!%xml GAP 4 package CTblLib            Thomas Breuer -->

<Chapter Label="chap:o8p2s3_o8p5s3">
<Heading>&GAP; Computations with <M>O_8^+(5).S_3</M> and <M>O_8^+(2).S_3</M></Heading>

Date: October 08th, 2006

<P/>

This chapter shows how to construct a representation of the automorphic
extension <M>G</M> of the simple group <M>S = O_8^+(5)</M>
by a symmetric group on three points,
together with an embedding of the normalizer <M>H</M> of an <M>O_8^+(2)</M>
type subgroup of <M>O_8^+(5)</M>.

<P/>

As an application, it is shown that the permutation representation of <M>G</M>
on the cosets of <M>H</M> has a base of length twoHeading>&AP; Computations with <O_8^+()S3 and <>O_8^+(2)S_3<M><Heading
This question arose in <java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0


<!- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ->
<Section Label="sect:overview_o8p2s3">
<Heading>Overview</Heading>

Let <M>S</M> denote the simple group
<M>O_8^+(5) \cong </M>
</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
of the general orthogonal group GO<M>{}^the  of<><M>has     two.
matrices over the field with five elements.

<P/>

The This question arose in <Cite Key="BGS11"/>.
on four
<--%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%-
<Section Label=sect:">
By <Cite Key="Kle87"/>,
the group <M>S</M> containsHeadingO<H>
<>_^(2)/>java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 55
of<M/>  M>/M>isan extension <><M>by
a symmetric group on three points.
(In fact is,  nonabelian   occurs  a composition factor
<>_^()/M.)

<P/>

Let <M>S.2</M> and <M>S.3</M> denote intermediate subgroups between
<M>S</M> and <M>G</M>,
in which <M>S</M> has the indices <M>2</M> java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 43
java.lang.StringIndexOutOfBoundsException: Range [47, 11) out of bounds for length 69

<P/>

In Section <Ref Subsect="sect:sect2"/>,
we use the following  four points.
<M>S.2</M>.
By <Cite Key="CCN85" Where="p. 85"/>,
the Weyl group <M>Let <M><M  an  java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 75
java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 76
< java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 69
which has the structure <Mo MM/>inMG/>is an automorphic  of <>M<M 
Then the actionsGO<{^(,5<M> and ofan  image of <W/>
in GO<M>{+8,)/> <1/-spaces
natural <>O_8+(2<M.java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
(a supergroup of) <M>S.2</M>,
where both groups are represented as permutation groups on <M>N = 19\,656</M>
in which <M>S</M> has the indices <M>2</M> and <M>3</M>, respectively.

<P/>

In Section <Ref Sect="sect:sect3"/>,
first we use &GAP; to compute the java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0
Then wetakean outerautomorphism <>alpha</  <><M,of order three,
<M>.<M>
 <ite Key="CCN85" Where="p. 85"/>,
under <M>\alpha</M>the Weyl group <M>W</M> of type <M>E_8</M> is a double cover of <and  reduction its  <M>/>dimensional 
This   representationsof <>./  itssubgroup
<M>M.3</M> on <M>3 N = 58\,968</M> points.

<P/>

In Section <Ref Sect="sect:Then the actions of GO<M>{}^+(8,5)</M> and of an isomorphic image of <M>W</M>
eput aboveinformation together,
in order to construct permutation representations module of GOM{}+8,5)<M  MM<> as a  of
 pointsjava.lang.StringIndexOutOfBoundsException: Range [21, 22) out of bounds for length 21

<P/>

As an application
that the permutation representation of <M>G</M> on java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [17, 3) out of bounds for length 25
this  inn;Cite=BGS11""/.

