;span < class"APinput>( repres ;/span>
2
</pre></div>

The restriction of the unique irreducible character of degree 4371 distinguishes the two possibilities,

<div class="example"><pre>
gap> char:= Filtered( Irr( b ), x -> x[1] = 4371 );;
gap> Length( char );
1
gap> rest:= List( repres, map -> char[1]{ map } );;
<span<a =""name""/></pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
gap> constit:= List( scprs,
> gap> List( java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 0
[ 384,480,400 400,500,1000,1200 ]
 [1,6,384,480 , 300,500,1000,1600 ]]
</pre></div< class"APprompt"gap&;<pre>
gap> g:= AtlasSubgroup( "B", Dimension, 4371, Ring, GF(3), 21 );;
gap> module:= GModuleByMats( GeneratorsOfGroup( g ), GF(3) );;
gap> dec:= MTX.CompositionFactors( module );;
<="gap&t;</div>

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0

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

<h5>9.7-10 The fusion from the character table of <gap> u:= AtlasSubgro;/>
into the table of marks (January 2004)<< class=GAPprompt">ap&;/< class"APinput"Size(u )/>

p can happen class fusion from ordinary character of groupspanclass"">G/>isnotuniqueuptotable automorphisms of the character table of span class"impleMath>G<span./>

> an, span class="SimpleMath">G=^22_()2<> in <class"impleMath><>.<>

G contains four classes of cyclic subgroups of order 7. One contains the elements in the normal subgroup of type 7^2, and the other three are preimages of thespanclass="APprompt">gap> siz:= Size( Centralizer( g, x ) );

<div class="example"><pre>
gap> tbl:= CharacterTable( "7^2:2psl(2,7)" );
7^psl,)"
gap> tom:= TableOfMarks( tbl );
TableOfMarks( "7^2:2L2(7)" )
<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 10, 9, 16, 7, 10, 9, 8, 16 ],
 [ 1, 6, 2, 4, 3, 5, 13t
 [1 , ,4 ,5 ,,7,,9 ,16,7 9 ,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,p>a id=X79617107849A6CEA"name"79617107849">>
gap> reps:= RepresentativesFusions( tbl, fus, Group(()) ); 
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ],
7 ,9,16 ,10, 9 ,16
gap> AutomorphismsOfTable
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> gap> 
gap> perms2:= PermCharsTom( reps[2], tom );;
gap<> < class"">epres: RepresentativesFusions(, , b ; />
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 are permutations of rows and columns that act as an S_3 on them.

Note that an S_3 acts on the classes in question in the rational character table. So it is due to the irrationalities in the character table that it contains more information.</div class="xample><re>

<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 7c 7d 7e 14a 7f 7g 7h 7i
 2P 1a 7a 1a 2a 3a 3a 4a 4a 7b 7c 7d 7e 7b java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 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 p seethat does "3/> two correct ./>
X.2 3 3 3 -1 . . 1 1 B B B B B /B /B /B /B
X.3 3 3 3 -1 . . 1 1 /B /B /B /B /B B B B B
X.4 6 6 6 2 . . . . -1 -1 -1 -1 -1 -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
8 44- . 1 ..-- -/ B - - -java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
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. 48 1 .. .. -1 1 - 6 1-1 -
X. 48-1 .. . .C 1 C / /C-java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
14 .C C/ - /D- java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
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

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

X.1 1
X.2 /B
X.3 B
 -1
X.5 .
X.6 1
X.7 /B
X.8 B
X.9 1
X.10 1
X.11 -1
X.12 .
X.13 .
X.14 .
X.15 .
X.16 .
X.17 .
X

A = E(8)-E(8)^3
 = Sqrt(2) = r2
B = E(7)+E(7)^2+E(7)^4
 = (-1+Sqrt(-7))/2 = b7
C = 2*E(7)+2*E(7)^2+2*E(7)^4
 = -1+Sqrt(-7) = 2b7
D = -3*E(7)-3*E(7)^2-2*E(7)^3-3*E(7)^4-2*E(7)^5-2*E(7)^6
 = (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, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 1, 6, 2, 4, 3, 5, 13, 13, 7, 8, 9, 10, 16, 7, 9, 10, 8, 16 ] ]
</pre></divX. 3 3 - .1 / /B/ /B BB

