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             <title>Some steps in the verification of the ordinary character table of the Monster group</title>
<h1 align="center">Some steps in the verification of the ordinary character table of the Monster group</4   java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 113
  <body bgcolor="FFFFFF">

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<h3 align="center"> T<font size="-2">HOMAS</font> B<font size="-2">REUER</font>, K<font size="->3n  ofelements of even </a>br /&;;;n;a =#4java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99

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<h3 align="center">December 12th, 2024 </h3>

<div class""</div>

<div class="p"></div>
We show the details8&;java.lang.StringIndexOutOfBoundsException: Range [22, 18) out of bounds for length 97

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<h1>Contents </h1><a href="#tth_sEc1"
>1  Overview</a><br /><a href="#tth_sEc2"
>2  Some restrictions of the natural character of <font size="+0">M</font></a><br /><a href="#tth_sEc3"
>3  The permutation character (1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></>4.9  Classes of elements of order divisible by 29</a><br />    <a href="#tth_sEc4.;  java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 112
>4  The conjugacy classes of <font size="+0">M</font></a><br />    <a href="#tth_sEc4.1"
>4.nOurtodescribethe of<font size"+>font</> >n;n;&bsp&;a =#."
>4.2  Utility functions</a><br />    <a href="#tth_sEc4.3"
>4.3  Classes of elements of even order</a><br />    <a href="#tth_sEc4.4"
>4.4  Classes of elements of order divisible by 3</a><br />    <a href="#tth_sEc4.5"
>4.5  Classes of elements of order divisible by 5</a><br />    <a href="#tth_sEc4.6"
>4.6  Classes of elements of order divisible by 11</a><br />    <a href="#tth_sEc4.7"
>4.7  Classes of elements of the orders 17, 19, 23, 31, 47</a><br />    <a href="#tth_sEc4.8"
>4.8  Classes of elements of order 13</a><br />    <a href="#tth_sEc4.9"
>4.9  Classes of elements of order divisible by 29</a><br />    <a href="#tth_sEc4.10"
>4.10  Classes of elements of order divisible by 41</a><br />    <a href="#tth_sEc4.11"
>4.11  Classes of elements of order divisible by 59</a><br />    <a href="#tth_sEc4.12"
>4.12  Classes of elements of order divisible by 71</a><br />    <a href="#tth_sEc4.13"
>4.13  Classes of elements of order divisible by 7</a><br /><a href="#tth_sEc5"
>5  The power maps of <font size="+0">M</font></a><br /><a href="#tth_sEc6"
>6  The degree 196 883 character χ of <font size="+0">M</font></a><br /><a href="#tth_sEc7"
>7  The irreducible characters of <font size="+0">M</font></a><br /><a href="#tth_sEc8"
>8  Appendix: The character table of 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub></a><br /><a href="#tth_sEc9"
9n; Appendix:The character table3<up112</sup><>+/sub>:6.span class="oman">Suz</span>.2</a><br />    <a href="#tth_sEc9.1"
>9.1  Overview</a><br />    <a href="#tth_sEc9.2"
>9.2  A permutation representation of H / X</a><br />    <a href="#tth_sEc9.3"
>9.3  A permutation representation of H</a><br />    <a href="#tth_sEc9.4"
>9.4  Compute the character table of H</a><br /><a href.&bsp;of   order 7<>/<href#tth_sEc5
>10  Appendix: The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2</a><br />

<div class="p"></div>

<div class="p"></div>

<h2><a name="tth_sEc1">
1</a>  Overview</h2>

<div class="p"></div>
The aim of [<a href="#Mverify" name="CITEMverify">BMW24</a>] is to verify the ordinary character table
of the Monster group <font size="+0">M</font>.
Here we collect,
in the form of an explicit and reproducible
<font face="helvetica">GAP</font> [<a href="#GAP" name6&bsp Thedegree 196& 83character &#pan style='color: green'>967; of <font size="+0">M</font></a><br /><a href="#tth_sEc7"
the relevant computations that are needed in that paper.

<div class="p"></div>
We proceed as follows.

<div class="p"></div>
Section <a href="#natural">2</a> verifies the decomposition of the restrictions of the
ordinary irreducible character of degree 196 883 of <font size="+0">M</font>
tothe.fontsize+"B<font>and .span r">Fi/pan>sub24/ub<sup&8242;/sup> (and<spanclassroman>/<24</>,
as stated in [<a href="#Mverify" name="CITEMverify">BMW24</a>,Lemma 1].

<div class="p"></div>
Section <a href="#suborbits">3</a> verifies the decompositions of
the transitive constituents of the permutation character
of the action of C<sub><font size="+0">M</font></sub>(a)  ≅ 2.<font size="+0">B</font> on the conjugacy class a<sup><font size="+0">M</font></sup>,
where a is a <tt>2A</tt> involution in <font size="+0">M</font>.

<div class="p"></div>
Sections <a href="#Mclasses">4</a> and <a href="#Mpowermaps">5</a> construct the
character table head of <font size="+0">M</font>, that is,
the lists of conjugacy class lengths, element orders, and power maps.

<div class="p"></div>
Section <a href="#sect:natcharM">6</a> constructs the values of the irreducible
java.lang.StringIndexOutOfBoundsException: Range [32, 6) out of bounds for length 57
and decides the isomorphism type of the <tt>3B</tt> normalizer in <font size="+0">M</font>.

<div class="p"></div>
With this information
of .font"0></, <>+24/>sub>+/ub.span class=">sub></ub>, .<spanclass"oman">Fi</span><sub>24</sub>, and
3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 of <font size="+0">M</font>,
computing the irreducible characters of <font size="+0">M</font> is then easy;
this corresponds to [<a href="#Mverify" name="CITEMverify">BMW24</a>,Section 5]
and is done in Section <a href="#sect:irreduciblesM">7</a>.

<div class="p"></div>
The final sections  of/>/<href#java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
<a href="#sect:norm3B">9</a>, <a href="#sect:table_N5B10&bsp;    5sup1+<sup>sub>+/sub>4< =><span<></.</< />
character tables of subgroups of <font size="+0">M</font>.

<div class="p"></div>
We will use the <font face="helvetica">GAP</font> Character Table Library
and the interface to the  A<font size
thus we java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<1<>nbsp;nbsp;</2

<pre>
    &62  "" false)
    true aim of&;< href#Mverify name="CITEMverify></> is to verify the ordinary charactertable
    gap62;( a,false );
    true

</pre>

<div class="p"></div>
Thefont face"elvetica">AGMA<font>system&bsp< href"# ="ITEMagma">/> willbe
<="<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 98
and for many conjugacy tests
the command tt><tt>
then thesesteps notwork

< class=p>!--<div

<pre>
    gap> CTblLibordinaryirreduciblejava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 81
    true

</pre the  2<fontsize"0><  3.spanclass=roman"Fispan<><sub<>#8242<sup and3< =""</>sub24/>,

<div class="p"></div>
Wesetthe line  to,likeotherstandard.

<div class="p"></div> < ="suborbits><a>verifies the decompositions of

<pre>
    gap#;SizeScreen[72  java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

</pre>

<div class="p"></div>

<h2><Sections&bsp< href"Mclasses"4/ andn;a="Mpowermaps5<a>constructthe
2</>    ofthe natural character of <font size="+0">M</font></h2><a name="java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 69
</a>

<p"
We assume the degree 196 883 character of <font size="+0">M</font>
of  196nbsp;883 of  Monster <font =+"M/,
and

<Withthis information with the (already )charactertables
Firstwe compute   of&967; to 2.font size"0"/.

<div class="p"></div>
The  degree 196&;83character of 2<font =+><font>
that has at most theirreduciblecharacters  fontsize=""Mfont>istheneasy
1a +4371a  96255a + 96256a,  + 96256a, as  in[ahref"Mverify"name"ITEMverify></>Lemma&;1]
This fromthe following about 2< size=+">/>java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70

<div class="p"></div>

<The  sections;a="sect:"8<a,
    &62 table2B=CharacterTable2B );
    62;cand:=Filtered(Irrtable2B) x-#62; x1]< 196883);
    gap> List( cand, x -> x[1] );
    [
    gap> inv:= Positions( OrdersClassRepresentatives( table2B ), 2 );
    [ 2, 3, 4, 5, 7 ]
   #java.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds for length 83
            ,       ,       ,      ,1        ]
      [    4371,    4371,    -493,     275,     275,      19 ],
      [   96255,   divp
      [   96256,pre>

<>

<div class="p">
Note that 96256a
isonlycandidate
and it can < java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
or 4371 ·1a + 2 ·96256a would be possible decompositions,
which haveIf thefollowingcommand ttjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 47
the  &. java.lang.StringIndexOutOfBoundsException: Range [44, 42) out of bounds for length 101

<div class="p">
f96255 b><b>occur  values #;>< =+">>
on the classes 3 and 7 would differ by 512 times the multiplicity of 4371,
but 2a&&Some  of   java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 110
Thus 96255 must occur exactly once.

<div class="p">/
The sum java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0
on,hence4371  .

<div class="p"></div>
We see div classp</div>
lyjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 28

<pre
    &62   {inv;
    [ 4371, 4371    &62 : (Irrtable2B  -62 x1 lt;= 196883 );;

</pre>

<div class="p"></div>
The restriction of χ to 3.<span java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 73
  [2 3,4 , 7 ]

<div classgap#;PrintArray(  -#;xConcatenation( 1 ]     ;
Exactly seven  1,       ,1 1       ,1]
constituents[4371,4371    -93,    275,275,19 ,

<[  ,96255    48632047    ,    255 ,

<pre>
    gap&#62,-2048       0 ] ]
    gap> cand:= Filtered( Irr( table3Fi24prime ), x -&#='color: green'>62; x[1] <= 196883 );;
    gap> inv:
    [ 4, 7 ]<pre>
    gap&#<ivclass""></div>
    gap>PrintArray  );
    [ [      1,      1,      1 ],
      [   8671,     once otherwise only 4371a2&18396256a
      [  57477,   1157,    133 ],
      [    783,     79,     15 ],
      [    783,    79,     15]java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
      [ 64584,  1352,     72 ]
      [  64584,   1352,     72 ] ]

</pre

<></div
Since χ is rational theclasses3and 7wouldjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 74
characters,that , the possibleconstituentsare1a 8671a,57377,
783ab, and 64584ab.

<div class="p"></div>

<pre>
    gap> List( cand, x -> x[2] T sum 96255 and 96256 has four
    [ 1, 8671, 57477, 783*E(3), 783*E involutions also 4371a  occur

</pre>

<div class="p"></div>
We 4371 aand 275,respectively
since all
and toolarge for other value.

<div class="p"></div>
Since
utions atmostequal   onthe  class,
and equal onlyfor1,
we conclude that the value on 4371 ,4371 275  ]

<div class="pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fromthe   the
and on the java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 11
java.lang.StringIndexOutOfBoundsException: Range [20, 17) out of bounds for length 57

<constituents #.

<pre>
    gap> Float( 196883 / 4371 );
    45.043
    gap> List( mat, v -> Float( v[1]    &62 : Filtered(Irr  ),x-&#;x1 &t 196883);
    [,991139 7692 . java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
    &62 (196883-2 64584    4371  2  1352 ) )java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
    40.6209
    gap> Float( ( 196883 - 57477 ) / ( 4371         ,,-33 ,
    43.3746
    gap> Float( ( 196883 - 2*57477 ) / ( 4371 - 2*1157 ) );
    39.8294
    gap> Float( ( 196883 -   64584,   1352,72 ]java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    27.1689

</pre>

<div class="p"></div>
Firstjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
degree57477
must occur.

<div class="p"></div>
However, then the degree admits at
hence  valueon the second  cannot 275,
a contradiction

=p>
This means that both 64584ab and 57477a occur     [ 1, 8671, 57477, 783*E(3), 783*E(3)^2, 64584(32

<div class="p"></div>

<pre>
    gap62 ] ] + mat;
    [ java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 58

</pre>

<div class  atmost    onfirst,
The  classforces   of  8671
( is only thatcan  a value,
and div ="</div>
lution thedegrees itsconstituents 1 +1566java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
decomposition
1a + 8671a + 57477a +div ">
as claimed in [<a href="#gap&62 (196883  4371)

<div gap#2 ( , v -62 (v[]/ 2  ;

<pre>
    gap#; Sum  )
    [ 196883, 4371, 275 ]

</pre>

<div class="p"></div>
The characters of the degrees 1,gap#62; Float((196883  - )/( 4371-1157 ))java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
from 3.span="oman"Fispan<>24/<&8242<supto3.span "oman"Fispan>sub><sub>
java.lang.StringIndexOutOfBoundsException: Range [18, 2) out of bounds for length 96
e to determine extensions 3.<span class=r"Fi/span>sub>4><>&8242;/> occur.
The following irreducible characters of 3.<span class="roman">Fi</span><sub>24</sub> can occur as
constituents of the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div =p</div>

<pre>
    gap> table3Fi24
    gap&62;cand=Filtered(Irr table3Fi24 ),x-&#; x[]&t=196883 ;java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    &;inv Positions ( table3Fi24 , 2)
    [ 3, 5, 172, 173 ] the on second  involutions  be275,
    gap> mat:=
    gap&62 ( mat );
    [ [       1,       T  that 64584b 57477a with .
      [       1,       1,       1,      -1,      -1 ],
      [    8671,
      [    8671,     351,     -33,   -1495,      41 ],
      [   57477,1157 133,5865,     233],
      [   57477,    1157,     133,   -5865,    -233 ],
      [    1566,     158,      30,       0,       0/re
      [  129168 java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 65

</pre>

<divclass""
We get the decomposition
1a + 86711a + 8671aa+783  64584

<div class="p"></div>

<pre>
    gap>    gap#62 (  )
    [ 196883, 4371, 275, 4371, 275 ]

</pre>

<div class="p"></div>

<h2><a name="from 3.<span class="roman">Fi</span><sub>24</sub><sup>′</sup> to 3.<span In order to decompose the restriction of χ to 3.<span class="roman">Fi</span><sub>24</ have to determine which extensions from 3.<span class="roman">Fi</span><sub>24</sub><sup>Rle='color:blue'>"/>>/>>#;/ java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 112
3</62 :java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 79
<#;: ( OrdersClassRepresentatives( table3Fi24) 2 )

<div     gap&;mat(cand  &62 {Concatenation[1], inv};
According to[a="GMS89"name=CITEGMS89>MS89/>,Tables VII,]
the character1>2<=+><java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 148
decomposes into nine transitive 1,       1      -,     1]java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
with the point stabilizers U listed in Table <a href    8671     ,    33,  -495,41

<div class="p"></div>

<div class="p"></div>
<a name="tth_tAb1">
</a, 1440,  ]
</a>

<center>

<table>
<tr><td>a c  ∈     </td><td>G<sub>a,c</sub>  </td><td align="right">&#e='color: green'>124;c<sup>G<sub>a</sub>  the
<tr><td><tt>1A</tt> </td><td>2.<font size="+0">B</font
<tr><td><tt>2A</   gap#;Sum {  ,4,,7   ;
<r<><t2<tt />td><up>+<sup>span =roman></span<>2<sub  /td>td align"ight"11707448673375 </td</>
<tr><td><tt>3A</tt> </td>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<tr><td><tt>3C</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<tr><td><tt>4A</tt> </td><td>2<sup>1+/a>n;nbspT permutation (1<>2<font size"0>B</>/ub>sup>font ="+0"M/font>/)><size=+"Bfont<sub</><a namesuborbits>
<tr><td><tt>4B</tt> </td><td>2.<span class="roman">F</span><sub>4</sub>(2)       </td><td align="right">1254793905192960000 </td></tr>
<tr>td>tt>5</tt /td><><span class"oman"HNspan>             <td>td ="">30434513446055706624 <td</>
<tr><td><tt>6A</tt> </decomposes into nine transitive permutation characters 1<sub>U</sub><sup>2.<font size="+0">B</font></sup>,

</center>

<div class="p"></div>
Here a denotes the central involutiona<center>Table 1: Suborbit information</center><a name="suborbitsTable">
the action is center>
and c

<div class="p"></div>
thissection wecompute   1<>U/>sup>2.<font=+0"B<font>/>,
where U is one of the abovetr>td>>A/ <td<td>< =+"B<font>            <td>td ="ight> <td<tr
Note thata&#8712 G<suba</>holds
(and thus the character is an><<java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 137
if and only if a and c commute;
this happens exactly for the first three orbits.