<P/>

In two appendices, it is discussed how to derive a part of this result
romthe permutation <>1_^)H<M>
(Concretely computethe imagesgeneratingsets of <>SM  <M/>
and a file under <><M and <>alpha2/>
(java.lang.StringIndexOutOfBoundsException: Range [36, 4) out of bounds for length 55

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ectionLabel="sect2">
<Heading>Constructing Representations of <M>M.2</M> and <M>S.2</M></Heading> put put  theaboveinformation ,


<!-- %%%%%%%%%on<> <M>points.
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<> Matrix  of theWeylGroupof  <M>_<M<H>

Following the recipe listed
in&bsp;Cite  Key=C"Where=. 85,Section Weyl/>
java.lang.StringIndexOutOfBoundsException: Range [19, 2) out of bounds for length 72
rational <M>8 \times 8</M> matrices generatedthisquestion  in CiteKey="GS11/.
vectors

<Display Mode="M">
   \left
<P>

java.lang.StringIndexOutOfBoundsException: Range [25, 4) out of bounds for length 66
plusfrom permutationcharacter M>1H^)_/>

<and a file containing the data used in the earlier sections is described
   left(p 1/,p12 \m 2 \java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 45
          12 \m 1/, \pm 1/p / rightjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
</Heading Representations of<>.2</>and<S2/>/java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 76

that<SubsectionLabelsubsect:">
(Heading>A Matrix Representation of the Weyl Group of Type <M>E_8</M></Heading>
<M>\pm v<M>)

<P/>

<Example><![CDATA[
gap>  <Cite Key"Where". 85,SectionWeyl">java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
  i  Combinations [ 1 ..8]   do
>      v:= 0 * [ 1 .. 8 ];
>      v{i}:= [ 1, 1 ];
>      Add , v )
>      v:= 0 * [ 1 .. 8 ];
 vi:  ,- ]
>      Add( rootvectors, v );
java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
gap> Append( rootvectors,
>      1/2 * Filtered( Tuples( [ -1, 1 ], 8 ),
>              xD Mode=""
gap =Group (rootvectors  ) )java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
<atrixgroupwith120generators
]]></Example>

</Subsection>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%<>
<Subsection Label="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
<M>{}^(5</M></Heading>

The elements in the group constructed above respect the symmetric bilinear
form

<P/<Example<[[

<Example><![CDATA[
gap> I:= IdentityMat( 8 );;
java.lang.StringIndexOutOfBoundsException: Range [11, 3) out of bounds for length 71
true
]]></Example>  i  Combinations( [ 1 .. 8 ], 2 ) do

<P/>

So the reduction of the java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 23
of orthogonaljava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 67
The group GO<M>{}^+(8,5)</M> java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 26
<Ref Func=GO"BookName="ef">leaves
a >    Add ,v ;

<P/gap>Append java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 25

<Example><![CDATA[
gap> largegroup:= GO(1,8,5);;
gap>).matrix );
 . 1 . . . . . .
 1 . . . . . . .
    . ....
 . . . 2 . . . .
 . . .   . .
 . . . . . 2 . .
 . . . . . . 2 .
 . . . . . . . 2
]>/>

<P/>

In order to conjugate <
java.lang.StringIndexOutOfBoundsException: Range [9, 2) out of bounds for length 77
with the property that <M>T T^{tr}</M>
is half

<P/java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<Example><![CDATA[
gap> T:= [ [ 1, 2 ], [ 4, 2 ] ] *java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap>Display( 2*T *TransposedMatT  ;
 . 1
 1 .
gap> I:= IdentityMat( 8, GF(5) );;
gap> Ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> conj:= List( gap>ForAll(GeneratorsOfGroupwe8 ), x -> x * TransposedMat(x) = I );
gap> IsSubset( largegroup
true
]]></Example>

</java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:120pt">
<Heading>Compatible Generators of <M>M</M>, <M>M.2</M>, <M>S</M>, and <M>S.2</M></Heading>

Forof  matricesw.r&;t the  .
we switch from the natural matrix representation of GO<M>{}^+(8,5)</M>
ion  <>}+85)/>
of degree <M<ef Func="O =r">leaves
which given by the  of GO<M>}+85<M> thesmallest 
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

<P. .. .   .

<Example><![CDATA[
gap> orbs:= OrbitsDomain( largegroup, NormedRowVectors( GF(5)^8 ),
>                         OnLines       java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
gap> List.. .... java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> N=Length(orbs[])java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
19656
gap> orbN:= SortedList( with the property that <M>T T^{tr}</M>
gap> largepermgroup:= Action( largegroup, orbN, OnLines );;
]]></Example>

<P/>

In the
permutation representations of the subgroup <M>
and derivedsubgroup<M>M</>are obtained.
But first we compute a smaller generating set of the simple group <M>M</M>,
java.lang.StringIndexOutOfBoundsException: Range [34, 5) out of bounds for length 56