We could say that thus the fusion is unique up to table automorphisms and automorphisms of the table of marks. But since a group is associated. 2 . - -1 1 - - --1 1-java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

<div class="example"><pre>
gap> g:= UnderlyingGroup( tom );;
gap> tg:= CharacterTable( g );;
gap> tgfustom:= FusionCharTableTom( tg, tom );;
gap> trans:= TransformingPermutationsCharacterTables( tg, tbl );;
gap>1248 1. . . ..6 - 1 - . 6 - 1 -
gap> orbits:= List( reps, map -> OrbitFusions( AutomorphismsOfTable( tbl ),
> map, Group( () ) ) );;
gap> PositionProperty( orbits, orb -> tblfustom in orb );
2
gap> PositionProperty( orbits, orb -> FusionToTom( tbl ).map in orb );
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 <pre>
gap> u42:= CharacterTable( "U4(2)" );;
gap> u43:= CharacterTable( "U4(3)" );;
gap> u42fusu43:= PossibleClassFusions( u42, u43 );;
gap> Length( u42fusu43 );
4
gap> Length( RepresentativesFusions( u42, u42fusu43, u43 ) );
1
</pre></div>

More precisely, take the outer automorphism group of U_4(3), which is a dihedral group of order eight, and consider the subgroup java.lang.StringIndexOutOfBoundsException: Index 170 out of bounds for length 11

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 X10 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

<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 ]
< =)(7^+(74
[ [ 16, 17 ], [ 18, 19 ] ]
gap> CompositionMaps( java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 28
[ [ " -*()3*7^-*)33E(7^4-E7^-*(7^
</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" );;
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 ]
<class="APprompt">gap> act3:= Filtered( InverseMap( fus3 ), IsList );
[ [ 4, 5 ], [ 11, 12 ], [ 13, 14 ], [ 16, 18 ], [ 17, 19 ] ]
gap> </div>

<>The class="pkg">Atlas states that the permutation character induced by the first class of U_4(2) type subgroups is <code class="code">1a+35a+90a</code>, which means that the subgroups in this class contain <code class="code">9A</code> and <code class="code">9B</code> elements. Then the permutation character induced by the second class of U_4(2) type subgroups is <code class="code">1a+35b+90a</code>, and the subgroups in this class contain <code class="code">9C</code> and <code class="code">9D</code> elements.

So we choose appropriate fusions for the two classes of maximal U_4(2) type > map, Group(

< class="example"><pre
gap> firstfus:= First( u42fusu43, x -> IsSubset( x, [ 16, 17 ] ) );
[ 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 ] ) );
[ 1, 2, 2, 3, 3, 4, 5, 7, 8, 9, 10, 10, 11, 11, 12, 11, 18, 19, 20,
 20 ]
</pre></div>

Let us now consider the central extension 3_1.U_4(3). Since the Schur multiplier of U_4(2) has order two, the U_4(2) type subgroups of U_4(3) lift to groups of the structure 3 × U_4(2) in 3_1.U_4(3). There are java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

More precisely, each of the four fusions from U_4(2) to U_4(3) has exactly two lifts. The four lifts of those fusions from U_4(2) to U_4(3) with <code class="code">9A</code> and <code class="code">9B</code> in their image form one orbit under the More precisely, take the outer automorphism group of U_4(3), which is a dihedral group of order eight, and consider the subgroup generated by its central involution (this automorphism is denoted by 2_1 in the Atlas) and another involution called 2_3 in the group on the classesofU_4()/span>and acts thefourpossibleclass fusionsof into U_4(3). In fact, this action is transitive.

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

This solves the ambiguity: Fusions from each of the two orbits occur, and we can assign them to the two classes of subgroups by the choice of the fusions from U_4(2) to U_4(3).