<div class="p"></div>
All subgroups U except 2<sup>2+<>tdtt3Ctt>/>td<="></>             />td ="91569524834304000<java.lang.StringIndexOutOfBoundsException: Range [117, 116) out of bounds for length 122
are  A<font size="-2">TLAS</font> groups whose character tables have been verified.
Thesubgroup <>2+22/. ="roman><span<>2<sub>is the  ofa maximal subgroup
2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub> of <font size="+0">B</font> under the natural epimorphism from 2.<font size="+0">B</font>,
and the computation/verification of the character table of 2<sup>1+22</sup>.<span class="roman">Co</span><sub>2</sub>
been [ =""=java.lang.StringIndexOutOfBoundsException: Range [71, 61) out of bounds for length 74
It will java.lang.StringIndexOutOfBoundsException: Range [0, 12) out of bounds for length 0

< a the involutionin<size"0></ont>
The nine characters will font size"0><font>-onjugacy a
tt<tt,ttpi2/> .,<><tt.

<div class="p"></div>
For U = 2.<font    compute java.lang.StringIndexOutOfBoundsException: Range [50, 47) out of bounds for length 99
">B  1sub< =+0><font<>java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100

div=p"

<pre>
    &62 :=TrivialCharacter  );

</pre>

<iv=""
For U = 2<sup> happens exactly for the  three .
thecharacter 1sub>/>sup>2.< ="0><font>/ is the of 1sub>&63717U<sub><size""B/font>/java.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143
from < size=+"B</font> to 2< =0"B/font>java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
for [U] = U / 〈a 〉 =  2.<sup>2</sup><span class="roman">E</span><sub>6</sub>(2).
(Note that the class fusion from [U] to <
determined by
but the permutation character is unique.)

<div class="p"></div>

<pre>
    gap> tableB:= CharacterTable( "B" );;
    gap> tableUbar:= CharacterTable( "2.2E6(2)" );;
    62;:=( tableUbar, tableB);
    gap> Notethatthe  fusion [63717;U fontsize"+"B notuniquely
    ;               -&#2 ( tableUbartableB,
           [ TrivialCharactertableUbar ) ,map[] )java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
    gap> Length( pi );
    1
    gap> pi2:= Inflated( tableB
    gap> multgap&62 :java.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 48
&;chi62 (table2B, pi2));
    gap> Maximum( mult );
    1
    #;Positionsmult,1 )java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    [ ,2 3 ,7,13 15 17 java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

<>

<div class="p"></div>
For U = 2<1
the character 1gap#;mult List(   ,
from <font size="+0">B</font> to 2&62                  -#;ScalarProduct ,pi2));
for [#63717U   / &9001; #9002   <>122/sup.spanclass=roman><<2<subjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
1 ,3 ,7 13,15,17 java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

<div class="p"></div>

<rejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    #62 : ( "N2B" )
    CharacterTable( "2^(1+22).Co2" )
    gap> fus:= PossibleClassFusions( tableUbar, tableB );;
    gap> pi:= Set( fus,
    &62               -&62 ( tableUbar tableB,
    >                         [ a maximal subgroup of <font size="+0">B</font>.
    gap&#62java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    1
    gap> pi3:= java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 5
    &#2 mult: ( Irr  )java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    >                 chi -> ScalarProduct( table2B, chi, java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 62
    >Maximum  )java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    1
    gap> Positions( mult, 1 );
    [ 1, 3, 5, 8, 13, 15, 28, 30, 37, 40 ]

</pre>

<div class="p"></div>java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
Next we consider U = <span class="romangap&62 mult:=List( (table2B )java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

<div class="p"></div>

<pre>
    &62 :=CharacterTableF" ;
    gap> [ 1, 3, 5, 8 ,28 30 37 40
    gap> pi:= Set( fus,
    >              map -> InducedClassFunctionsByFusionMap( tableUprejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    &62 [ TrivialCharacter tableU ),  )1]);
    gap> Length( pi );
    1
    gap> pi4:= pi[1];;
    gap> mult:= List( java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 5
    >                 chi -> ScalarProduct( table2B, chi, pi4 ) );;
    gap> Maximum(gap62 :(
    1
    gap> Positions( mult, 1 );

      63, 68, 77, 81, 83, 185, 186, 187, 188,gap#;Length ;
      208, 220 ]

</pre                  -62;ScalarProduct ,pi4));

<div     gap& (mult;
Next we consider U = <span class="roman> Positions(,1 ;

<iv class=p>!--<div>

    gap> tableU:= CharacterTable208 220 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    &62;: PossibleClassFusions tableU,  )java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
    &62;=Setfus
    >              map -> InducedClassFunctionsByFusionMap
    >                         [ TrivialCharacter( tableU ) ], map )[1] );;
    gap> Length( pi);
    1
    gap> pi5:= pi[1];;
    gap> mult:= List &62 = ,);java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
    >                 chi -&&#2                          (java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 78
    gap#=ListIrr ,
        #;                 chi &62 ( ,chi,pi5 ;

     3 ,12 13,,19, ,28,34 38 41 , 68 70,,78
      85, 89, 113, 114, 116, 129, 133, 142, 143, 145, 155, 156, 185, 187,
      188, 193, 195, 196, 201, 208, 216, 219, 225, 232, 233, 235, 236,
      237, 242 ]
    gap> Positions( mult, 2 );
    [ 62 ]

</pre>

<div class=gap&62  ,2 ;
For U = 2<sup>1+22</sup>.<span class="<div class="p"></div>
[<a href="#ctblpope" name="CITEctblpope">Breb</a>,Section "A permutation character of 2.<font size="+0">B</font>"].
Weknowthat Uisa  of2<>2+2/up. =roman><span>>2<sub,
and that 〈U, a 〉 has the structure 2<sup>2+22</sup>.<span class="roman">McL</span>.

know   java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 91
As a first step, we induce the trivial character of that #9001,&;has the structure2sup222/><pan class"oman>/pan>
to 2.<font div =p</div>
inducingthetrivial  of span=r></span to < class""Co<><ub<subjava.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136
then to inflate this character to 2<sup>1+22</sup>.<span class="roman">then to inflate this character to 2<sup>1+22</sup>.<span class="roman">Coclass=>o<><>2<sub,
then to inducethis to<size"+></ont>,
and then to inflate this character then toinduce   fontsize"0"></>

<div class="p">

<pre>
    gap> mcl:= java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0
    gap> co2:= CharacterTable( "gap#;:= java.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 44
    gap> fus:= PossibleClassFusions( mcl
    62;Length )
    4
    gap> ind:= Set( fus, map -&#&62                               java.lang.StringIndexOutOfBoundsException: Range [60, 58) out of bounds for length 81
    &#java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    gap> Length( ind );
    1
    gap> bm2g&62 :=Inflated,,ind);
    CharacterTable( "2^(1+22).Co2" )
    gap&;infl:Inflated co2 , );
    gap> ind:= Induced( bm2, tableB, infl );;
    #;infl: (tableB ,ind )[];;

</pre>

<div class="p"></div>
As a second step,
we compute 1<sub>U</sub><sup>2.<font size="we compute 1sub></sub><sup>2.<font size="+0">B</font></sup> with the <font face="helvetica">GAP</font> function <tt>PermChars</tt>,
using that we can java.lang.StringIndexOutOfBoundsException: Range [25, 15) out of bounds for length 60
the permutation character induced from the closure of U with
the normal subgroup &div=">!->/>

<div class="There isa  ,and its computation is quite fast.)
(We are div ""</div>
There is a unique java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 5

>

<pre>
    gap># nonfaithful ;
    [ 1, 2 ]
    gap> 1
   #;=
    >                             nonfaithful:= infl ) );;
    gap> Length( pi );

    gap> pi6:= pi[1];;
    gap> List( Irr( table2B ), chi -> ScalarProduct( table2B, chi, pi6 ) );
     1,1, 2 1,2 0,2 3,,1 ,1,2, 0, 3 2 0 , 0 0,
      2, 2, 0, 0, 2, 3, 1, 5, 0, 4, 3, 2, 0, 0, 3, 2, 0, 6, 4, 0, 1, 1,
      0, 0, 0, 0, 3, 0, 1, 0, 0, 5, 0, 5, 2, 0, 0, 2, 0, 0, 4, 1,0,4 2 , , ,4 ,,0 ,,  ,,,,,1
      0, 4, 2, 4, 4, 3, 0, 2, 4, 2, 4, 0, 3, 0, 3, 2, 5, 0, 1, 0, 3, 1,
      0, 1, 1, 2, 5, 3, 1, 1, 4, 5, 1, 1, 0, 3, 0, 0, 3, 2, 1, 1, 2, 1,
      1, 4, 0, 3, 2, 3, 1, 3, 0, 1, 3, 0, 2, 2, 1, 3, 3, 0, 0, 2, 0, 0,
      0, 0, 3, 0, 3, 3, 3, 1, 0, 3, 0, 4, 0, 1, 0, 0, 21 ,3 0 0 , 1 1 ,1 ,3, 2,0,0,2 2 ,3,5 ,4
      ,1 ,0, 0 ,0,1,2 ,1 ,0 1,1 1,1,1 ,0 ,1
      1, 3, 3, 0, 0, 0, 1, 1, 1, 1, 2, 3, 2, 0, 0, 2, 2, 4, 3, 5, 2, 4,
      ,,0,0,5 ,0 0,, ,1 , ,0 ,0 ,,0 ,1,,
      7, 0, 0, 0, 1, 6, 4, 5, 0, 0, 3, 0, 0, 0, 0, 0, 4, 1, 1, 3, 8, 3,
      2  1java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

<>

<div class="p"></div>
Next  =2<"java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 67
ejava.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 85

<div class="p"></div>

<pre>
    &#;= 22");
    gap> fus:= java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 26
: (   - java.lang.StringIndexOutOfBoundsException: Range [79, 71) out of bounds for length 89
    >             [
    gap> Length&62 (  java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 83
    2
    gap> pi:= Filtered( pi, x -> ClassPositionsOfKernel( x ) = [ 1 ] );;
    gap> Length(  2,1 ,0 ,2 , ,0 , ,,,3 20 ,4 ,,0
    1
    gap> pi7:= pi[1];;
    gap> List( Irr( table2B       0, 0,1, 0,,0,0, 1,0 0,2,1 ,3 0, , 3,0, 1 4,0,0, ,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
    [ 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 0, 2, 4, 1, 3, 0, 2, 1, 0, 0, 0, 0,
      2, 1, 0, 0, 2, 2, 1, 4, 0, 2, 1, 2, 0, 0, 3, 2, 0, 4, 4, 00,2,0 3 3,3,0 4,,0,4 ,11 1 1 2,1,1 , 3, 0,
      0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 3, 3, 0, 0, 3, 0, 1      2, 3,1 0, 2,0 ,2, 1,0 ,0,0 ,,0 ,0 0,1 ,
      0, 6, 0, 3, 2,       ,0 1, 1,,0,0 ,3 ,0 ,0 ,0 ,,0 ,1 ,
0, ,1,3 ,0 0,4,5 ,1 0,3 0 ,1 ,2 ,0 2,
      0, 2, 0, 3, 3, 3, 0, 4, 1, 0, 4, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 0,
      0, 2, 2, 0, 5, 5, 3, 0, 1, 5, 1, 4, 0,="p"</div>
      2,
      1,4 4,0 0, 0, 3,1,1,1, 2 2 2, 0 ,1 2,3, ,3 1, 2,
      0, 0, 1, 1, 4, 2, 0, 0, 0, 3,    gap#62;;tableU=CharacterTable"N");
      5, 0, 1, 1, 2, 2, 4, 4, 0, 0, 3, 1, 1, 1, gap&62;pi=Set   -&62 InducedClassFunctionsByFusionMap,table2B
      2, 5, 5, 0, 1 ]

</pre>

<ivclass=p"></div>
Next we consider     [ 1, 1, 2, 1,

<div class="p"></div>

<pre>
    gap> tableU:= CharacterTable( "HN" );;
    gap> fus:= PossibleClassFusions( tableU, table2B,02 ,,,7 ,,7 ,9 ,0 0 6 ,2,13 ,,4
    gap> pi:= Set( fus      ,12,11,16,9, 7 ,11 13,12,20,5 10, 6 11,13,17,4
    &62;             [TrivialCharacter( tableU)], map [1] );
    gap> Length( pi );
    1
    gap> pi8:= pi[1];;
gap&62 (table2B,chi-#;ScalarProduct( , chi, pi8 ) );
    [ 1, 1, 2, 1, 2, 0, 3, 4, 2, 1, 1, 4, 4, 2, 1, 1, 3, 3, 1, 3, 0, 0,
      5, 3, 0, 0, 6, 4, 5, 6, 1, 7, 4, 7, 0, 0, 3, 8, 2, 6, 11, 2, 5, 5,
      0, 0, 2, 1, 3, 4, 7, 0, 0, 7, 3, 9, 5, 0, 0, 6, 4, 2, 13, 6, 0, 4,
      4, 12, 11, 16, 9, 7, 3, 11, 13, 12, 20, 5, 10, 6, 11, 13, 17, 4,
      10, 7, 19, 7, 7, 8, 10, 14, 18, 19, 5, 10, 12, 23, 7, 12, 6, 24, 6,
      4, 17, 16, 8, 9, 17, 11, 12, 23, 8, 24, 18, 26, 21, 29, 10, 18, 31,
      10, 24, 21, 17, 27, 35, 13, 14, 29, 19, 12, 7, 18, 26, 15, 34, 34,
      5,, 14,36,14,39 8, , 24,15 40,13, 9,38 24,17 ,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
,,24,22,17,31 39, 29, 30, 30, 19,44,,37,28,30,
      31, 29, 42, 40, 40, 56, 56, 30, 30, 42, 50,
      0, 4, 6, 10, 10, 12, 8, 12, 0, 0, 2, 4, 16We that  doesnotcontainthecentral involutionof2.fontsize=+0"font>,
0, 14, 34, 40, 2,/java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 6
        ,16   ,28 ,9072 , 56,
      90 ]

<

< =">
Finally, we consider U = 2.<#62;             [ TrivialCharacter ,map[] ;;
are candidates thepermutationcharacter (1sub</sub>)sup2.font size=+0"B/font><sup>,
according to the possible class fusions.
One of the two characters gap> pi:= Filtered( pi -#;ClassPositionsOfKernelx)= [1 ]);;
ofgap&62 Length pi)
We know     1
hence we candecidewhich character is correct.

<div="p"</div>

<pre>
    # java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 50
    gap> fus:= PossibleClassFusions( tableU, table2B );;
    gap> pi:23,,1,8 ,23 18,27 18, ,7,22 ,21 ,6 ,7
    >             [ TrivialCharacter( tableU 22,18 33 3 ,10 34,12,,  ,28,,34,,20,,
    gap&#40 15 ,3 40,9 6,34 25,18 2130 21 ,43,12 ,
    2
    gap> pi:= Filtered( pi, x -> ClassPositionsOfKernel( x ) = [ 1 ] );;
    gap> Length( pi );
    1
    gap> pi9:= pi[1];;
    gap> List( Irr( table2B, 49,,75 ,63,68 ,63 90 87,,118 118 74,,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
    [ 1, 2, 3, 1, 4, 1, 5, 5, 5, 1, 1, 5, 8, 4, 4, 1, 7, 6, 0, 5, 0, 0,
      10, 7, 0, 0, 10, 6, 6, 13,       2 ,28 19 ,0, 5 16 ,0 ,,0 , ,,26, ,
      6, 6, 5, 0, 0, 0, 3, 6, 7, 11, 0, 0, 17, 2, 20, 9, 0, 0, 12, 8, 1,
      23, 11, 1, 8, 7, 23, 18, 27, 18, 12, 7, 22, 29, 21, 34, 6, 22, 7,
     22,, 33, 3 19 ,34 , 12,15,17,28 34 ,7 ,26
      40, 15, 25, 3, 40, 9, 6, 34, 25, 18, 21, 30, 21, 18, 43, 12, 45,
      39,"
      29, 19, 40, 47, 31,
      75, 29, 24, 72, 50, 41, 72, 68, 58, 52, 54,pi6,pi8, ]java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
      65, 49, 85, 75, 75, 63, 68, 65, 63, 90, 87, 83, 118, 118, 74, 71,
      90, 109, 109, 2, 3, 6, 9, 8, 0, 0, 7, 10, 18, 16, 22, 12, 23, 0, 0,
      2 6 , 19 ,0,5, 16,18,0,0 0,0,0 0 ,1,1 26 ,
      76, 11, 18, 18, 39, 77, 80, 77, 22, 22, 66, 27, 27, 33, 20, 20, 87,
      60, 60, 103, 175, 148, 152, 187, 215, 140, 201 ]

</pre>

<div class="p"></div>
Now we can form the restriction of (1<sub>2.<font size="+0">B</font></sub>)<sup><font size="+<pre>

<div class="p"></div>

<pre>
    gap> constit:= [ pi1, pi2, pi3, pi4, pi5, pi6, pi7, pi8, pi9 ];;
    gap&#java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

</pre>

<="

h2<name">
4</a>  The conjugacyanduse that   classrepresentative ,
</

< class""</div>
We know the order of <font size="+0">M</font> and its prime divisors.
Let us check whether this fits to our data computed up to now.