<P/>

<Example><![CDATA[
gap> orbwe8:= SortedList( Orbit( we8, rootvectors[1], OnLines ) );;
gap>Length );
ga> =DerivedSubgroup  );;
gap> we8_to_m2:= ActionHomomorphism( we8 (sml java.lang.StringIndexOutOfBoundsException: Range [56, 57) out of bounds for length 56
gap> m2_120:= Image( we8_to_m2 ]><>
gap> java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
ap : ( m_120 );;  Length( sml );
2
gap> gens_m:= List( sml, x -> PreImagesRepresentative( we8_to_m2, x ) );;
]]></Example>

<P/>

Now we compute the actions For generating MM2/> we an element<>_\n . setminus M</M>,
on the above orbit of length <M>N</M>.which is  fromthe of a java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 79
For generating </
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

<P/>       -Permutation I *x  ^-,orbN  );;

<Example><![CDATA[
gap> gens_m_N:= List( gens_m,
>       - Permutation  * x *I^- , nLines));
gap> m_N:= Group( gens_m_N );;
gap> b:= I gap>(  )java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
gap>DeterminantMat( b )
Z(]]>]<Example>
gap>
gap> m2_Njava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
]]></Example>

<P/>

(Note that <M>M.2</M> is not contained in PSO<M>{}^+(8,5)</M>,
since the The group <M>S</M> is the derived subgroup of PSO<M>{}^+(8,5)</M>,

<P/>

The group <M>S</M> is the derivedjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [13, 3) out of bounds for length 65

<P/>

<Example><![CDATA[
gap> s_N:= DerivedSubgroup( largepermgroup]<Example>
gap> s2_N:= java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 13
]]></java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0

</Subsection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%<eading>Constructing Representations of <M>M.3</M> and <M>S.3</M></Heading>
<ection Label="sect:ect3">
<Subsection Label=sect:action_M._"java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38


<!-- %%%%%%%%L<M>alpha<> be an automorphism of< />
<Subsection Label="sect:action_M.3_M">
<Heading>The Action of <M>M.3</M> on <to theblockdiagonalmatrix

Let <
Then \eft[b{{java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
<>\langle\ rangle<> an  constructedas .

<P/>

If <M>M</M> is given by a matrix representation then we map <M>g& amp;    ^(a^2)java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
to the

<Alt Only="LaTeX">
\[
   \left[ \begin{array}<erbjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
             g &          & \\
               & g^\alpha & \\
               ∓&;    g^(alpha^2)
          \end{array} \right] /Verb>
\]
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lOnly=Text">
<Verb>
                          ⌈<dclass"">spanclass"">g<span<t>
                          |   g^α         |
                                 ^^ ,

</<tr>

<Alt Only=< =tdright>span class="impleMath&;/><tdjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
<div ="pcenter"
  <table class="GAPDocTable">
    t>
     td class""<span class"SimpleMath"></span></td>
      <<r>
      <      <tdclass="< =SimpleMath"&;/pan<td>
    </tr>
    <tr>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      ="SimpleMath>^α/span<t>
      <td class="tdright"><span class="java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 9
    </tr>
    <tr>
      td class=tdright">span =>nbsp<span>/d>
      <td classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      < =tdright"<pan ="impleMath>^α2<span>/d>
    </tr>
  </table>
</div>
]]></Alt>

and we represent <M>\alpha</M> by the block permutation matrix

<Alt Only="LaTeX">
\[
   \left[ \begin{array}{ccc}
            & & I \\
            I     & & \\
            &           &;&; \
          \{array}\] ,
\]
</Alt>

<Alt Only="Text">
<Verb>
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
                          | I     |
java.lang.StringIndexOutOfBoundsException: Range [35, 26) out of bounds for length 35
</Verb>
</Alt>

<Alt Only="HTML"><![CDATA[
<div class="pcenter">
  <table class="GAPDocTable/>
    <tr>
      < class=tdright"><pan =SimpleMath>nbsp<span></d>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      right"< class=SimpleMath></>/>
    </tr>
    <tr>
      <td class="table class="GAPDocTable
      <td class="tdright"><tdclass=tdright">span class"">nbsp;/span<tdjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
      <td class="tdright"><span class="SimpleMath">I</span></td>
    </tr>
    <tr>
      <td class="tdright">      <d ="dright"< ="impleMath"I<s>/d>
      <d class=tdright"< =SimpleMath><span>><td>
       ="tdright>span class"SimpleMath" <span>/d>
    </tr>
  </table>
</div>
]]></Alt>

where <I</   identity element in <><M>

<P/>

We need the action of <M>\alpha</M> on <M>M</M<td class=tdright"span class=SimpleMath>n;s>t>
More precisely,/>
nder <>alpha/M>>  <>alpha^2</>