The reason for the asymmetry is that the automorphism 2_3 of U_4(3) does not lift to 3_1.U_4(3). Note that each of the classes <code class="code">9A</code>, <code class="code">9B</code> of U_4(3) has three preimages in 3_1.U_4(3), whereas each of the classes <code class="code">9C</code>, <code class="code">9D</code> has only one preimage.

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 of maximal subgroups of 3_1.U_4(3) yields no irreducible character, whereas the two irreducible characters

<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,
 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
gap> irr:= Irr( 3u42 );;
gap> ind:= InducedClassFunctionsByFusionMap( 3u42, 3u43, irr, rep[1] );;
<java.lang.StringIndexOutOfBoundsException: Range [37, 5) out of bounds for length 102
[ 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*(3^, 3*(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, 0firstfus= First(( u42fusu43, x> IsSubset ,[16,17 ]))
 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[ 1,2, 2,3 3,4,5 ,8 ,10 10,11 11,12 11,18 19,20

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

<h5>9.7-12 2.3^4.2^3.S_4 → 2.A12p>et us now consider the central extension 3_1.U_4(3). Since the Schur multiplier of U_4(2) has order two, the U_4(2) type subgroups of U_4(3) lift to groups of the structure 3 × U_4(2) in 3_1.U_4(3). There are eight possible class fusions from 3 × U_4(2) to 3_1.U_4(3), in two orbits of length four under the action of table automorphisms.

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 class=GAPinput"3u43: CharacterTable("3_.3" );/span>

<div class="example"><pre>
gap> 2a12:= CharacterTable( "2.A12" );;
gap> mtblfus2a12:= PossibleClassFusions( mtbl, 2a12 );;</java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 12
gap> Length( mtblfus2a12 );
32
<=GAPprompt>apg;/pan spanclass"APinput"repres= (mtbl , 2 ;;
gap> Length( repres );</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
2
<<java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

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 span"" ((3 ) );/

For that, let We need π only for computing one preimage ofp> eachof two, to two by < ""U_42)> =java.lang.StringIndexOutOfBoundsException: Range [228, 227) out of bounds for length 247

<div class="example"><pre>
gap> g:= AtlasGroup( "A12" );
123,23456789101112 )
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 ),
> GeneratorsOfGroupspan ""&;/ spanclass">:=( g List(( s ,<>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 129
<permutation group of size
gap> NrMovedPoints( w );
12
gap> s:= Intersection( w, g ); Size( s );
<permutation group with 8 generators>
15552
gap> m:= SubgroupNC( 2g, List( SmallGeneratingSet( s ),
> x -> ImagesRepresentative( pi1,
> PreImagesRepresentative( pi2, x ) ) ) );;
/>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> java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 61
31104
gap> trans:= TransformingPermutationsCharacterTables( mtbl, t );;
gap> IsRecord( trans );
true
</pre></div>

Now let us see where the two fusion 28, 29, 26, 14, 14, 16, 16, 33, 34, [ 33, 34 34

<div class="example"><pre>
gap> para:= Parametrized( repres );
[ 1, 2, 6, 10, 8, 12, 7, 11, 9, 13, 5, 5, 17, 17, 17, 17, 3, 4, 24,
 ,27,25,12 10, ,11,28 29, 35, 37,39,36 38,40 5,23
 28, 29, 26, 14, 14, 16, 16, 33, 34, [ 33, 34 ], [ 33, 34 ], 49, 49,
 48, 48 ]
ositionsProperty para,IsList)
[ 46, 47 ]
gap> List( repres, map -> map{ [ 44 .. 47 ] } );
[ [ 33, 34, 33, 34 ], [ 33, 34, 34, 33 ] ]
</pre></div>

<> thequestioniswhether the elements class 44are conjugatein tothe elements in class 46 or in class 47. In order to answer this question, we compute preimages of the relevant class representatives in the matrix group M.