<div class="p"></div>

<pre>
    gap> sizeM:= pi[1] * Size( table2B );
    808017424794512875886459904961710757005754368000000000
    gap> In order not to count the same class several times,
    "246320*5^*7^6*11^2*3^*17*9**1****"
    gap> sizeM = Size( CharacterTable( "M" ) );
    true

</pre divisor of theelement order.

<div class="p"></div>
For each prime p dividing |<iv=p"</div>
we classifytheconjugacy of elements order p in <font size="+0">M</font>
and use the facts that use the charactertableof <sub<fontsize="0>M</font>/sub>〈 )java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
theclasses of roots of  inthe centralizernormalizerof x
are in bijection with the corresponding classes in <font size="+0">M</font>,
and that this bijectionand ad  arguments.

<div class="p"></div>

<div class="p"></div>
For
we will use the  <h3<aname"th_sEc4..2"java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
to java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
In ordernotto count the sameclass several times,
we proceed by increasing p,
and collectonlythoseclassesof  ofx for  is smallest
prime divisor of the element order.

<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Range [16, 12) out of bounds for length 75
it is not necessary to use the characterwhich isa with  following components.
we will use the permutation character values (1<sub>2.<font size="+0">B</font></sub>)<sup><font java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 0
and ad arguments.

<divclass""<--><div>

     h3<a ="tth_sEc4.2">
4.2</a>  Utility functions</h3>    the group order&124;font size=+"M/font>#124,/dd>
</a

<div class="p"></div>
During of <font size"+"Mfont>,
we record our knowledge about the characterestablishedup to now,/>
in a global <font face="helvetica">GAP</font> variable <tt>head</tt>,
which is a record with the<dd  <r>

<div class="p"></div>

<dl compact="compact">
<dt><b><tt<d  <br/
a  thatcollectsthe currently knownpartialclass fusions into<fontsize="0"M/>
    the java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 46
>>tt</tt<b/>
<dd>  <br />
    the list and <>map/t t list known images
    established upunknownpositions areunbound)</ddjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
<dt><b><tt>OrdersClassRepresentatives</tt></b></dt
<dd>  <br />
    the list of corresponding representative orders,</dd>
<and that thereisan identity element.
<dd>  <br />
     list thatcollects the currently known partial class fusions into <font size="+0">M</font>;
    each
    <tt>subtable</pre>
    and <tt>mapgap#62 head:= rec Size=sizeM
    unknown positions are unbound).</dd>
</dl>

<
We initialize this variable, using the group order <font size="+    #62                fusions:= [],
and that there is an identity element.

<div class="p"></div> function <ExtendTableHeadByRootClasses</tt> takes

<pre>
    gap> head:= rec( Size:= sizeM,
    &62 : [ ]
    >                OrdersClassRepresentatives:= [ 1 ],
    >                fusions:= [],
    &62;             ;

</pre>

<div and such >/> java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 59
The function <tt>ExtendTableHeadByRootClasses</tt> takes
the object <tt>head</tt>,
thecharactertable<stt ofa H font size"0></ont>
and an integer <tt>pos</tt> as its arguments,
where it is assumed    gap&62;ExtendTableHeadByRootClasses=function head,s  )
contains an element x of prime order p
such that N<sub&62 local fus,orders, ,cents oldnumber , ord;
and such that <tt>head</tt> contains information only about
those classes of <font size="+0">M</font> whose elements&#62    = rec, map: [1])java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
that is smaller than p.

<div class="p"></div>#;#Computethe positionsof  classes of pos.

<pre>
    &;function,s pos ))
    >    local fus, orders, p, cents, oldnumber, i, ord;
    >
    >    #Initialize fusioninformation
    >    fus:= rec( subtable:= s, map&#;   : (  ;
    >    Add( head.fusions, fus );
    >
    >    # Compute the positions of root classes of 'pos'.
    >    orders:= OrdersClassRepresentatives62;   #Run over the classes of 's'
    >    p:= orders[ pos ];
    >    cents:= SizesCentralizers( s );
    >    oldnumber:= Length( head.OrdersClassRepresentatives );
    >
    >    # Run over the classes of 's'
    >    # are already contained in head
    >    for i in [ 1 .. NrConjugacyClasses( s ) ] do
    >      ord:= orders[i];
    >      if ord mod p = 0 and
    &62;         Minimum( PrimeDivisors( ord ) ) = p and
    >         PowerMap( s, ord / p, i ) = pos then
    >        # Class 'i' is a root class of 'pos' and is new in 'head'.
    >        Add( head.SizesCentralizers, cents[i] );
    >        Add( head.OrdersClassRepresentatives, orders[i] );
    &62        map]=Length head.SizesCentralizers );
    >      fi;
    >    od;
    >
    >    Print( "#>        Add( h. i;
    >           Length( head.OrdersClassRepresentatives ) - oldnumber,
           "classes,now have"
    >           &62
    #62 end;

</pre

<div class="p"></div>
In several cases, we will establish a conjugacy class g<sup><font size="+0">M</font></sup> without
knowing the character table of a suitable subgroup of/pre>
<tt>ExtendTableHeadByRootClasses</tt> can be applied, where g is<div class""</div>
root classes.
That is, we may know just element order <tt>s</tt>
and centralizer order <tt>cent</tt>.

<div class=p>!-><div>
We are a bit better off if we know the character table <tt>s</tt>
of a subgroup of <font size="+0"> centjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
in table  the gsup =+"<>/>,because then
store a   "0>>  <></>  java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 89
that is stored in <tt>head</tt>.

>-

<pre>
    gap> ExtendTableHeadByCentralizerOrder:= function( head, s
    >    local ord, fus, i;
    >
    >    if IsCharacterTable( s ) then
    >      ord:= Set( OrdersClassRepresentatives( s ){ poss } );
    >      if Length( ord ) <> 1 then
    >        Error( ":= Set( OrdersClassRepresentat java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 68
    >      fi;
    >      ord:= ord&62       [1
    >    elif IsInt( s ) then
    >      ord:= s;
    >    fi;
    >    Add( head.SizesCentralizers, cent );
    >    Add( head.OrdersClassRepresentatives, ord );
    >
     order,ord,"" )
    >    if IsCharacterTable( s )     >    if IsCharacterTable( s ) then
    #62;       stored from s
    >      fus:= First( head.fusions,
    >                   r -> Identifier( r.subtable ) = Identifier( s ) );
    >      for i in poss&62         java.lang.StringIndexOutOfBoundsException: Range [31, 32) out of bounds for length 31
    >        fus.map[i]:= Length( head.SizesCentralizers );
    &62;      od
    >      Print(  #;fi;
    >    fi;
    &#    &;Print : ,
    >           Length( head.>           Length( head.OrdersClassRepresentativesjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
    >    end;;

</pre>

<div class="p"></div>The permutation character 1subHsub>supG<sup>where &8804;G  two,
The   1<>H/><supG</>,whereH ̤  aretwo groupsjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
has the property 1<sub>H</sub><sup #;><><>>/>)&183&124# &;<>G>#8745H124;
8712H
this  we which offuse g>/>
can be computed from the character (1<sub>HThe <></tt the information
ffuse gsup<sup.
The function <tt>where <tt>s</tt> is the character</>isthe table Hjava.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
in <tt>head</tt> by the <tt>ttis  ofpositionsof those classesin<><tt
java.lang.StringIndexOutOfBoundsException: Range [42, 5) out of bounds for length 45
<tt>pi_rest_to_s</tt> is (1<sub>H
and <tt>posspre>
that to<<fontsize""M/><>java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

<div class="p"></div>

<pre>
    gap:= functionhead,s  )
    >    local pival, cent;
    >
    &#;    =Setpi_rest_to_s  };
    >    if Length( pival ) <> 1 then
    &62;     Error ccannot ;
    >    fi;
    >
    >    cent:= pival[1<prejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    >    ExtendTableHeadByCentralizerOrder( head, s, cent, poss );
    &62 end;

/>

<div class="p"></div>

     <h3><a name="tth_sEc4.3">
4./&bsp& of  ofeven<h3>a ="lements_2"
</a>

<div class="<font size=+0"B>(the<tt/t java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
By [<a href="#Mverify" name="CITEMverify">BMW24</a>],
we know that <font size="+0">M</font> has exactly two conjugacy classes of involutions,
and that the involution centralizers have the structures
2.<font size="+0">B</font> (for the classMoreover,   tables   thatare
2<sup>1+24</supsub+/.spanclass""Cospan<sub1/sub (   tt2</>,respectively

<div class=p>!--><div>
Moreover thecharacter ofthese  thatare
stored java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 6
2 =0>>java.lang.StringIndexOutOfBoundsException: Range [68, 65) out of bounds for length 116
as shown in [<a href="#BMverify" name="CITEBMverify">BMW20</a>] and the computations
in [].
For 2<sup>1+24</sup><sub>+<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
in Section <a href="#sect:table_c2b">8</a>.

<div class="p"></div>
Thus we can determine the <font size="+0">M</font>-conjugacy java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 0
as follows.

<div class="p"></div>

<pre>[1 2]
    ;=java.lang.StringIndexOutOfBoundsException: Range [34, 31) out of bounds for length 42
    gap> ClassPositionsOfCentre( s );
    [ 1, 2 ]
    gap> [ ,2]
    #I  after 2.B: found 42 classes, now have 43
    gap> s:=     Iafter^+.o1 91classes   134
    gap> ClassPositionsOfCentre( s );
    [ 1, 2 ]
    gap> ExtendTableHeadByRootClasses<ivclass=""</div>
er 21+24.Co1: found 91 classes,nowhave 134

<>

<div class="p"></div>

     < know=+>/java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 85
4.4</a>  Classes and that normalizers the structures
<>

<div=">
We know that <font size="+0">M</font> has exactly three conjugacy classes of elements
of 3,
and that their normalizers have the structures
3.<span class="roman">Fi</span><sub>24</sub> (for the class <tt>3A</tt>),
3<sup>1+12</sup><sub>+<the font =helvetica></>charactertables 3<class"oman"Fi/>sub24/>and < class"">hspanareA "2><font> tables
and S<sub>3</sub> ×<span class="roman">Th</span> (for the class <tt>3C</tt>), respectively.

<div class="p"></div>
Moreover,
the <font face="helvetica">GAP</font> java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 28
and have been verified#;s(". ;java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

<div class="p"></div>
We determine the <font size="+0">M</font>-conjugacy classes of elements of odd order
that are roots of <tt>3A</tt> or <tt>3C&;=  T")

<div class="p"></div>

<pre; ,,  ;
    & :  " ;
    gap> ClassPositionsOfPCore( s, 3 );
    [ 1, 2 ]
    gap> ExtendTableHeadByRootClasses( head, s, 2 );
    #I  after 3.F3+.2: found 12 classes, now have 146
    gap#62;:( CharacterTable( "h")
    >                                     CharacterTable( "Symmetric", 3 ) )  href="sect:"><a     character
    gap#62;ClassPositionsOfPCore( s 3 )
    [ 1, 3 ]
    gap>  the <tt>3B</tt> normalizer two
    #  after ()  7,  153

</pre>

<>/>
The situation with the <tt>3B</tt> normalizer the
Section <a href="Note  3core of  =3sup>112<sup>>/>6< =roman"<>2
a  java.lang.StringIndexOutOfBoundsException: Range [26, 23) out of bounds for length 104
of the <tt>3B</tt> normalizer, and gives two
candidates for the character table of the <tt>3B</tt> normalizer.

< =""
It will turn out that each of these candidates leads to "the same"
root classes,
in the sense that theFirst we java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 72
and their centralizer orders are java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 28
java.lang.StringIndexOutOfBoundsException: Range [25, 4) out of bounds for length 109
X ×N,
where X has order 3
and N  ≅ 3<sup>1+12</sup><sub>+    ,1819  java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
We are interested 1819 20java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
the factors of H by the one of the java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0
that are not equal to X or Z(exactly one factor of H by a n subgroupof  3 admits a
(See the picture in Section <a href="#sect:norm3B">9</a> for the details.)

<div class="p"></div>
First we pre
nique normal subgroup N of order 3<sup>13</sup>.

<div class="p"></div>

<pre>
    gap> exts:= CharacterTable( "3^(1+12):6.Suz.2" );;
    gap> kernels:= Positions( SizesConjugacyClasses( exts ), 2 );
    [ 2, 18, 19, 20 ]
    gap> order3_13:= Filtered( CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" ) ]
    >        l -> Sum( SizesConjugacyClasses(    gap&62 =java.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 46
       .4]java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    gap> kernels:= Difference("^(+)6.2/1,20]) java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
    [ 18, 19, 20 ]

</pre>

<div class="p"></div>
The
exactly one factor of H by java.lang.StringIndexOutOfBoundsException: Range [0, 28) out of bounds for length 0
class java.lang.StringIndexOutOfBoundsException: Range [0, 12) out of bounds for length 0
of 3<sup>1+12</supgap#;:=(,

<div class="p"></div>

<div class="p"></div>

<pre>
    gap> facts:= List( java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 72
    [ haracterTable( "3(+2g&#   [,2 ))java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
      span.overacc1 {position: relative;  left: .6em
      ^(1+/>
    &62;f=CharacterTable
    gap> facts:= Filtered( facts,
    >        x -> Length( java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    [CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ),
("(+)java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 54

/>

<div class="p"></div>;",java.lang.StringIndexOutOfBoundsException: Range [45, 41) out of bounds for length 65
We
For that,
efirstcreatecopytt</> of informationin<>head/>.

<div class="p"></div>.     oforder13/< /&bsp&;nbsp&bsp;a    java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

<411nbsp    order by<abr/&bsp&;nbsp#;ead5.  ead2f5.java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    gap> kernels:= List( facts,
    &;       f-#;Positions (  , 2 ))java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
    [[  ,[2 ] java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    #;head2=(head;
    &; java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 63
    Iafter3(+2)6Suz 1 19]:found classes
    gap> java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 16
    #I  after 3^(1+12):6.Suz.2/[ 1, 20 ]: found 12 classes, now have 165

</pre>

<div to the sub 2<font size=+"B< =p>java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
We observe that <tt>head</tt> and <tt>head2</tt> differ only by the
two character tables in the last fusion record.

<div class="p"></div>

<
    &;(;
    [ "Size", "OrdersClassRepresentatives", "java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0
      "fusions" ]
    gap> ForAll( Difference( nams, [ "fusions" ] ),
    >            nam -> head.( nam ) = head2.( nam ) );
    true
    gap> Length( head.fusions );
    5
    gap> ForAll( [ 1 .. 4 ], i ->-/
    java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
    &;.w java.lang.StringIndexOutOfBoundsException: Range [27, 16) out of bounds for length 155
    true 196nbsp883   size"0">M/>

</pre>

<class>--><java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
We
bles belongs toasubgroup
be answered inSectionn;ahref"gap#; [  Size (""))

< class=p>!-->/iv

     <h3><a name=/rejava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
4.    java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 28
</div class

< ">
The
ofjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<divclass="p"></div>

< "!----></div>

<
    &;s (".";java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    gap> pos:= Positions( OrdersClassRepresentatives( s ), 5 );
    [23 25 ]
    &62 {pos }
    [ 1539000,   degree196nbsp883 character 2< =+><iv=>!