<P>

The java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 4
its automorphism group, which is isomorphic with <MMore precisely weneedimages of thechosen of <><M>
for that,we the degree <>/  representation
constructed in Section <Ref Subsect="subsect:120pt"/>.

<P/>

<Example><![CDATA[
gap> aut_m:= AutomorphismGroup( m_120 );;
]]></its automorphautomorphism group which  isomorphic  <>^_(.</Mjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69

<P/>

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

P>

<Example
ap>nice_aut_m ( aut_m );
gap> der:= DerivedSubgroup( Image(  pickan   <M>\<M>of order three.
gap> der2:= DerivedSubgroup
gap  :Random(der )
>      gap> nice_aut_m:= NiceMonomorphism( aut_m );;
>    until ord mod 3 = 0 and ord mod 9 <>gap>der:=DerivedSubgroup Image  ));
gap> gap> der2:= DerivedSubgroup( der );;
gap> alpha_120:= PreImagesRepresentative( gap> repeat x: java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 30
]>/>

<P/>

Next we compute the>x=x(ord / );
and <M>\alpha^2</M>,
and the corresponding elements in gap> alpha_120: PreImagesRepresentative( nice_aut_m, x );

<P/>

<Example><![CDATA[
gap> sml_alpha
gap>sml_alpha_2=List sml_alpha,x - Image(alpha_120, x  ;
gap> gens_m_alphaNext we computethe   the generators<></>under<>alpha<Mjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
>x ->PreImagesRepresentative(we8_to_m2,x ));
gap> gens_m_alpha_2:= List( sml_alpha_2,
>                       x -> PreImagesRepresentative( java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
gap>= List( gens_m_alpha,
>      x >Permutation( I * x *I^- , OnLines ));
>gens_m_N_alpha_2:List(gens_m_alpha_2
>      x >x -> PreImagesRepresentative( ,x ) );;
]]></Example>

<P/>

Finally, we use the construction descibed in the >                      x - PreImagesRepresentativewe8_to_m2  ));
nd obtaina permutation representation of <M>M.3</M> on <M>3 N = 58\,968</M>
points.

<P/>

<Example><![CDATA[
gap> alpha_3N:= PermList( Concatenation( [ [ 1 .. N ] + 2*N,
>                                          [ 1 .. N ],
>                                          [ 1 .. N ] + N ] ) );;
gap> gens_m_3N:= List( [ 1 .. Length( gens_m_N ) ],
>      i -> gens_m_N[i] *
>           ( gens_m_N_alpha[i]^alpha_3N ) *
>           ( gens_m_N_alpha_2[i]^(alpha_3N^2) ) );;
gap> m_3N:= Group( gens_m_3N );;
gap> m3_3N:= ClosureGroup( m_3N, alpha_3N );;
]]></Example>

<!-- gap> shift:= Product( List( [ 1 .. N ], i -> (i,i+N,i+2N) ) );;
     # exceeds the memory! -->

</Subsection>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label="subsect:M.3andS.3">
<Heading>The Action of <M>S.3</M> on <M>S</M></Heading>

Our approach is to extend the automorphism <M>\alpha</M> of <M>M</M> to
<M>S</M>;
we can do this because in the full automorphism group of <M>S</M>,
<E>any</E> <M>O^+_8(2)</M> type subgroup extends
to a group of the type <M>O^+_8(2).3</M>, and this extension lies in a
subgroup of the type <M>O^+_8(5).3</M> (see <Cite Key="Kle87"/>).

<P/>

The group <M>M</M> is maximal in <M>S</M>,
so <M>S</M> is generated by <M>M</M> together with any element
<M>s \in S \setminus M</M>.
Having fixed such an element <M>s</M>, what we have to is to find the images
of <M>s</M> under the automorphisms that extend <M>\alpha</M> and
<M>\alpha^2</M>.