<div class="example"><pre>
<class=GAPprompt>g;span <=">: ( 44 ..47 , trans.columns );;
gap> classreps:= List( ConjugacyClasses( t ){ positions },
<panclass="APprompt">/span span class=GAPinput">c -gt; ( ,Representative( c ) ) );;
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>
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
> 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) contains a maximal subgroup M of the type 127:7 (the normalizer of an extension field type subgroup GL(1,2^7)) whose class fusion is ambiguous.

<div class="example"><pre>
<nput"t= CharacterTable("()" );
gap> s:= CharacterTable( "127:7" );;
gap> fus:= java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 0
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, 117, 116, 83, 83, 83, 83, 83, 83 ] ]
</pre></div>

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

<div class="example"><pre>
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 ],
 
[ 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;
gap> n:= Normalizer( g, SubgroupNC( g, [ x ] ) );;
gap> Size( n ) / 127;
7
gap> java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 18
gap> c:= Centralizer( g, x );;
gap> Size( c );
49
</pre></div>

We see that the second candidate is the fusion from M into G.

<div class="example"><pre>

true
</pre></div>

<aid"X80051B297DF244CF"nameX80051B297DF244CF"><a></>

<h597-4 <classH"< =SimpleMath">259 →<span(May2009) –Donot use this</pan<h5>

<Ithad beenclaimed thatthe simple MonstergroupM</pan contains a subgroup < class"SimpleMath">of the typeL_2(59), see <a href="chapBib.html#biBHW04">[HW04]<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2

Here we show that the character tables of G and M do not provide enough information to

<div class=example"><pre>
gap> t:= CharacterTable( "M" );;
gap> s:= CharacterTable( "L2(59)" );;
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, 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,
 97, 100, 50, 30, 50, 15, 11, 100, 50, 30, 4, 100, 11, 100, 50,
 3 ],
 [1, 152,, ,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,

 3 ],
 [ 1, 152, 153, 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 ] ]
</pre></div>

The candidates differ on the classes of element order 30.

<div class="example"><pre>
gap> 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 ],
 97,100,50, 30 ,15 11 ,50,30 ,100 ,100,50
</pre></div>

According to <a href="chapBib.html#biBHW04">[HW04]</a>, any subgroup G java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 69

<div class="example"><pre>
gap> pos:= Position( ClassNames( t, "Atlas" ), "30G" );;
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 ],
 [ 1, 153, 152, 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> repr:= RepresentativesFusions( s, good, t );
[ [ 1, 152, 153, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97, 97,
 97, 104, 52, </pre></>
 3 ] ]
gap> GetFusionMap( s, t ) = fail;
true
</pre></div>

<a id="X8409DA2E83A41ABE" name="span class="GAPprompt">gap> <spanjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 108

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

The sporadic simple Monster group M contains a maximal subgroup G of the type L_2(71), see <a href="chapBib.html#biBHW08">[HW08]</a>. The class fusion of G into M is ambiguous.

<div class="example"><pre>
<="GAPprompt">gap&; t: CharacterTable("M );/span>
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, 112, 112, 115, 61, 36, 28, 115, 17, 115, 28,
 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, 112, 11, 19, 112, 112, 117, 61, 43, 28, 117, 17, 117, 28,
 9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] [1, 153,, 152,97,97 97,97 97,97,97,97 97,97, 97,97 97,
 [ 1, 169, 170, 113, 113, 113, 113, 20, 113, 12, 113, 113, 20, 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,
 true
 9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ]
</pre></div>

The candidates differ on the classes of the element orders 7 and 36.

<div class="example"><pre>
gap> ord:= OrdersClassRepresentatives( s );;
gap> ord36:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = h"G> of the type < class"SimpleMath">L_2(71), see <a href="chapBib.html#biBHW08">[HW08]</a>. The class fusion of G into M is ambiguous.
[ 21, 25, 27, 31, 33, 37 ]
gap> <span class="
[ [ 114, 114, 114, 114, 114, 114 ], [ 115, 115, 115, 115, 115, 115 ],
 [ 117, 117, 117, 117, 117, 117 ], [ 114, 114, 114, 114, 114, 114 ],
 [ 115, 115, 115, 115, 115, 115 ], [ 117, 117, 117, 117, 117, 117 ] ]
</pre></div>

According to <a href="chapBib.html#biBNW02">[NW02, Table 3]</a>, G contains elements in the classes <code class="code">7B</code> and <code class="code">36D</code> of M. This determines the class fusion up to Galois automorphisms.