</pre this the case  already for their  2< size+"  fromfollowing   2<size=+"<>

<div class="p"></div>
establishestwo classes <>5A/t,<tt>B</tt> <re
of elements of order 5 in    ,,-,     ,     275      ]
ith  5&124<class"><[1  ]

<div class="p"></div>

<pre>
    gap> cents:= List( pos,
    &62;      &62 [i*(s / SizesConjugacyClasses  )i ;
    [1365154560000000 94500000000 ]
    gap&#62
    >              5^7 * Sizeo involutions  also4371 mustoccur
    true

</pre>

<div class""!--<div

<  irreducible  .span"oman></span><>24<sub>sup>&8242<sup>can  as
By [<a href="#Mverify" name="CITEMverify">BMW24</a>],
">< gap#2s:=  ";

<ttA</t with centralizer 5 ×<spanclass="roman">HN</span> and normalizer
(gap#2;PrintArray (mat )
and <tt>5 ,      133]
4584   ,72,

<div class="
he two classes rational
because this is java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0

<div class="p"></div>

<="></div>
   62 :java.lang.StringIndexOutOfBoundsException: Range [34, 31) out of bounds for length 42
    (2
    gap> ClassPositionsOfPCore
    [1,45  ]
    ( head , 45 )
    #I  after (D10xHN).2: found 5 classes, now have 170

</pre>

< =p>
The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</271689
with <font face="helvetica">MAGMA</font>, see gap#62;mat[3] mat[6] +mat[7]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
get
the

<div class="p"></div>

>
    gap&In  to  the java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
    CharacterTable( "5^(1+6):2.J2.4" )
    gap#62; 5core=ClassPositionsOfPCore( s, 5 );
    [ 1 ..4 ]
    gap&#62<pre
    [1, 4,     =FilteredIrr( )x-#2 [1 &; 196883)java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    gap> ExtendTableHeadByRootClasses( head, s, 2     gap&#le='color: green'>62; mat:= List( cand,x-&#62 x{Concatenation(
    #I  java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0

</pre 1157 ,

<div

     <h3><a name="tth_sEc4.6">
4.6</a> &div class="p"></div><2>a ="tth_sEc3"java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
The group 2.<font size="+0">B</font> contains a rational class of elements of order 11.
The permutation character (1<sub>2.<font size="+0">B</font></sub><sup><font size="+0">M</font></sup><div class"p"></div>
f 11 withorder 11&124;sub12</>&124 <java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 93

<div class="p"></div>

<pre>
    gap&62 := CharacterTable "2.B" );
    gap> pos:= Positions( tr><td>a c  ∈     </td><td>G<sub>a,c</sub="right">&#n style='color: green'>124;c<sup>G<sub>a</sub></sup>| </td></tr>
    [ 71 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

</pre>

<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 75
<font size="+0">M</font> has := Positions( OrdersClassRepresentatives;

<div class="p"></div>

<pre>
    gap&62;ExtendTableHeadByPermCharValue>
    #I  after order 11 element from 2.B: have 174 classes

<>

<</div>

java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 30
4.I this,  compute nine 1<>U/><sup2< size"0></<sup>,
</a>

<div class="p"></div>
The elementsThe equation<iv=p>--->/>
Sylow  thatimplies

< =p></div>
The elements of order 17 and 19 are rational in 2.<font size="+0">B</font>
and hence also in <font size="+0">M</font>.

<div class="p"></div>

<
    gap> s:= CharacterTable( "2has been  in [<a href="#BMverify" name="CITEBMverify">BMW20</a>]
    gap> pos:= Positions( OrdersClassRepresentatives( s ), 17 );
     118 ]
    java.lang.StringIndexOutOfBoundsException: Range [49, 7) out of bounds for length 64
    Iafter17  element 2B have 175 classes
    gap> pos:= Positions( OrdersClassRepresentativeswe have [N<sub><font size="+0">Mwe have >U/><up2.font size=+0"</>/>  <>2.font size=""B/>/>
    [>
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos );
    #I  after order 19 element from 2.B:</re>

<>

<div class="</div>
Forelements g order p  ∈; { 3, 31,47 },
the group2.<size+"B</ont> contains exactly two Galois conjugate classes that contain
the nonidentity powers of g,
which means that [N<sub>2.<font size="+0">B</font></sub>(〈g &#n style='color: green'>9002;):C<sub>2.<font size="+0">B</font></sub>(g)] = (p−1)/2 holdsdo notfuse.
The equation &#
implies

<brclear=all"/< border=""width="100%>tr>>
<table align="center" cellspacing="0"  cellpadding="2"><tr><td nowrap=    
&gap> pi2= Inflated(tableB,,)1];
<>/<table

Note that
in0>font
in   ,
we have [N<sub><font size="+0">M</font></sub>(〈g 〉):C<sub><java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 60
&java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
whereas  have
#9001g&9002;:<("^+)o2java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
|g<sup><java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 8
Thus we can compute     &62 =(,java.lang.StringIndexOutOfBoundsException: Range [46, 44) out of bounds for length 55
di consider =class/<>23/java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

div=""</div>
p23.

<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<pre>
    gap> s:= CharacterTable( "2.B" );
    CharacterTable( "2.B" )
    gap> ord s );
    izesConjugacyClasses( s );;
    gap>208,220 ]
    gap> pos:= Positionsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ 147, 149 ]
    gap> n:= (p-1)/2 * Size( s ) * java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 0
    6072
    gap#62 ( Factors  ))java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    [ [ 2, 3 ], [ 3, 1 ], [ 11, 1 ], [ 23, 1 ] ]

/>

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
In to thatthe  classes ofelementsof  23in 2.font="+div "">-->/>
do  fuse in< size"0><font>, it suffices to show that the centralizer order is
<>3/>java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
We seethat this is the casealready in the <tt>2B</tt> centralizer     gap#62;ExtendTableHeadByPermCharValuehead s,pi,pos]  )

<iv class=p</div>

<pre>
    gap> u:= CharacterTable( "MN2B" );
    CharacterTable( "2^1+24.Co1" )
    2 upos: Positions(java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 6
    [ 289, 294 ]
    gap&62; SizesCentralizers( u ){ upos } / 2^3;
    [ 23, 23 ]

<pre>

<div class="p"></div>
Thus we have<iv     gap&62;mcl=CharacterTable( "McL" );;

<iv class="p"></div>

<pre>
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 71
    #I  after order 23 elementgap> Length( ind);
    gap> ExtendTableHeadByPermCharValue
    #afterorder 23 element 2.:have classes

<>

<div class class=p></div>
The ompute 1<ub</sub<sup>.fontsize="+0">B</font></sup> with the <font face="helvetica">GAP</font> function <tt>PermChars</tt>,
Here the necessary 2-part of the centralizer occurs already in 2.<font size="+he normal subgroup &9001a &9002;of .<font size=+0"><font>

<ivclass"">!----></div>

<pre
ap62 p:31;
    gap> pos:= Positions( OrdersClassRepresentatives( s ), p );
    [ 190, 192 ]
    gap> n:= (p-1)/2
    2790
    gap> Collected( Factors( n<divclass="    java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 83
    [ [ 2, 1 ], [ 3, 2 ], [ 5, 1 ], [ 31, 1 ] ]
    gap> SizesCentralizers( s ){ pos };
     ,62 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    gap> ExtendTableHeadByPermCharValue( head, s, pi,      0,, 1, 12 5 3, 1,1,4,5, 1,1 , 3 0 0,  p#62; p= 47;;
    #I  after order 31 element from 2.B: have 179 classes
    gap> ExtendTableHeadByPermCharValue( head, s, pi, pos{ [2] } );
    #  after order 31 elementfrom 2.:  180 classes

/pre>

<div class="p"></div>
Finally,we  with p= 47.

<iv class=p"</div>
The class <tt>13A</tt> of <font size="+0">M</font> arises from the rational class of
of order 13 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
We use the permutation character to enter the information about
java.lang.StringIndexOutOfBoundsException: Range [19, 3) out of bounds for length 23

<=p>

<pre>
    gap> p:= 13;;
    gap> pos:= Positions( OrdersClassRepresentatives( s ), p );
      java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
    gap> c:= Size( s ) * pi[ pos[1] ] / classes[
    73008
    gap> Factors( c );
    [ 2, 2, 2, 2, 3, 3, 3, 13, 13 35 ,,29  ,  ,38,24, 17,35,
    gap>; ExtendTableHeadByCentralizerOrder( head c2b, 13^ * 24, ,61414 8 28 ,  , ,92 ,56
           2B:  183

<>

<div class="p"></
The class <tt group 3[( ) ,map1] ;java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
Here we just know the centralizer order 13<java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 28

<div class="p"></div>

<pre>
    gap> c2b:= CharacterTable(     gap&>62;pos: Positions((       10,7 0 ,10 6  ,, ,,0  ,,,4 ,
    gap> pos:= Positions( OrdersClassRepresentatives( c2b ), 13 );
    [ 220 ]
    gap> ExtendTableHeadByCentralizerOrder( head, c2b, 13^3 * 24, pos );
    #I  after order 13 element from 2^1+24.Co1: have 184 classes

</pre>

<div class="p"></div>

      ,,40 47 ,69,69,,42 ,68 ,73 ,53,,38
bleby29<h3>a=">
/a>

<div class="p"></div>
The group 3.<span class="roman">Fi</span><sub>24</sub> 2, 6, 28, 19, 0, 0, 5, 16, 18, 0, 0, 0, 0, 0, 0, 52, 1, 1
with centralizer order 3 ·29.

<div class="p"></div>

<pre>
    gap#62;u CharacterTable 3. ;java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    gap&#2 pos: Positions OrdersClassRepresentatives  ) 29)java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    [ 142 ]
    gap&#62    #2hSizesCentralizers;
    [ 87 ]

<

<div class="p"></div>
The list of classes of <font size="+
elements of the orders 2, 3, 5, 11,  <h3> name">
and 29 occurs as a factor of the centralizer order only for the
tt/>> <>3A> <>87AAtt><>87B.

<class""</div> font=""/font ofanelement order29  741,59 71

<pre
    gap> poss:= PositionsProperty( head.SizesCentralizers,
    >                              x -> x mod 29 = 0 );
    [ 1, 135, 144, 145 ]
    gap> head.OrdersClassRepresentatives{ poss };
    [ 1, 3, 87, 87 ]
    java.lang.StringIndexOutOfBoundsException: Range [28, 7) out of bounds for length 44
    [ 808017424794512875886459904961710757005754368000000000,
      3765617127571985163878400, 87, 87 ]

</pre>

<div class="p"></div>
Thus the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ont size=+"M<font>of an element x of order 29 are 7, 41, 59, and 71.

<---></div

<pre>
    p   the order.
    &62;                     [ 2,3 5, 11,13 17 19,23 29 31, 47 ]);
    [ 7, 41, 59, 71 ]

</pre>

<div class="p"></div>
The centralizer order
3 ·29 ·7<sup>i<42/    p#62;parts=( ( (headSize ,
with  0  ≤ i  &#8804java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class global=java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 69

<pre>
    gapdlcompact">
    >                      x -> x[ />
    [ [ 7, 6 ], [ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> poss:= List( partst list of  ordersof  conjugacyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    &62;cart=Cartesian  );
    gap&62;possord::=3  29 *List cart,roductjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55

</pre>

<div class="p"></div>
Only 3 ·29 and 3 ·29 &#yle='color: green'>183;59 each   a with components
that is,     <java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
Note that we have [N<sub><font size="+0">M</font></>

< p>

<pre>
    gap>List(good, <ivclass""></div>
    &62;                     x -#;( head
    gap#62; head:=recSize:= sizeM,
    gap> List( good, Factors );
    [  3, 29 ] [ 3,    &62                : 1 ]

</pre>

<div The functi<t>xtendTableHeadByRootClasses>takes
Now we can exclude the possible the character table <tt>s</tt> of a subgroup H of <font size="+0">M</font>,
the  thatSylow59  would
and thus would be normalized and hence centralized by an element ofcontains an element x of prime p
order 3, a contradiction.

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<pre>
    gap> java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 5
    [ 1 ]

</pre>

<div class="p"></div>
Thus we have established a rational class of elements of order 29,
with centralizer of order 3 ·29.

<=java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 28

<pre>
    gap> java.lang.StringIndexOutOfBoundsException: Range [0, 46) out of bounds for length 35
    #  after order 29  from3F32:have185 classes

<pre

<div

     <h3#;       .map] of  3&183;.
4.10</a>  Classes of  >     fi;
</a#;java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

<div class="p"></div>
We assume that <font size="+0">M</font> contains a subgroup of the structure 3<sup>8</sup>.O    #I  after order29element from3.3.:have185classes
The fact that an element x of order 41 in <font size="+0">M</font> is normalized
by an element of order 4 can be java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
O<sub>8</sub><sup>−</sup>(3).

<div class="p"></div>

<
    gap> t:= CharacterTable( "java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 4
    CharacterTable( "O8-(3)" )
    gap> Length( Positions( OrdersClassRepresentatives( t ), 41 ) );
    10

</pre>

<In several ,we will establisha conjugacyclass <><fontsize"0>/   oforder 4 anbe    the
By the above arguments, the only possible odd prime divisors of
&<>8<ubup&#722</>().
where 5 cannot divide the centralizer order.
That is we  know just element order <tt>s</tt>
andget
|N<sub><font size="+0">M</font></java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 28

<div class=""</div>get
Suppose that 71tsize="+"Mfont<sub(&9001;&62     IsCharacterTable  then
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
  element 8
and thus centralized by pre

<div class="p"></div>

<pre>
    gap> Filtered( DivisorsInt( good[2] ), x -> x mod 71 = 1 );
    [ 1]

/>

< <head> the for class <sup><size"0"M/>/sup
Thus we tt>pi_rest_to_s> is(<sub><sub><sup>G</sup>)<sub>H</sub>,
of order 41.

<div class=Tthe    N>size="0

<pre>
    gap&;ExtendTableHeadByCentralizerOrder, 41 41, );
    #I  after order 41 element: have 186 classes

</pre>

<classp>!--</iv>

     <h3><a name="tth_sEc4.11
4.11</a>  Classes of java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 30
</a>

<div class="p"></div>
thejava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 23
the normalizer order of an
58 ·7<suppre  java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 58
Sylows admitsjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 61

<div as shown < ="BMverify"Cjava.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 84

<pre>
    gap> possord:= 59 * DivisorsInt( 58*7^6*71 );;
    gap> good:= Filtered( possord,

    [ 1711 ]
    gapjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 5
    [ [ 29, 59 ] ]

</pre>

<div class="p"></div>
Thus we have established a pair ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
self-centralizing elements 3;9&18371java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

<div class="p"></div>

    gap> ExtendTableHeadByCentralizerOrder( head, 59, 59java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    #I  after order 59 element: have 187 classes
    gap> ExtendTableHeadByCentralizerOrder( head, 59, 59, fail );
    #I  after order 59 element: have 188 classes

<java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

<class"-->/iv>

     <h3><a name="tth_sEc4.12">
4.12</a>  Classes of elements of order divisible by 71</h3><a name="elements_71">
</a>

<div class="p"></div>
By java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [14, 3) out of bounds for length 54
70 ·7<sup>5</sup> ·71.
Sylow admits normalizer71&;.

<div<iv =java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<pre>
    gap> possord:= 71 * DivisorsInt( 70gap&62 ExtendTableHeadByRootClasses(, s  )
    gap> good:= Filteredgap&62 : ( CharacterTable" ,
    >                     x -> ( head.Size / x ) mod 71 = 1 );
    [ 2485 ]
    gap> List    gap#62;( head,59, 59  )
    [ [ 5, 7, 71 ] ]

</pre>

< java.lang.StringIndexOutOfBoundsException: Range [0, 7) out of bounds for length 0
Thus we have established a pair of Galois conjugate classes divclass=p>
self-centralizing elements of order 71.

<div class="p"></div>

<pre>
    gap> java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    #I  after order 71 the normalizer order of an element orderof element order 71 divides
    not toX ()
    after order 71element   190 classes

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc4.13">
4.13</a>  Classes of elements of order divisible by 7</h3><4.13</a>  Classes of elements of order divisible by 7</h3><a
</a>

=p
.size+></>yieldsa  <>7Att
with java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 6

<div divjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<pre>
    &62s=(2"java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
;:=PositionsOrdersClassRepresentatives(s 7 )
    [ 41 ]
    gap>gap&62 facts:Filtered,
    #   order7   2B have191classes
    gap> Last( head.SizesCentralizers ) = 7 * Size( CharacterTable( "He" ) );

<>

<div class="p"></div>
By
structure 7 ×<span class="roman">He</span>,gap<a
and the normalizer has the structure (7:3 ×<span class="roman">He<[[2 ] [2]]
a subdirect product of 7e=+0>B/font>    gap&62;ead2=StructuralCopy  );;

<div class="p"></div>
Since"oman"Hespan>hasa ofGalois#  3(+)6S./1  ]found12 classes   165
we get also a pair ofIafter 3(+)6Suz.2[1 20]  12 classes
&183;17  119.

=>

<preI  order elementfrom .: havejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> "Size", "OrdersClassRepresentatives"

    <pre
    #I  after

</pre>

<div class="p"></div>
second  java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 8
is java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
two classes of elements of order 7, We continue with
degreeediv=p>!--</>

<divSince span class"roman">He<span> hasapairofGaloisconjugateclasses of element order

<>
    gap> u:= CharacterTable( "3.Fi24" );;
    gap&#</>
    gap> rest:= Sumdiv class"p"</div>
    gap> pos:= Positions( OrdersClassRepresentatives( u ), 7 );
    [ 41, 43 ]
    gap> rest{ pos };
    [ 50, 1 ]

/re

<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28
Note that class 43 fuses to <tt>7B</tt> because the restriction of <pre>
to 2.<font size="+0">B</font> has the value 50 on <tt>7A</tt>.

<div class="p"></div>

    (u5 ()  43] )
    #I  after order 7 element from 3.F3+.2: have 194 classes

</pre>

<div class=divclass=p></div>
  size+"/.