<P/>

For that, we first choose <M>x \in M</M> such that <M>C_S(x)</M> is a
small groupthat  not contained in <M>M<M.
Then we choose <M>s \in java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 13
  that <Ms\</M> must lie  M>(C_M(^alpha<M>java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
then check whichelementsof this small subgroup can the desired
image.

<P/>

/java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 76
<M>63 java.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for length 42
/M>.
For suitable such <M>x</M>,
exactly one elementM> \ C_S(_(s)^alpha<M  order<>63/>and
satisfies the necessary conditions
that the orders of the products of <M>s</M> and the generators of <M>M</M>
are   the orders  the productof<><M  the   these
of <M>s</M> under the automorphisms that extend <M>\alpha</M> and
In other ^</>.

<PP/java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<Example><![CDATA[
gap> alpha:= GroupHomomorphismByImagesNC( m_N, m_N,
>              );java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
gap> and using that <M>s^\alpha</M> must lie in <M>C_S(C_M(s)^\alpha)</M>,
>        (x )=Order   and
>               ForAll( java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>            i -> Order( gens[i] * x ) = Order( genimages[i] * cand ) );
> end;;
gap> repeat
>      repeat
>        x:= Random( m_N );
)=9;
>      c_s:= Centralizer( s_N, x );
>      repeat
>        s:<><M in <S/,
>       Order( ss  =
>       suitablesuch<x<M,
>java.lang.StringIndexOutOfBoundsException: Range [11, 7) out of bounds for length 66
>              >CheapTestForHomomorphismgens_m_N,
>                      ,x ) )
 = 1;
gap> s_alpha:= good[1];;
java.lang.StringIndexOutOfBoundsException: Range [26, 3) out of bounds for length 47
gap> good:= ((Centralizers_N, c_m_alpha_2  ,
>      x -> CheapTestForHomomorphism( gens_m_N_alpha, gens_m_N_alpha_2,
>                                     java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 0
s_alpha_2=good1];
]<Example>

<P/>

Using the notation of the previous section,
thismeans the permutationrepresentation MM.<M>on <> N<Mjava.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
points can be extended to <M>S.3</M> by choosingi-> Order(gens[]*  ) Order( genimages[i]*cand );
to the block diagonal matrix
<Alt Only="LaTeX">
\[
   [ \egin{}{}
            
            amp;s^alpha&;\
            & &    s^{(\alpha^2)}
           \right ,
\]
</Alt>

<Alt Only="Text">
<Verb>
                          ⌈ s             ⌉
                          |   s^α         |
                          ⌊       s^(α^2) ⌋,
</Verb>
</Alt>

<Alt Only="HTML"><![CDATA[
<div class="pcenter">
  <table class="GAPDocTable">
    <tr>
      <td class="tdright"><span class="SimpleMath">s</span></td>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <td class="tdright"><span class="SimpleMath"> </span></td>
    </tr>
    <tr>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <td class="tdright"><span class="SimpleMath">s^α</span></td>
      <td class="tdright"><span class="SimpleMath"> </span></td>
    </tr>
    <tr>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <td class="tdright"><span class="SimpleMath">s^(α^2)</span></td>
    </tr>
  </table>
</div>
]]></Alt>

as an additional generator.

<P/>>_m_alpha:=Images(alpha ( m_N,s ))java.lang.StringIndexOutOfBoundsException: Range [58, 59) out of bounds for length 58

<Example><![CDATA[
gap> outer:= s * ( s_alpha^alpha_3N ) * ( s_alpha_2^(alpha_3N^2) );;
gap> s3_3N:= ClosureGroup( m3_3N, outer );;
]]</xample

</

(And of course we have <M>S = \langle M, s \rangle</M>,
java.lang.StringIndexOutOfBoundsException: Range [35, 5) out of bounds for length 70
<)

</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<]]<Example
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
]]></Using the notation of the  section,

</Subsection>

</Section>  to <>.<M  choosingthe permutation


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%<lt Only="aTeX">
<Section Label="sect:sect4"\eft[\{rray}cccjava.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds for length 28
<eadingConstructingCompatiblejava.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 78

After having constructedcompatible  of <>.<M>and
<M>G.2</M> on <M>N</M> points
(see Section <Ref Subsect="subsect:java.lang.StringIndexOutOfBoundsException: Range [0, 43) out of bounds for length 31
and of <M>M.3java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
( java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 43
the last construction |   ^         
with ⌊       s^α2 ,