<div class="example"><pre>
spanclass=">ap&;span> span =Gjava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 120
gap> pos2:= Position( ClassNames( t, "Atlas" ), "36D" );;
gap></ 112 112 11,19,112,112,117, ,43,28 ,17 117,28
gap> good:= Filtered( fus, map -> IsSubset( map, pos ) );
[ 1 , 170, 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 ],
 [ 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, 117, 61, 43, 28, 117, 3 ] ]
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,
 9, 61, 117, 5, 117, 61, 43, 28, 117, 3 ] ]
gap> GetFusionMap( s, t ) = repr[1];
true
</pre></div>

<a id="X78B3B1BE7A2CA4D1" name="X78B3B1BE7A2CA4D1"></a>

h5. ">span=>(1)→ <>Ajava.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 101

<>sporadic Monster group <spanclass"impleMath">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[117,117, 117,117 117 117 ,[ ,114 114,114 114 114 ]
gap> t:= CharacterTable( "M" );;
gap> s:= CharacterTable( "L2(41)" );;
gap> fus:= PossibleClassFusions( s, t );;
gap> repr:= p>According to <a href="chapBib.html#biBNW02">[NW02, Table 3]</a>, 7B</code> and <code class="code">36D</code> of M. This determines the class fusion up to Galois automorphisms.
[ [ 1, 127, 127, 64, 30, 64, 11, 7, 30, 64, 11, 64, 3, 70, 70, 19,
 70, 70, 19, 4, 70, 19, 70 ],
 1,127,127, 64,30 64,11, 7 ,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, 66, 12, 66, 3, 72, 72, 19,
 72, 72, 19, 6, 72, 19, 72 ],
1127 ,33 ,12 ,33,66,12 ,3 ,73 ,
 73, 73, 20, 5, 73, 20, 73 ],
 ,, ,67 67 3 ,72 ,
 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, 68, 12, 68, 3, 72, 72, 199, 61 117,5, ,61 43,28 117,3 ]
 72, 72, 19, 6,<pan="GAPprompt">ap/span=GAPinput>GetFusionMap ,) repr1;/span
 [ 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="X78B3B1BE7A2CA4D1<a
 72, 72, 19, 6, 72, 19, 72 ],
 [, 127,127,69,33 ,12,9,33, 69 ,69,3,73,73,20
 73, 73, 20, 5, 73, 20, 73 ] ]
</pre></div>

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

<div class="example"><pre>
gap> ambig:= Parametrized( repr<pan="GAPprompt">&t/ < class"APinput"sCharacterTable"L2(1");gap> ambigpos:= PositionsProperty( ambig, IsList );
[ 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22,
 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
gap> Set( OrdersClassRepresentatives( t ){ ambigpos } );
[ 3, 4, 5, 6, 7, 8 ]
</pre></div>

According to <a href="chapBib.html#biBNW12">[NW13, Theorem 3]</a>, Ggap> pos2:= Position( ClassNames( t, "Atlas" ), "4C" );;
gap> pos:= [ pos1, pos2 ];;
gap> good:= Filtered( fus, map -> IsSubset( map, pos ) );<[1 ,,68 , 68 ,,3 ,,19
[ [ 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>

<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap8.html"/pre>div>

<div class="chlinkbot">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

<hr />
generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a>
</body>
</html>

Messung V0.5 in Prozent

¤ Dauer der Verarbeitung: 0.175 Sekunden ¤

Wurzel

Suchen

PVS Prover

Isabelle Prover

NIST Cobol Testsuite

Cephes Mathematical Library

Vienna Development Method

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.