<="p">!-->/div elements order5 fontsize"0></ont>java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

<pre>
    gap> Sum( head.SizesCentralizers, x -> head.Size / x/pre>
    true

</pre>

<div class="p"></div>
We initialize the character table head of <font size="+0">M</font>.

<div class="p"></div>

<
    mConvertToCharacterTableNC rec
    >      UnderlyingCharacteristic:= 0,
    >      Size:= head.Size,
    >      SizesCentralizers:= head.SizesCentralizers,
    >      OrdersClassRepresentatives:= head.OrdersClassRepresentatives,
  ;

</pre>

<div class="p">

#; (s, 5)
5;&bsp;he  mapsof < size"+0">M<font<h2>a name="powermaps"java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
class">

<div class="p"></div>
Using the element orders of the class representatives of the table
of <font size="+0"the <=h"GAP</font>character table library.
previous sections, we compute approximations of the p-th power maps classpre
for primes    #UnderlyingCharacteristic=0java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44

<div class="p"></div>
otethat havenot yet determined 1.4 ]
tables of the <tt>3B</tt> normalizer belongs to a subgroup of <font size="+0">M</[1,,37800,40320 java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
fusionjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49

<div class="p"></div>

sjava.lang.StringIndexOutOfBoundsException: Range [28, 25) out of bounds for length 51
    >        r -> not IsIdenticalObj( r.subtable,     >        r -> not IsIdenticalObj( r.subtable, facts
    gap> Length( safe_fusions );
    6

</pre>

<div class="p"></div>
First we initialize the class fusions, compatible with the definitions of
theclassesasgivenbythe partialfusions whichwe havejava.lang.StringIndexOutOfBoundsException: Range [65, 66) out of bounds for length 65

<div class="p"></div>

    thus   correspondingpartialfusionjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
    >      fus:= InitFusion( r.subtable, m );div=""</div>
    >      for i in [ 1 .. Length( r.map ) ] do
    >        if IsBound( r.map[i] ) then
    >          if IsInt( fus[i] ) java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 6
    >            if fus[i] <> r.map[i] then
    &    gap#62;Length  ;
    >            fi;
    >          elif java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
    &62             notr.map] ini
    >              Error(Sylowsubgroups  First we initialize the classfusions,compatible thedefinitions
    >            fi;
    >          else
    >            if not IsSubset( fus[i], r.map[i] )also<size+"<>java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
    >              Error( "fusion problem" );
    >            fi;
    >          fi;
    >          fus[i]:= r.map&;= java.lang.StringIndexOutOfBoundsException: Range [58, 56) out of bounds for length 68
    #62;fi
    >      od;
    >      r.fus:= fus;
    >    od;

</pre>

<div class="<pre>
Next we initialize approximations of the power maps of the table of <font size="+0">java.lang.StringIndexOutOfBoundsException: Range [0, 85) out of bounds for length 28
and improve them using the the group 2.<font size="+0">B</font> contains exactly two#else
of the subgroups w. r. twhich meansthat [<sub2.<font size=+0"Bfont>/> #9002):sub>2< size"0"B</font>/>()] =)2holds.
Note that also the knowledge about the class fusions increases this way.

<div class="p"></div>

<pre>
    gap> maxorder:= Maximum( head.OrdersClassRepresentatives );
   java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    gap62;powermaps: ];
    gap> primes:= Filtered( [ 1 .. maxorder ], IsPrimeInt );;
    gap> for p in primes do
    >      powermaps62    ;
    >      for r in safe_fusions do
    >        subpowermap:= PowerMap( r.subtable, p );
    #;        if( subpowermap,r. powermapsp )  then
    62          Error inconsistency" );
    >        fi;
    >      od;
    &62;;    odjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16

</pre>

<div class="p"></div>
Werepeatapplyingthe  until further
improvements found.

<div&62      []:InitPowerMap, ;

<pre>
    gap> found:= true;;
    gap> res:= "dummy";;  # avoid a syntax warning
    gap> while found do
    >      Print( "#I  start a round\java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 18
    ></d/tr>
    >      for p in primes Note that either the two classes  of
    >        for r in safe_fusions do
    >          subpowermap:= PowerMap( r.subtable, p );
    >          res:= TransferDiagram( subpowermap, r.fus, powermaps[p] );
    &62;if = fail then
    >            Error( "    &62 : dummy";#avoidwarning
    >          elif ForAny( RecNames( res ), nam -> res.( nam ) <> [] ) then
    >            whereas found=false;
    >          fi;
    >        od;
    &#awe then  whythe &;         : ( r, [] )java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    >    od;
    #Istart &2 elif(RecNames res, -&;es am  &;62 ]
    #I  start a round
    #<ivclass="">!--<div
    #I  start a round

</pre>

<div java.lang.StringIndexOutOfBoundsException: Range [21, 22) out of bounds for length 21
see themaps
starting with the 5-th power map.   3, ,1 , , ,[ 23   java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

< map

<>
     class"</div>
    [ 157, 158, 163, 164, 187, 188, 189, 190,    gap#62;; pos:= PositionsProperty(powermaps[5], IsList );
    gap> head.OrdersClassRepresentatives{ pos };
    [ 15, 15, 39, 39, 59, 59, 71, 71, 119, 119 ]

</pre>

<div class="p"></div>
The/pre>
areunderstandable:
For each  theseelement orders,there a ofGalois
classes, and the subgroups whose class fusions we have used do not contain
these elements.

<div class="p"></div>
For each of the primes l  &these elements
  - java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 70
which   order
- java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
fixes aclass ofelementorder if and only if
the Galois automorphism that raises l-th roots of unity to the pth power
fixes √{−l}.

< ="p>
In the case of element
thefieldInorderto whichof them ,
<font size="+0we look at  subgroup that  elements of order 119.
In order to decide which of them actually occurs,<re>
we look at a subgroup that contains The <tt>7A</tt> centralizer in <font size="+0">M</font> has the structure 7 ×<span class="roman">He</span>,
  &62; 31;java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
and the java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
n; #java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 48
see Section <

<div class="p"></div>
The     [ 62, 6
java.lang.StringIndexOutOfBoundsException: Range [69, 70) out of bounds for length 62
and the character values in the subgroup lie in the field <font size="+0">Q</font>(√{−119}).

<div class="p"></div>

<pre>
    gap> u:= CharacterTable( "(7:3xHe):2" );;
    gap> ConstructionInfoCharacterTable( u );
    [ "ConstructIndexTwoSubdirectProduct", "7:3", "7gap&62; u:= CharacterTable( "(7
      [ 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129,
          130,,,133,134 ,207 208,209,210,211 ,
          213, 214, 215, 216, 217, 218, 219, 220, 221,      "java.lang.StringIndexOutOfBoundsException: Range [14, 10) out of bounds for length 16
          225, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307,
          308,309,310, 311,312,313, 314 315], (,( ]
    gap> pos:= Positions>(  ;
    [ 52, 53 ]
    gap> f:= Field(,298,, 299, 300 301,302, 303 304, 305, 306,307[94,94 ]
    gap> Sqrt(-119) in f;
    true

</pre>

<div class="p"></div>
We insert the relevant power map values.

<div class="p"></div>

<pre>
    gap> for l in [ 59, 71, 119 ] do
    >      val:= Sqrt( -l );
    >     poss:= Positions( head.OrdersClassRepresentatives, l );
    >      for p in primes do
    #;        if(   )= 1then
    >          if GaloisCyc( val, p ) = val then
    >            powermaps[p]{ poss }:= poss;
    >          else
    >            powermaps[p]{ poss }:= Reversed( poss );
    >          fi;
    >        fi;
    >      od;
    >    od;

</pre>

<div class="p"></div>
Now p-th power maps, for p  ≥ 17,
java.lang.StringIndexOutOfBoundsException: Range [56, 3) out of bounds for length 60
of element order 39.
These classes had been found as roots of <tt    gap&62;Factors( c );

<div class="p"></div>

<pre>
    gap> PositionsProperty( powermaps[17], IsList );
    [ 163, 164 ]
    gap> head.OrdersClassRepresentatives{ [ 163, 164 ] };
    [ 39, 39 ]
     (  fusions
    >                      java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 83
    >          div"
    [ CharacterTable>

</pre>

<div classgap#2 ( head c2b 133* 24,pos;
In order to decide whether the p-th power map fixes or swaps
the two classes, we consider elements of order 78,
which are the square roots of the order 39 elements.
There are three> =java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 30
a rational class that powers to <tt>2A</tt>
and a pair of Galois conjugate classes that power to <divclass">-->/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<div class="p"></div>

<pre>
    gap> 78pos:= Positions( head.OrdersClassRepresentatives, 78 );
    [37, 132 133 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    gap> head.fusions[1].subtable;
    CharacterTable( "2.B" )
    gap> Intersection( 78pos, head.fusions[1].map );
    [ 37 ]
    gap> s:= head.fusions[2].subtable;
    CharacterTable( "2^1+4.Co1")
    gap> Intersection( 78pos, head.fusions[2].map );
    [ 132, 133 ]
    gap> Positions( head.fusions[2].map, 132 );
    [ 342 ]
    gap> Positions( head.fusions[2].map, 133 );
    [ 344 ]
    gap> PowerMap,7)]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    344

</pre>

<div class="p"></div>
Since the <tt>3B</tt> normalizer in <font size="+0">M</font> contains a pair of Galois conjugate
classes of element order 78 which power to the generators of the normal
subgroup of order 3,
these two classes fuse to the non-rational <font size="+0">M</font>-classes of elements
of order 78,
and their squares are the classes of element order 39 we are interested java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 0

<div class="pThus the  possible additional prime divisors of the centralizer order

<pre>
    gap> poss:= Filtered( head.fusions
    gap> List( poss, r -> r.subtable );
    [ CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" ) ]
    gap> 62[3  , ,,23 ,,  java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
    173
    gap> Position( poss[1].map, 164 );
    174
    gap> List( facts, s -> Positions( OrdersClassRepresentatives( s ), 39 ) );
    [ [ 173, 174 ], [ 173, 174 ] ]
    gap> List( facts, s -> PowerMap( s, 7 )[173] );
    [ 174, 174 ]

<;java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 61

<div class="p"></div>
d   on twoclasses fontsize="0"M>is
to the corresponding field of character values in the <ttgap#;poss=List(parts  -62 ([0 .l]] i#;[1^  ;java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
which is <font size="+0">Q</font>(√{−39}).
(Note java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0
to the <tt>3B</tt> div class="p"></div>
utgetsamejava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 48

<div class="p"></div>

<pre
    &62 := Listiv=p</div>
    >                   s -> Field( Rationals, List( Irr( s ),
    >                                                x<iv=">
s;
    1
    gap> Sqrt(-39) in fields[1];
    true

</pre>

<gap#62 ( Filtered .fusions
Now we#;r &62 ( r., [ 163    we  excludethecentralizer 3 &183;&83;java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69

<div class 3

<pre>
    gap> val:= Sqrt( -39 );;
    gap> poss:= [ 163, 164 ];;
    gap> for
    >      if Gcd( 39, p ) = 1 then
    >        if GaloisCyc( val, p ) = val then
    >          powermaps[p]{ poss }:= poss;
    &There are classesof java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
    >          powermaps[p]{ poss }:= Reversed( poss );
     fi
    >      fi;
    &<iv=""</div>

<pre>
    gap> r2:= First( head2.fusions, r -> IsIdenticalObj( r.subtable, facts[2] ) );;
    f= .  )java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
    gap> for i in [ 1 .. Length( r2.map ) ] do
    >      if IsBound( r2.map[i] )
    >        if IsInt( fus2[i] ) then
    >          ifgap&62 java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 6
    >            Error( "fusion problem" );
    >          fi;
    >        elif IsInt( r2.map[i] ) then
    >          if not r2.map[i] in fus2[i] then
    >            Error( "fusion problem" );
    &62;          fi
    >        else
    >          if not IsSubset( fus2[i], r2.map[i] ) then
    >            Error( "fusion problem" );
    >          fi;
    >        fi;
    >        fus2[i]:= r2.map[i]< classp"
    >      fi;
    >    od;
    gap> r2.fus:= fus2;;

</pre>

<div class="p"></div>
gap62 - in[]
of power maps, andtrue
power maps in

<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

<pre>
    gap> powermaps2:= StructuralCopy( powermaps );;
    gap> s:= r.subtable;
    CharacterTable( "pre>
    ;  in primesjava.lang.StringIndexOutOfBoundsException: Range [31, 32) out of bounds for length 31
    >      if TransferDiagram( PowerMap( s, p ), fus, powermaps[p] ) = fail then
    >        Error( "    >      if Gcd( 39, p ) = 1#62;     if Gcd( 39, p ) = 1 then
    >      fi;
    #;od< name"12>
    62 :   &#62 powermaps] }=;
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
    gap> for p in primes do
    >      if TransferDiagram( PowerMap( s2, p ), fus2, powermaps2[p] ) = fail then
    &62;         ")    #;      ;
    #;      ;
       #;
    gap> powermapsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    true
    gap> List( powermaps, Indeterminateness );
    [ , 32, 64,, 1,, 1,,,, 1,, 1,,,, 1,, 1,,,, 1,,,,,, 1,, 1,,,,,, 1,,,,
      1,, 1,,,, 1,,,,,&62;                    x#;(head  x )  71=)
      , 1,,,,,1,,1,1,,1,1&62;(, ;

  ,

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
bout squarestherational
of element order 78.

<div class="p"></div>

<pre>
    gap> powermaps[2]{ [ gap#;  ,71  java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
    [ [ 163, 164 ], [ 163, 164 ] ]
    gap> pos78:= List( facts,
    >                  s ->     gap>;fus= InitFusion.subtable ;
    [ [ 235, 236 ], [ 235, 236 ] ]
    gap> fus{ [ 235, 236 ] };
    [ [ 132, 133 ], [ 132, 133 ] ]
    gap> fus2{ [ 235, 236 ] };
    [ [ 132, 133 ], [ 132, 133 ] ]

<>

<div class="p"></div>
We may identify one class of element order < class"p"</div>
#;

<div class="p"></div>&62;:  (  ,7 ;

<
    gap> fus[235]:= 132;;
    gap> fus2[235]:= 132;;
    gap> TransferDiagram( PowerMap( s, 2 ), fus, powermaps[2] ) <&#pan style='color: green'>62; fail;
    true
    gap&;    odjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    true
    gap> powermaps = powermaps2;
    true
    < class="">-->/>
    [ 8, 64 ]

</pre>

<div class="p"></div>
classes of   39
,this  now using the
2nd and the 3rd power map commute.

<div class="p"></div>

<pre>
    gap> powermaps[3]{ [ 163, 164 ] };
    [ [ 183, 184 ], [ 183, 184 ] ]
    gap> TransferDiagram( powermaps[2], powermaps[3], powermaps[2] ) <> fail;
    true
    gap> List( powermaps{ [ 2, 3 ] }, Indeterminateness );
    [ 8, 16 ]

</pre>

<div class="p"></div>
The next open question is about >
The classes of element order 93 -a pair of Galois conjugate classes-
have been found inside subgroups
of the type S<sub>3</sub&62 od
we have considered.
Thus we may choose which of them cubes to thegap&#; { ;
31.

<div class="p"></div>

<pre
    gap&#62  two .
    >                     r -> 93 in OrdersClassRepresentatives( r.subtable ) );;
    gap> List( poss, r -> r.subtable )<divclass=">
    [ CharacterTable( "ThxSym(3)" ) ]
    gap> pos93:= Positions( head.OrdersClassRepresentatives, 93 );
    [ 152, 153 ]
    gap> powermaps[3]{ pos93 };
    [ [ 179,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> powermaps[3][152]:= 179;;
    gap> TransferDiagram( PowerMap( poss[1].subtable, 3 ), poss[1].fus,
    #;                    powermaps[3 )&t&62 ;
    true
    gap> List(    gap> forp in primes do
    [ 8, 4 ]

java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 6

<div class="p"></div>
The next open question&62;m=ConvertToCharacterTableNC(rec(
The classesgap>powermaps =powermaps2
have beentrue
of the type 3.<span class="gap#62; List(powermaps,Indeterminateness ;
we have considered.
Thus we may choose which of them cubes to the first class of element order
23.

<div class="p"></div>

<pre>
    gap> poss:= Filtered( head.fusions,
    >                     r -> 69 in OrdersClassRepresentatives( r.subtable ) );;
    gap> List( poss, r -> r.subtable );
    [ CharacterTable( "3.F3+.2" ) ]
    gap> pos69:= Positions( head.OrdersClassRepresentatives, 69 );
    [ 142, 143 ]
    gap> powermaps[3]{ pos69 };
    [ [ 177, 178 ], [ 177, 178 ] ]
    gap> powermaps[3]{ [ 142, 143 ] }:= [ 177, 178 ];;
    gap> TransferDiagram( PowerMap( poss[1].subtable, 3 ), poss[1].fus,
    >                     java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 28
    <div cla="p"></div>
    gap> List( powermaps{ [ 2, 3 ] }, Indeterminateness );
    [ 8, 1 ]

</

<div class="p"></div>
The next open question is about the squares of certain elements of order 46.
There are two pairs of Galois conjugate classes of element order 46,
and the 2nd power map is not yet determined for those classes which power
to the class <tt>2B</tt>.

">--/>

<pre>
    gap>  givenjava.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 65
    [ 26, 27, 118, 120 ]
    gap> powermaps[2]{ pos46 };
    [ 177, 178, [ 177, 178 ], [ 177, 178 ] ]
    gap> powermaps[23]{ pos46 };
    [ 2, 2, 44, 44 ]

<>

<div62           java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 46
We have defined the two classes of element order 23 as squares of those
two classes of element order 46 that power to#            ;
and we have not yet distinguished the other two classes of element order 46.
Thus we    #if  ( [] [i then

<div class="p"></div>

<pre>
    gap> powermaps[2]{ [ 118, 120 ] }:= [ 177, 178 ];;
    gap> Indeterminateness ( powermaps#;r:= java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    2

</pre>

<""></div>

<
    gap> <div class="p">div>
    [ 155, 156 ]
    gap> head.OrdersClassRepresentatives[78];
    18
    62 .[78]
    3888

</pre>

<div class="p"></div>
There are two classes with this property in <font size="+0">M</font>,
both are roots of the generators of the normal subgroup of order 3 in
3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2, and the corresponding two classes in this subgroup
have the same square.

<div class="p"></div>

<pre>
    gap> Filtered( [ 1 .. Length( head.OrdersClassRepresentatives ) ],
    >              i -> head.OrdersClassRepresentatives[i] = 18 and
    >                   head.SizesCentralizers[i] = 3888 );
    [ 78, 79 ]
    java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 27
   ,
    gap> powermaps[2][52];
    154
    gap&62;First .usions r#62forp  do
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]" )
    gap> s:= facts[1];
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 19 ]"     >          res then
    gap> pos18:= Filtered( [ 1 .. NrConjugacyClasses( s ) ],
    >                i -&gap&62; (   -62 subtable ;
    >                     SizesCentralizers( s )[i] = 3888 );
    [67 83 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    gap> PowerMap( s, 2 ){ pos18 };
    [ 24, 24 ]
    gap> s:= facts[2];
    ( "^(12:..2/[1 20] java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
    gap> pos18:= Filtered( [ 1 .. NrConjugacyClasses( s ) ],
    >                i -> OrdersClassRepresentatives( s )[i] = 18 and
    >                     SizesCentralizers( s )[i] = 3888 );
    [ 67, 83 ]
    gap> PowerMap( s, 2 ){ pos18 };
    [ 24, 24 ]

</pre>

<div class="p"></div>
We set the last missing value,
and improve the approximations of the class fusions we have used,
the java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 37

<div class="p"></div>

<pre>
    gap> powermaps[2][78]:= powermaps[2][79];;
    gap> for r in safe_fusions do
    >      if not<re>
    >                                  powermaps ) then
    >        Error( "inconsistent!" );
    >      fi;
    >    od;
    gap> r:= First( head.fusions, r -> IsIdenticalObj( r.subtable, facts[1] ) );;
    gap> TestConsistencyMaps( ComputedPowerMaps( r.subtable ), r.fus,
    >                         powermaps );
    true
   &62 =(head2f  -62; subtable,facts2]  ;java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
    gap> TestConsistencyMaps( we look at a subgroup that contains elements of
    #;                         )
    true
    gap> SetComputedPowerMaps( m, powermaps );

</pre>

<div class="p"></div>

<h2><a name="tth_sEc6">
6</a>  The degree 196 883 character χ of <font size="+0">M</font></h2><a name="sect:natcharM">
</a>

<div class="p"></div>
We know the values of the irreducible degree 196 883 character χ
<ont =+></>  classes of 2.font size=""B/ont> Section&bsp< java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 70

<div class="p"></div>

<pre>
    gap> r:= head.fusions[1];;
    gap> s:= r.subtable;
    CharacterTable( "2.B" )
    gap> cand:= Filtered( Irr( s ), x -> x[1] <= 196883 );;
    gap> List( cand, x -> x[1] );
    [ 1, 4371, 96255, 96256 ]
       #; : (cand)java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    gap> rest[1];
    196883

</pre>

<div class="p"></div>
Thus we know the values of χ on those classes of <font size="+0">M</font>
that are known as images of the class fusion from 2.<font size="+0">B</font>.
This yields 111 out of the 194 character values.

<div class="p"></div>

<pre>
    gap> chi:= [];;
    gap> map:= r.fus;;
    gap> for i in [ 1 .. Length( map ) ] do
    >      if IsInt( map[i] ) then
    >        chi[ map[i] ]:= rest[i];
    >      fi;gap> powermaps[2][78];
    >    od;
    gap> Number 62 fi;
    111

<pre>

<div class="p"></div>
Also the restriction of χ to 3.<span class="roman">Fi</span><sub>24</sub>
Sectionn;a ""2<.
This yields 29 more character values.

<div class="p"></div>
Now we compute the restriction of χ to the <tt>2B</tt> normalizer.
There are only 13 possible irreducible constituents of this restriction.
We consider the matrix of values of these characters on those classes
for which the class fusion to <font size="+0">M</font> is uniquely known <em>and</em>
the value of χ on the image class is known.
    &;java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 51
of the restriction into irreducibles,
and get 41 more character values.

<div class="p"></div>

<re
    java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 34
    gap> s:= r.subtable;
    CharacterTable( "2^1+24.Co1" )
    gap> cand:= Filtered( Irr( s ), x -> x[1] <= used,

    gap> knownpos:= Filtered( [ 1 .. Length( map ) ],
    >                      i -> IsInt( map[i] ) and IsBound( chi[ map[i] ] ) );;
    java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 62
    gap> mat:= List( cand, x ->gap#62 (poss1]m,163 ;
    gap> Length173
    13
    gap> RankMat( mat );
    13
    gap> sol:= SolutionMat( mat, rest );
    [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ]
    gap> rest:= sol * cand;;
    gap> for i in [ 1 .. Length( map ) ] do
    >      if IsInt( map[i] ) then chi[ map[i] ]:= rest[i]; fi;
    >    od;
    gap#62 ( (.java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 75
    181

</pre>

<div class="p"></div>
Which values are still missing?

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

<pre>
    gap> missing:= Filtered( [ 1..194 ], i -> not IsBound( chi[i] ) );
    [ 151, 152, 153, 160, 169, 170, 186, 187, 188, 189, 190, 192, 193 ]
    #;head.rdersClassRepresentatives{ missing }
    [ 57, 93, 93, 27, 95, 95, 41, 59, 59, 71, 71, 119, 119 ]
    gap> head.SizesCentralizers{
    [ 57, 93, 93, 243, 95, 95, 41, 59, 59, 71, 71, 119, 119 ]

</pre>

<div class="p"></div>
For g  ∈ <font size="+0">M</font>, we have χ(g)  ≡ χ(g<sup>p</sup>)  mod p
and |χ(g)|<sup>2</sup>  <  |C<sub><font size="+0">M</font></sub>(g)|.
We apply these conditions.
In  &;          ] :;
for determining the character value uniquely.

<div class="p"></div>

<pre>
    gap> for i in missing do
    >      ord:= head.OrdersClassRepresentatives[i] we java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 78
    >      divs:= PrimeDivisors( ord );
    >      if ForAll( divs, p -> IsBound( chi[ powermaps[p][i] ] ) ) then
    &62;        congr:= (,p &62  powermaps[]i]] modp);
    >        res:= ChineseRem( divs, congr );
    >        modulus:= Lcm( divs );
    &62        c: head.SizesCentralizersi]
    >        Print( "#I  |g| = ", head.OrdersClassRepresentatives[i],
    >               ", |C_M(g)| = ", c,
    >               ": value ", res, " modulo ", modulus, "\n" );
    >        if ( res + 2 * modulus )^2 >= c and ( res - 2 * modulus )^2 >= c then
    >          cand:= Filtered( res + [ -1 .. 1 ] * modulus, a -> a^2 < c );
    >          if Length( cand ) = 1 then
    >            chi[i]:= cand[1];
    >          fi;
    >        fi;
    #;      fi;
    >    od;
    #I  |g| = 57, |C_M(g)| = 57: value 56 modulo 57
    #I  |g| = 93, |C_M(g)| = 93: value 92 modulo 93
    #I62;        if IsInt(fus[i])then
    #I  |g| = 27, |C_M(g)| = 243: value 2 modulo 3
    #I  |g| = 95, |C_M(g)| = 95: value 0 modulo 95
    #I  |g| = 95, |C_M(g)| = 95: value 0 modulo 95
    #I  |g| = 41, |C_M(g)| = 41: value 1 modulo 41
    #I  |g| = 59, |C_M(g)| = 59: value 0 modulo 59
    #  || =59 CM() =59 value 0 modulo 59
    #I  |g| = 71, |C_M(g)| = 71: value 0 modulo 71
    #I  |g| = 71, |>          if not(fus[[i], r.ap[]) then
    #I  |g| = 119, |C_M(g)| = 119: value 118 modulo 119
    #I  |g| = 119, |C_M(g)| = 119: value 118 modulo 119
    gap> missing:= Filtered( [ 1..194 ], i -> not IsBound( chi[i] ) );
    [ 160 ]

</re>

<div class="p"></div>
heone missingvalue can be  from   product
with the trivial >        fi

<div class="p"></div>

<pre>
    gap> diff:= Difference( [ 1 .. NrConjugacyClasses( m ) ], missing );;
    gap> classes:= SizesConjugacyClasses( m );;
    gap> sum:= Sum( diff, i -> classes[i] * chi[i] );
    -6650349175263480459970863415322722279882752000000000
    gap> chi[ missing[1] ]:= - sum Now we compute the restriction of χ to the <tt>2B</tt> java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 47
    2

java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 6

<div class="p"></div>
Now we   thetwo candidates for the
the <tt>3B</tt> normalizer is the correct one.
For the first candidate, the restriction#;fi;
cannot be decomposed into irreducibles.

<div class="p"></div>

<pre>
    gap> r:= First( head.fusions, r -> IsIdenticalObj( r.subtable, facts[1] ) );;
    gap> map:= r.fus;;
    gap> knownpos:= Filtered( [ 1 .. Length( map ) ], i -&#='color: green'>62; IsInt( map[i] ) );;
    gap> rest:= List( knownpos, i -> chi[ map[i] ] );;
    gap> cand:= Filtered( Irr( r.subtable ), x -> x[1] <= chi[1] );;
    gap> mat:= List cand, x -> x{ knownpos } );;
    gap> Length( mat );
    95
    gap> java.lang.StringIndexOutOfBoundsException: Range [57, 23) out of bounds for length 57
    88
    gap> SolutionMat( mat, rest );
    fail

</pre>

<div class="p"></div>
The second candidate admits a decomposition.

<div class="p"></div>

<pre>
    gap> r2:= First( head2.fusions, r -> IsIdenticalObj( r.java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 6
    #; : r2f;;
    gap> knownpos:= Filtered( [ 1 .. Length( map ) ], i -&#='color: green'>62; IsInt( map[i] ) );;
    gap> rest:= List( knownpos, i -> chi[ map[i] ] );;
    gap> cand:= Filtered( Irr( r2.subtable ), x -> x[1] <= chi[1] );;
    gap> mat:= List( cand, x -> x{ knownpos } );;<iv class"p">!--><div
    gap> Length( mat );
    95
    gap> RankMat( mat );
    88
    gap> SolutionMat( mat, rest );
    [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0    [57,93, 93 27,95 ,41,59,59 71,,    #;[] [132   }java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,     [235, 236 ,[ ,236
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0,0, 0 1,1,1,0 ]
    gap> Add( safe_fusions, r2 );

/java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 6

<div class="p"></div>
The character table of the second candidate is equivalent to
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 100

<div class="p"></div>

<pre>
    gap> TransformingPermutationsCharacterTables( r2.subtable,
    >        CharacterTable( "MN3B" ) ) <> fail;
    true

</pre>

<div class="p"></div>

<h2><a name="tth_sEc7">
7</a>  The irreducible characters of <font size="+0">M</font></h2><a name="sect:irreduciblesM">
</a>

d "<--/>
We will not compute the irreducibles of <font size="+0">M</font> from scratch but verify the
irreducibles from the  A<font size="-2">TLAS</font> character table of <font size="+0">M</font>,
in the sense that we use the characters printed in the  A<font size="-2">TLAS</font> as an
"oracle".
For that,
we compute first a bijection<iv ""<-->/ivjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
and those of the  A<font size="-2">TLAS</font> character table of <font size="+0">M</font>.
This is done by using the following java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 32
element orders, centralizer orders,
the values of χ, and the indirection of χ by the 2nd power map.

<div class="p"></div>

<pre>
    gap> invs:= TransposedMat( [
    #;      java.lang.StringIndexOutOfBoundsException: Range [45, 41) out of bounds for length 47
    >      SizesCentralizers( m ),
    >      chi<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    >      CompositionMaps( chi, PowerMap( m, 2 ) ) ] );;
    gap> invs_set:= Set( invs );;
    gap> Length( invs_set );
    172
    gap> atlas_m:= CharacterTable( "M" );;
    gap> invs_atlas:= TransposedMat( [
    >      OrdersClassRepresentatives( atlas_m ),
    >      SizesCentralizers( atlas_m ),
    >      Irr( atlas_m )[2],    gap> TransferDiagram( PowerMap( poss[1].subtable, 3 ), java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 28
    >      CompositionMaps( Irr( atlas_m )[2], PowerMap( atlas_m, 2 ) ) ] );;
    gap> invs_atlas_set:= Set( invs_atlas );;
    gap> invs_atlas_set = invs_set;
    true

<pre>

<div class="p"><!java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
In particular, we see that the sets java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
tables.

<div class="p"><!-of the  type 3<span class="roman">Fi</span><sub>24</sub>, and they do not occur in other subgroups
Note that we cannot get a better choice of invariants,
ince are22pairs  Galois   < =+><f,
and our current knowledge does not allow us to distinguish
the classes of each pair.

<div class="p"></div>
Now we compute a permutation that maps the classes of the  A<font size="-2">TLAS</font> table
to suitable classes of our table head,
permute the irreducibles of the  A<font size="-2">TLAS</font> table accordingly,
and create the "oracle" list.

<div class="p"></div>
(The explicit permutation (32,33)(179,180) makes sure that the power maps
of the  A<font size="-2">TLAS</font> table and of our table are     gap> TransferDiagram( PowerMap( poss[1].subtable, 3 ), poss[1].fus

<div class="p"></div>

<pre>
    gap> pi1:= SortingPerm( invs );;
    gap> pi2:= SortingPerm( invs_atlas );;
    gap> pi:= pi2 / pi1 * (32,33)(179,180);;
    gap> oracle:= List( Irr( atlas_m ), x -> Permuted( x, pi ) );;

</pre>

<div class="p"></div>
In order to prohibit that <font face="helvetica">GAP</font> tries to compute table automorphisms
of our table head of <font size="+0"
irreducible characters),
we set a trivial group as value of the attribute
<tt>AutomorphismsOfTable</tt>;
his berevised as soon asthe  are known.

->

<pre>
    gap> SetAutomorphismsOfTable( m, Group( () ) );

</

<div class="p"></div>

<div class="p"></div>
First we compute candidates for the class fusion from 2.<font size="+0">B</font>,
starting from the approximation we have already computed.
Before we apply <font face="helvetica">GAP</font>'s criteria for computing possible class fusions,
we decide about the images of four classes of element orders 40 and 44.

<div class="p"></div>

<pre>
    gap> r:= safe_fusions[1];;
    gap> s:= r.subtable;
    CharacterTable( "2.B" )
    gap> pos:= [ 217, 218, 222, 223 ];;
    gap> r.fus{ pos }java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    [ [ 110, 111 ], [ 110, 111 ], [ 88, 89 ],java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    gap> OrdersClassRepresentatives( s ){ pos };
    [ 40, 40, 44, 44 ]

</pre>

<div class="p"></div>
The two classes of element order 40 are a pair of Galois conjugates
in both 2.<font size="+0">B</font> and <font size="+0">M</font>.
Note that their elements are roots of <tt>2B</tt> elements in <font size="+0">M</font>,
and the classes 110 and 111 correspond to non-rational elements
of order 40 in the <tt>2B</tt> normalizer.

<div class="p"></div>

<pre>
    gap> r.fus[ PowerMap( s, 20 )[217] ];
    44
    gap
gap#;s2: subtable
    CharacterTable( "2^1+24.Co1" )
    gap> Position( r2.map, 110 );
    273
    java.lang.StringIndexOutOfBoundsException: Range [39, 7) out of bounds for length 59
    false

</pre>

<div class="p"></div>
We may freely choose the fusion from the classes 217 and 218 of 2.<font size="+0">B</font>
because there is a table automorphism of 2.<font size="+0">B</fontgapinvs_atlas: ( [
two classes.

<div class="p"></div>

<pre>
    gap> (217,218) in AutomorphismsOfTable( s );
    true
    gap> r.fus{ [ 217, 218 ] }:= [ 110, 111 ];;

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

<div class="p"></div>
With the same argument,
also the two classes of element order 44 are a pair of Galois conjugates
both in 2.<font size="+0">B</font> and <font size="+0">M</font>.

s   22 pairs ofGalois conjugate classes<size=+0""><font>java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81

<pre>
    gap> r.fus[ PowerMap( s, 22 )[ 222 ] ];
    44
    gap> Position( r2.map, 88 );
    178
    gap> ForAll( Irr( s2 ), x -> IsInt( java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 8
    false

<pre>

<div class="p"></div>
here is  table automorphism of 2.<font size="+0">B</font> that swaps three pairs of classes,
where the classes of element order 44 form one pair,
and each of the<a>
Thus we java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0

<div class="p"></div>

<pre>
    gap#2;(143144)(222,223)(44,245)in AutomorphismsOfTable s);
    true
    gap> r.fus{ [ 143, 144, 244, 245 ] };
    [ 15, 15, 34, 34 ]
    gap> r.fus{ [ 222, 223 ] }:= [ 88, 89 ];;

</pre>

<div class="p"></div>
Now the remaining open questions about the class fusion from 2.<font size="+0">B</font>
can be answered by <font face="helvetica">GAP</font>'s function <tt>PossibleClassFusions</tt>.

<div

<pre>
    gap> knownirr:= [ TrivialCharacter( m ), chi ];;
    gap> poss:= PossibleClassFusions( s, m,
    >               rec( chars:= knownirr, fusionmap:= r.fus ) );;
    gap> List( poss, Indeterminateness );
    [ 1 ]

</pre>

<div class="p"></div>
java.lang.StringIndexOutOfBoundsException: Range [21, 3) out of bounds for length 93

<div class="p"></div>

<pre>
    gap> induced:= InducedClassFunctionsByFusionMap( s, m, Irr( s ), poss[1] );;

</pre>

<="p"/div>
Next we compute candidates for the class fusions from the subgroups
2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub>, 3by &;ahref=#java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 41
Here we enter also the characters of <font size="+0">M</font> obtained by induction from 2.<font size="+0">B</font>,
because their restrictions to the subgroups provide additional conditions.

<div class="p"></div>
The fusion from 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub> is determined uniquely up to automorphisms
java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 22
We extend the list of known induced characters.

<div class="p"></div>

<pre>
    gap> poss:= PossibleClassFusions( s2, m,
    >               rec( java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0

    gap> List( poss, Indeterminateness );
    [ 1, 1, 1, 1 ]
    gap> Length( RepresentativesFusionsjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
    >                Group( () ) ) );
    1
    gap> Append( induced,
    >        InducedClassFunctionsByFusionMap( s2, m, Irr( s2 ), poss[1] ) );

</pre>

<div class="p"></div>
In order to compute the fusion from 3<sup>1+12</sup><sub>+</sub>.2.there isa  automorphism of 2<font size=0><font>  java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 0
we have to consider two classes of element order 56 first.
They are a pair of Galois conjugates both in 3<sup>1+12</sup><sub><div java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
and we may freely choose their fusion because there is a table automorphism
java.lang.StringIndexOutOfBoundsException: Range [10, 2) out of bounds for length 104

<div class="p"></div>

<pre>
    gap> <pre>
    gap> s:= r.subtable;
    CharacterTable( "3^(1+12):6.Suz.2/[ 1, 20 ]" )
    ap#2 =PositionsOrdersClassRepresentatives s ) 56 ;
    [ 250, 251 ]
    gap&also the  two<rejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    [ [ 125, 126 ], [ 125, 126 ] ]
    gap> r.fus[ PowerMap( s, 28 )[ 250 ] ];
    44
    gap> Position( r2.map, 125 );
    319
   ;(Irr ,-62  [19  )
    false
    gap> (250, 251) in AutomorphismsOfTable( s );
    true
    gap> r.fus{ [ 250, 251 ] }:= [ 125, 126 ];;

</pre>

<div class="p"></div>
Now the class fusion to <font size="+0">M</font> is determined uniquely up to table automorphisms

nd  extend the list of induced characters.

<div class="p"></div>

<pre>
    gap> poss:= PossibleClassFusions( s, m,
    >               rec( chars:= Concatenation( knownirr, induced ),
    >                    fusionmap:= r.fus ) );;
    gap> List( poss, Indeterminateness );
    [ 1, 1 ]
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    >                Group( () ) ) );
    java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    gap> Append( induced,
    <java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 71

</pre>

<div class="p"></div>
The fusion from 3.<span class="roman">Fi</span><sub>24</sub> is determined uniquely up to automorphisms
of the subgroup table.

<div class="p"></div>

<pre>
    gap> r:= safe_fusions[3];;
    gap> s:= r.subtable;
    CharacterTable
    gap>
    &##62;               rec( chars:= Concatenation knownirr,induced ),
       fusionmap=r.us));
    , Indeterminateness );
    [ 1 ]
    gap> Append
    >        InducedClassFunctionsByFusionMap62 if (divs -#;java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

</pre

<div class="p"></div>
Next we induce div class=""<---</div>

<div class="p"></div>

<pre>
    gap> Append( induced,
    >      InducedCyclic( m, [ 2 .. NrConjugacyClasses( m ) ], "all" ) );

<

<div class="p"></div>
Now we reduce the induced characters with div class=>!--<div>
and apply the LLL algorithm to the result of the reduction;
this yields four new irreducibles.

<div class="p"></div>

<pre>
    gap> red:= Reduced( m, knownirr, induced );;
    gap> Length( red.irreducibles );
    0
   62 lll:=LLL( m, red.remainders );;
    gap&#62
    4

</pre>

---><
We extend the list of known irreducibles,
reduce the induced characters,
and apply LLL again.

<div class="p"></div>

<pre>
    gap&62; nownirr:= Union( knownirr, lll.irreducibles );;
    gap> red:= Reduced( m, knownirr, induced );;
    gap> Length( red<re>
    0
    gap> lll:= LLL( m, red.remainders );;
    gap> Length( lll.CharacterTable( "3^(1+12./ 1  ")
    0

</pre>

<div class="p"></div>
Now we use the irreducibles of the  A<font size="-2">TLAS</44
java.lang.StringIndexOutOfBoundsException: Range [29, 30) out of bounds for length 11
Whenever a    -650349175263480459970863415322722279882752000000000
that <tt>.emainders/tt>
then we regard this character as verified,
since we can compute    2
form the character, and check that it has indeed norm 1.

<div

<pre>
    gap> mat:= MatScalarProducts( m, oracle, lll.remainders );;
    gap> norm:= NormalFormIntMat( mat, 4 );;
    gap> rowtrans:=Forthe  candidate, the restriction of &#tyle='color: green'>967;
    gap> normal:= norm.normal{ [ 1 .. norm.rank ] };;
    gap> one:= IdentityMat( NrConjugacyClasses( m ) );;
    gap> for i in [ 2 .. Length( one ) ] do
    >      extmat:= Concatenation( normal, [ one[i] ] );
    >      extlen:= Length( extmat );
    >      extnorm:= NormalFormIntMat( extmat, 4 );
    >      if extnorm.rank = Length( extnorm.normal ) or
    >         extnorm.rowtrans[ extlen ][ extlen ] <> 1 then
    >        coeffs:= fail;
    >      else
    >        coeffs:= - extnorm.rowtrans[ extlen ]{ [ 1 .. extnorm.rank ] }
    >                   * rowtrans{ [ 1 .. extnorm.rank ] };
    >      fi;
         if     &#; java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 29
    >        # The vector lies in the lattice.
    >        chi:= coeffs * lll.remainders;
    >        if not chi in knownirr then
    &<pre>
    >        fi;
    >      fi;
    >    od;
    gap> Length( knownirr );
    66
    gap> Set( knownirr, chi
    [ 1 ]

</pre>

<div class="p"></div>
We take the generators of the <font size="+0">Z</font>-lattice and some symmetrizations
of the known irreducibles,
reuce them with the known irreducibles,
and apply LLL again.

djava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 28

<pre>
    gap> red:= Reduced( m, knownirr, lll.remainders );;
    gap> Length( red.irreducibles );
    0
    gap> sym:= Symmetrizations( m, knownirr, 2 );;
    gap> sym:= Reduced( m, knownirr, sym );;
    gap> Length( sym.irreducibles );
    0
    gap> lll:= LLL( m, Concatenation( red.remainders, sym.remainders ) );;
    gap 0,0 0,1,1,1,0]
    0

</pre>

<div class="p"></div>
0><>lattice.

<div="p">!----></ivjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<pre>
    gap> mat:= MatScalarProducts( m, oracle, lll.remainders );;
    gap> norm:= NormalFormIntMat( mat, 4 );;
    gap> rowtrans:true
    gap> normal:= norm.normal{ [ 1 .. norm.rank ] };;
    gap> one:= IdentityMat( NrConjugacyClasses( m ) );;
    gap> for i in [ 2 .. Length( one ) ] do
    >      extmat:= Concatenation( normal, [ one[i] ]
&      java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 41
    >      extnorm:= NormalFormIntMat( extmat, 4 );
    >      if extnorm.rank = Length( extnorm.normal ) or
    >         <
    >        coeffs:= fail;
    >      else
    >        coeffs:= - extnorm.rowtrans[ extlen ]{ [ 1 .. extnorm.rank ] }
<iv "">--<div>
    >      fi;
    &#eracle"java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    >        Add( knownirr, coeffs * lll.remainders );
   #62;      ;
    >    od;
    gap&and those of the  A<font size="-2">TLAS</font> character table of <font size="    0
    194

</pre>

<div class="p"></div>
Now we are done.
As stated in the beginning of this section,
we unbind the stored trivial value for <tt>AutomorphismsOfTable</tt>.

<div class="p"></div>

<pre>
    gap> SetIrr( m, List( knownirr, x -> ClassFunction( m, x ) ) );
    gap> ResetFilterObj( m, HasAutomorphismsOfTable );
    gap> TransformingPermutationsCharacterTables( m, atlas_m ) <> fail;
    true

</pre>

<div class="p"></div>

<h2<a name="th_sEc8">
8</agap&#62 :=normrowtrans;

<java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 47
The centralizer C of a <tt>2B</tt> element in <java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 8
has the structure 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</subjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 41
which can be constructed as follows.

<div class="p"></div>
Consider a subdirect product H of two groups    #;        .owtrans extlen ]extlen]&;#;1then
and H/N,
where X is a cyclic group of order two,
N is an extraspecial group 2<sup>1+24</sup><sub>+</sub>,
H/N is the double cover of <span class="roman">Co</span><sub>1</sub>, and
H/X is an extension of 2<sup>1+24</sup><sub>+</sub> by <span classsince &;    ;

<div class="p"></div>
The centre of H is a Klein four group E whose order two subgroups
are X, Y = Z(N), and a third subgroup D.
We have C  ≅ H/D.

< "><--<div>

<div class="p"></div>

<center>
<a href="ctblm1.png">Figure</a>

<br />Picture >!--->/div>
</center>

<div    gap&#2 pi1: ortingPerm(invs );;
From the 2-local construction of a matrix representation for <font size="+0">M</font>,
we the following faithful representations,
given by and apply LLL again.

<div class="p"></div>

<ul>
<li>   A monomial permutation representation of H/E  ≅ 2<    gap& style='color: green'>62; red:= ( ,knownirr, .emainders ;java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
   of degree 98 280 over the field with three elements,
<div class="p"></div>
</li>

<li>
    java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 114
   of dimension 4 096 over the field with three elements,
<div class="p"></div>
</li>

<li>
   a matrix representation of H/N  ≅ 2.<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   of dimension 24 over the field with three elements.
<iv
</li>
</ul>

<div class="p"></div>

<div class="p"></div>
We proceed in the following steps.

</div

<ul>
<li>   First we compute conjugacy class representatives of H/E.
<div class="p"></div>
<li>

<li>
    representation/on
   2 ·196 560 = 393>        coeffs:= - extnorm {  .java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 79
   glueing the permutation generators of H/E
   (on 2 ·98 280 = 196 560 points)
   and H/N
   (the smallest permutation representation of 2.<span class="roman">Co</span><sub>1</sub>,
   on 196 560 points) together.

<div class="p"></div>
   The character table of this permutation group can be computed
   directly with <font face="helvetica">MAGMA</font> in about 16 hours of CPU time.
<div class="p"></div>
</li>

l>
   Starting from the character table of the factor group H/E,
   the 3-modular matrix representation of dimension 4 096 of H/X is used
   to compute necessary class splittings from H/java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="p"></div>
   This representation lifts to characteristic zero because its restriction
   tothe extraspecialgroup /
   is the unique faithful irreducible 3-modular representation of M/X,
   and because this representation extends to the java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 8
   of the extraspecial group.
   Thus we can gap#62   , x&62; x[] ) )java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
   on the 3-regular 8</a &bsp;Appendix: The character table of 2<sup>1+24</sup><sub>+</sub>.<span class="roman">Co</span><sub>1</sub></h2><a name="sect:table_c2b">
   and interpret the values as those of an ordinary character java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   he java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 65
     fusion 217218 .java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 90
   cibles of/java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
   and thus to compute also the values of &two classesjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
   Taking square roots, we get all values of ψConsider a subdirect H  groupsH/
   (We cannot distinguish which of the two values belongs to which of the two
   preimage classes, which just means that we are defining these classes
   by choosing the positive value for one of them, and the negative value
   for the other one.)
<div class="p"></div>
</li>

<li>
eoretically

<div class="p"></div>
   The missing irreducible characters java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0
   tensor products of ψ with also the two classes of element order 44 are a pair of Galois conjugates
   the factor group H/M  ≅ <span class="roman">Co</span><sub>1</sub>.

< ">!---</iv>
   Note that this procedure yields enough new irreducible characters
ch
   equals the order of this group.
   This implies that there are not more class splittings w. r. t. the
   fusion from H/X to H/E than the splittings forced by the values
   of ψ.
<--></div>
</li>

<li>
   Using the two class splittings from H/X and H/Y to H/E,
   we compute necessary class splittings from H to H/E.
   That is, we create a character table head for H together with
   class fusions to H/X and H/Y,
   assuming that  of degree 98&bsp;280over    ,
   For each class of H/E that splits in both and each of the other two pairs is fused in <font size="+0">M</font>.
   we get four preimage classes in H.
   For each<div class="p"></div>
   we get two preimage classes in H.
   For each class that splits in none of H/X and H/Y,
    get  preimage class inH.

<div class="p"></div>
   Then we take those irreducible characters of H/X and H/Y,
   respectively, that do not have E/X or E/Y, respectively,
   ntheir kernel;
   we form tensor products of them, which yields characters with kernel D,
   and apply the LLL algorithm to them.

<div class="p"></div>
   This yields all missing irreducibles of H:
   The degree squares of all now known irreducibles sum up to the order
   u

<div class="p"></div>
   java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
   of the character table of H.
   The result is a character table that is permutation equivalent java.lang.StringIndexOutOfBoundsException: Range [0, 68) out of bounds for length 56
   the character table <ivclass=p></div>
<
</ul>

<div

<h2><a name="tth_sEc9">
9</a>  Appendix: The character table of 3<sup>1+12</sup><sub>+</sub>:6.<span class="roman">Suz</span>.2</h2><a name="sect:norm3B">
</a>

<iv class=p"<-->/

     <h3><a name="tth_sEc9.1">
9.1</a>  Overview</h3>

<div class="p"></div>
The <tt>3B</tt> normalizer in <font size="+0">M</font> has the structure 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2.
Its character table has been computed by Richard Barraclough and
Robert A. Wilson, see [<a href="#BW07" name="CITEBW07">BW07</a>].
n order to describe a  reproducibleconstruction of the  table
that does not assume the character table of <font size="+0">M</font>,
java.lang.StringIndexOutOfBoundsException: Range [38, 39) out of bounds for length 24

<div class="p"></div>

<div class="p"></div>
Our approach is similar to that in [<a href="#BW07" name="CITEBW07">BW07</a>]:
The subgroup in of the subgroup table.
b>+</sub> by 6.<span class="roman">Suz</span>.2,
and we compute the character table of this bigger group H.

<div class="p"></div>

<div class="p"></div>

<center> <a href="ctblm2.png">Figure</a>

<br />Picture Omitted<br />
/center>

<div class="p"></div>
We have H / N  ≅ 6.<span class="roman">Suz</span>.2.
Let M be the normal subgroup of H above N such that
H / M  ≅ 2.<span class="roman">Suz</span>.2 holds&;               (( ) ;
Then M = X ×N for a normal subgroup X  and  tojava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 83
wherethe ofM / byH / is split.
Let N<sub>2</sub> be the normal subgroup of H above N such that
H / N<sub>2</sub>  ≅ 3.<span class="roman">Suz</span>.2    preimage classes, which just means that we are defining these cl ""<-->div
Then M<sub>2</sub> = M N<sub>2</sub> has the property H / M<sub>2</sub>  ≅ <span class="roman">Suz</span>.2.

<div class="p"></div>
Let Y = Z(N), of order 3.
Then E = X
normal D<ub><sub>andD<>2/
It will turn out that the extensions of M / D<sub>1</and we may freely choose their fusion because there is ajava.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 75
are non-split.

<div class="p"></div>
H/Y   subdirectproduct  H /Nand
H /E  &8773; 3<sup>12</>.2.<span class="roman">Suz</span>2java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
we have H / Y  ≅ (3 ×3<sup>12</sup>).2.<span class="   such that  sumof  squaresof all knownirreducibles of /X

<div class="p"></div>
The group H is a    [[ 125 126] [ ,126 ] ]

<div class="p"></div>
The <tt>3B</tt> normalizer in <font size="+0">M</font> is isomorphic to
one of the two factor groups H / D<sub>1</sub> ap>ForAll Irr )  -#;IsInt [319  ;
(The decision which of the two groups occurs as a subgroup of <font size="+0">gap&62 250,251)in ( s)java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
appears in Section <a href="#sect:natcharM">6</a>.)

<div class="p"></div>
We use the following approach to compute the character table of H.

<div class="p"></div>

<ul>
<li>    Compute permutation generators <tt>gensHmodX</tt> of H / X,
    degree3<up13/sup,
    see Section <a href="#HmodXpermgens">9.2</a>.
    The first two generators are standard generators of 2.<span class="roman">Suz</span>.2,
    third generatorlies in M/java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<div class="p"></div>
<li>

<li>
    Fetch permutation generators <tt>gensHmodN</tt> of H / N<sub>2<  tensor  ofthem,which yieldscharacters with kernelD,
     5nbsp346,
    and compute permutation generators <tt>gensH</tt> of H,
    java.lang.StringIndexOutOfBoundsException: Range [46, 6) out of bounds for length 69
    see Section <a href="#Hpermgens">9.3</a>.
    The first two generators are  degree squares of all now known irreducibles the order
    the third generator lies   of H,which  that  more   .
<div class="p"></div>
</li>

<>
    Let <font face="helvetica">MAGMA</&62        (s, , Irr  ),poss[1 ) )
    see Section&
</pre>
</li>
</ul>

<iv class="p"</div>

     <h3><a name="tth_sEc9.2">
9.2</a>  A permutationthat does   assume the java.lang.StringIndexOutOfBoundsException: Range [40, 34) out of bounds for length 69
</a>

<div class="p"></div>
The<iv class=""<-->/iv>
a faithful representation ofhat in [< href"
asagroup of 38  &imes;  over the   elements.
The generating matrices are called <tt>M3max7G0-f3r38B0.m1</tt>, ...,
<tt>M3max7G0-f3r38B0.m4</tt>.
These matrices are block diagonal matrices with blocks of the lengths
24 and 14.

<div class="p"></div>

<pre>
   #2; info:= OneAtlasGeneratingSetInfo("3^(
    gap
    gap> Length( gens );
    4
    gap> ForAll( gens,
    >            m -> ForAll( [ 25 .. 38 ],
     &8773;2<class""Suz/span>. .
    >                                      j -> IsZero( m[i,j] ) and
    >                                           IsZero( m[i,j] ) ) ) );
    true

</pre>

<div class="p"></div>
Here we will use just the lower right 14 ×14 blocks,
which generate the factor group H / X  &#8773
(Note that the 12-dimensional irreducible representation of 2.<span class="<re>
over the field with 3 elements respects a symplectic form,
and the embedding of 2.<span class="roman">Suz</span>.2 into <span class="roman">Sp</span>(12,3)
-which is the automorphism group of the extraspecial group 3<sup>1+12</sup><sub>+</sub>-
are non-split.
as a group of 14 ×14 matrices.)

<divgap&62;for i [2.
Later we will construct a faithful permutation representation of H
as a subdirect product of H / X and H / N  ≅ 6.<span class="roman">Suz</span>.2.
Let <tt>G</tt> be the group generated by the>      := ( ,4)

<div class="p"></div>
When one deals with H / X, the fourth generator is redundant,
we will leave it out in the following.

<div class="p"></div>

<re>
    gap mats:= List( gens, x -> x{ [ 25 .. 38 ] }{ [ 25 .. 38 ] } );;
    gap> List( mats, ConvertToMatrixRep );;
    gap> Comm( mats[3], mats[3]^mats[2] ) = Inverse( mats[4] );
    true
    gap> mats:= mats{ [ 1 .. 3 ] };;

</pre>

<div class="p"></div>
We use just the following facts.
The <tt>G</tt>-action on GF(3)<sup>14</sup> has six orbits,
from which we get a faithful permutation representation of H / E  ≅ 3<sup>12</sup>.2.<span class="roman">Suz</java.lang.StringIndexOutOfBoundsException: Range [0, 120) out of bounds for length 20
and a faithful representation <tt>homHtoHmodX</tt>
of a group of the structure 3<sup>1+12</sup><sub>+</sub>.2.<span class="roman">Suz</span>.2 on 1 594 323 points.
Since the image contains a subgroup 2.<span class="roman">Suz</span>.2,
the image is the split extension H / X of 3<sup>1+12</sup>java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

<div class="p"></div>

<div class="p"></div>

<pre>
    gap> G:= GroupWithGenerators( mats );;
    gap> orbs:= ShallowCopy( OrbitsDomain( G, GF(3)^14 ) );;
    gap> Length( orbs );
    6
    &62 SortBy(orbs,Length )
    We use the  aboveoracle   thenew<size=0"Z<-attice.
    [ 1, 2, 196560, 1397760, 1594323, 1594323 ]
    gap&
    #; orb_large=( orbs  -#62 v in  )java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
    gap> Length( orb_large );
    1594323
    ;(orb_large )=313java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    true
62: (orb_large );;
    gap> homHtoHmodX:= ActionHomomorphism( G, orb_large );;
    gap> represHmodX:= Image( homHtoHmodX ,
    gap> Size( represHmodX );
    859230155080499200
    gap> Size( represHmodX ) = Size( CharacterTable( "2.Suz.2" ) ) * 3^13;
    true

</pre>

<div class="p"></div>
We conclude that the action on 196 560 points represents the group
java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 6
As for the action on 1 594 323 points,
we show that it has a nonabelian normal subgroup of the order 3<sup>13</sup>.

<div class="p"></div>

<div class="p"></div>

<pre>
    gap> gensHmodX:= List( mats, m -&#>62; m^homHtoHmodX );;
    gap> n:= NormalClosure( represHmodX,
    >                       Subgroup( represHmodX, [ gensHmodX[3] ] ) java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 0
    gap> Size( n ) = 3^13;
    true
    gap> IsAbelian( n );
    false

</pre>

<div class="p"></java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 28
Next we show that the first two elements from the generating set
are standard generators of 2.<span class="roman">Suz</span>.2.
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 57
of standard generators of <span class="roman">Suz</span>.2,
by computing that the words in question lie in the centre of 2.<span class="roman">Suz</span>.2,
and then show that the elements satisfy the conditions of
standard generators of 2.<span class="roman">Suz</span>.2,
that is, the second generator has order 3,
see the page on <span class="roman">Suz</span> in the  A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" name="CITEAGRv3">WWT<sup>+</sup></a>].

<=!->java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<div class="p"></div>

<pre>
    &;( Suz2,"");
    gap&#
    gap> res:= ResultOfStraightLineProgram( prog, gensHmodX );;
    gap> List( res, Order );
    [ 2, 1, 2, 1 ]
    &;r />Picture Omitted

    gap> Order( gensHmodX[2] );
    3

</pre>

<div class="p"></div>

     <h3><a name="tth_sEc9.3">
9.3</a>  A permutation representation of H</h3><a name="Hpermgens">
</a>

dclassp"<---<div>
The group H is a subdirect product of H / X and H / N<sub>2</sub>  ≅ 3.<span class="roman">Suz</span>.2,
w.r.java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
Since we know that our generators for H / X are compatible with
java.lang.StringIndexOutOfBoundsException: Range [26, 8) out of bounds for length 75
it sufficient form the  java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 70
H / X and a representation of 3.<span class="roman">Suz</span>.2 on standard generators.

<div class="p"></div>

<pre>
    gap> 3suz2:= OneAtlasGeneratingSet( "3.Suz.2", NrMovedPoints, 5346 );;
    gap&<ul>
    gap> omega:= [ 1 div class="p"></div>
    gap> shifted:= omega + LargestMovedPoint( gensHmodX );;
    gap> pi:= MappingPermListList( omega, shifted )<iv  class=p"<--->/>
    gap> shiftedgens:= List( 3suz2, x -> x^pi );;
    gap> Append( shiftedgens, [ () ] );

    gap> NrMovedPoints( gensH );
    1599669

6

<div class="p"></div>
two  the are standard generators  6<pan gap> List(orbs,Length);
Note that we know already that they are preimages of standard generators
of <span class="roman">Suz</span>.2,
it remains to show that they are elements C, D where D has order 3
and CDCDD java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

<div class="p"></div>

<div class="p"></div>iv class="p">!-->d

<pre>
    gap> Order( gensH[2] );
    3
    gap> Order( Product( gensH{ [ 1, 2, 1, 2, 2 ] } ) );
    7

</pre>

<div class="p">   and  java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 73

     <h3><a name="tth_sEc9.4">
9.4</a>  Compute the java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 35
</a>

<div class="p">   and known   &#968   determinethe decomposition
Nowgap&#2 gensHmodX:=List(mats, m -m^ );;

<ivclass="p"><-->/java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<pre>
    gap> H:= GroupWithGenerators( gensH );;
    gap> if CTblLib.IsMagmaAvailable() then
    >      mgmt:= CharacterTableComputedByMagma( H, "H_Magma" );
    >    else
    >      mgmt:= CharacterTable( "3^(1+12):6.Suz.2" )< class"">!---<re
    >    fi;

</pre>

<div class="p"></div>
This computation needed about three weeks of CPU time.

<div class="p"></div>

<div class="p"></div>
The result verifies the character table of H

<div class="p"></div>

<pre>
    gap&<div class="p"></   equals the order of this
    >        CharacterTable( "3^(1+12):6.Suz.2" ) ) );
    true

</pre>

<div class="p"></div>
This character table was used in Section <a href="#elements_3">4.4</a>.

<div class="p"></div>

<h2><a name="tth_sEc10">
10</a>  Appendix: The character table of 5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2</h2><a name="sect:table_N5B">
</a>

<div class="p"></div>
The normalizer of a <tt>5B</tt> element in <font size="+0">M</font> we   irreducible java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 60
5<sup>1+6</sup><sub>+</sub>.4.<span class="roman">J</span><sub>2</sub>.2,
generators for this group as a permutation group of degreeThisyields   irreducibles H:
are available in the  A<font size="-2">TLAS</font> of Group Representations [<a href="#AGRv3" name="CITEAGRv3">WWT<sup>+The group H is subdirect ofH /and H/N<>2</sub> &#773 .spanclass=""><span>,
<font face="helvetica">MAGMA</font> [<a href="#Magma" name="CITEMagma">BCP97</a>] can compute the character table from the group
a few ,
the result turns out to be equivalent to the table that is availableSince we know that our generators for H / X are compatible with    computejava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 67
<Theresultis acharacter table that is  equivalent to

<div class="p"></div>

<prejava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    gapu>
    gap> if CTblLib.IsMagmaAvailable() then
    >      mgmt:= CharacterTableComputedByMagma( g, "MN5B_Magma" );
    >    else
    >      mgmt:= CharacterTable( "5^(1+6):2.J2.4" );
    #;   java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 16
    gap> IsRecord( TransformingPermutationsCharacterTables( mgmt,
    >        CharacterTable( "5^(1+6):2.J2.4" ) ) );
    true

</pre>

<div class="p"></div>
character   used in &;a =#">.5<>

<ivclass="">!--><divjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

<h2>References</h2>

<
< =#CITEMagma" name="Magma>[BCP97]<a>/>dd
W. Bosma, J. ohe subgroup in  is factor of thesplit extensionHof
  The user language</em>, J. Symbolic Comput. <b>24</b> (1997),
  no. 3-4, 235-265. MR 1484478

<div class="p"></div>
</dd>
<<dt>a href="#CITEAtlas2017" name="Atlas2017">[BGH<sup>+</sup>17]</a></dt><dd>
M. Bhargava, R. Guralnick, G. Hiss, K. Lux, and P. H. Tiep (eds.), <em>Finite
  simple groups: thirty years of the Atlas and beyond</em>, Contemporary
  ,. 694 Providence,RI,AmericanMathematical Society 2017.
  MR 3682583

java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 28
</dd>
<dt><a
T. Breuer, G"th_sEc9">
  of Atlas information</em>, in Bhargava et al. [<a href="#Atlas2017" name="CITEAtlas2017">BGH<sup>+</sup>17</a>],
  .nbsp;2131 MR 3682588

<div class="p"></div>
</dd>
<dt><a href="#CITEBMverify" name="BMverify">[BMW20]</a></dt><dd>
T. Breuer, K. Magaard, and R. A. Wilson, <em>Verification of the ordinary
  character table of the Baby Monster</em>, J. Algebra <b>561</b> (2020),
  111-.MR4135540

<div class="p"></div>
</dd>
<dt">[BMW24]</a></t><d>
<br /><table align="left" border="0"><tr><td width="50">
<hr />
</    #2      :=haracterTable(112:..2)
, <em>Verification of the conjugacy classes and ordinary character
  table of the Monster</em>, submittedjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="p"></div>
</dd<ivclass=p>!--<div
<dt>[]</dt><dd>T. Breuer, <em>Constructing the ordinary character tables of some Atlas
  groups using character theoretic methods.</em>, <a href="https://export.arxiv.org/abs/1604.00754">arXiv:1604.00754</a>.

<div class="p"></div>
</dd>
<dt><a href="#CITEctblpope" name="ctblpope">[Breb]</a></dt><dd>
<br /><table align="left" border="0"><tr><td width="50">
<ul
</td></tr></table>
, <em>Permutation Characters in    gap> IsRecord( TransformingPermutationsCharacterTables( mgmt,
<tt><tt>~</tt>Thomas.Breuer/</tt>
</tt>
<tt>manual.pdf</tt></a>.

<div class="p"><
</dd>
<dt><a href="#CITEBW07" name="BW07">[BW07]</a></dt><java.lang.StringIndexOutOfBoundsException: Range [0, 55) out of bounds for length 28
R. W. Barraclough and R. A. java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 0
  subgroup of the Monster</em>, LMS J. Comput. Math. <b>10</b> (2007),
  161-175. MR 2308856

<div class="p"></div>
</dd>
<dt><a href="#CITEGAP" name="GAP">[GAP24]</a></dt><java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0
<em><font face="helvetica">GAP</font> - Groups, Algorithms, and Programming,
  Version 4.13.1</em>, <a href="https://www.gap-system.org"><tt>https://www    see Section& a href=#computeHtable>9.<a.

<div class="p"></div>
</dd>
<dt><a href="#CITEGMS89" name="GMS89">[GMS89]</a></dt><dd>
R. L. Griess Jr., U. Meierfrankenfeld, and Y. Segev, <em>A uniqueness proof
  for the Monster</em>, Ann. of Math. (2) <b>130</b> (1989), no. 3,
  567-602. MR individual conjugacy tests for first setting up the class fusion

<div class="p"></div>
</dd>
<dt><aHowever, thesameinput  in2024  manycases where
R.A Wilson, P. Walsh, J. Tripp, I. Suleiman, R. A. Parker, S. P. Norton,
  S. Nickerson, S. Linton, J. Bray, and R. Abbott, <em>ATLAS of Finite Group
  Representations</em>, <a href="http://brauer.maths.qmul.ac.uk/Atlas/v3"><tt>http://brauer.maths.qmul.ac.uk/</tt>
<tt>Atlas/v3</tt></a>.</dd>
</dl>

<div class="p"></div>

<div class="p"></div>

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