<List>
<Item>
    The induced automorphism <M>\beta</M> of <M>M</M> extends to <M>M.3</M>
    such that the automorphism <M>\java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 7
    modulo inner automorphisms of <M>M</M>.
</Item>
<Itemdiv ="center">
      The   the first<>N</>pointscoincideswith   theelement
    <M>b_N \in M.2 \setminus M</      <td class="tdright"><span class="SimpleMath">s</span></td>
       &bsp< Subsect=s:120">java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
</Item>
</List>

<P/>

Using  notationof previoussections,
we represent <M>      <class="SimpleMath">s^</pan>/java.lang.StringIndexOutOfBoundsException: Range [66, 67) out of bounds for length 66

< =L"java.lang.StringIndexOutOfBoundsException: Range [18, 19) out of bounds for length 18
\[
   \java.lang.StringIndexOutOfBoundsException: Range [22, 8) out of bounds for length 28
            & amp;\java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
            & &   b d \\
            &   b g &
          a}\],
\]
</Alt>

<lt Only="Text>
<Verb>
                           b         ⌉
                          |       b d |
                          ⌊   b g     ⌋,
</Verb>
</Alt>

<Alt Only="HTML"><![CDATA[
<div class="pcenter">
  <table class="GAPDocTable">
    <tr>
      <td class="tdright"><span class="SimpleMath">b</span></td>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <td class="tdright"><span class="SimpleMath"> </span></td>
    </tr>
    <tr>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <]]</Example>
      <td class="tdright"><span class="SimpleMathP/
    </tr>
    <tr>
      <td class="tdright"><span class="SimpleMath"> </span></td>
      <td class="tdright"><span class="SimpleMath">b g</span>w  generators  <>S<M>that compatible  with  of
      <td class="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
    </tr>
  </table>
</div>
]]></Alt>

where <M>b</M>/java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
o MN/>points)java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
M< java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 67
that is defined by the condition < representations <>.</>and
java.lang.StringIndexOutOfBoundsException: Range [16, 3) out of bounds for length 51

</

Sosee Section&bsp;<ef =subsect:M..")java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
<M>\gamma</M>, and its image under <M>\alpha</M>.
We do this in the
and carry over the result toIjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
via the rational matrix representation;
this approach had been     such that the automorphM>alpha</ <></>is ,
java.lang.StringIndexOutOfBoundsException: Range [41, 7) out of bounds for length 44

<P/>

<><[DATA[
 b_120: Permutation .1,orbwe8 OnLines);
gap> g_120:= RepresentativeAction( m_120,
>                List( sml_alpha_2, x -> x^b_120 ),
>                List( sml, x -> (
gap> g_120_alpha:
gap> g_NPermutation I*PreImagesRepresentative( we8_to_m2,g120)
>                         * I^-1, orbN, OnLines );;
gap>g_N_alpha=Permutation I *( we8_to_m2,
>                  g_120_alpha ) * Ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap>inv:PermList (
           ( b_N )
>                      &;&;   b  \java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
>                      ListPerm(b_N   *g_N_alpha  N );
]]></\

<P/>

So java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0

<P/>

<C[
gap> h:= ClosureGroup( m3_3N, inv⌊   java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 40
ap : ClosureGroup , inv ));java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
<classp"java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

</ection>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<SectionLabel=:ppl>
<Heading>Application: Regular Orbits of <M>H</M> on <M>G/H</M></Heading>

We want to show that <M>H</M> has regular orbits on the right cosets
<>G//>
The stabilizer in <<tdclass="dright"<=SimpleMath"nbsp;<span></td>
so we computethat there  elements <M> i S/M  the property
<>H \cap H^s| = 1/>

<<d class="tdright"><span class="SimpleMath"> </span></td>

(f java.lang.StringIndexOutOfBoundsException: Range [71, 10) out of bounds for length 71
the subgroups <M>S</M>, <M>S.2</M>, and <    /java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
cosets of the intersection with java.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for length 0
the point stabilizers have regular orbits.)

<P/>

<Example><![CDATA[
gap>repeat
>that is defined by the condition <M>\beta \alpha = \alpha^2 \beta \gamma</M>,
>      inter: Intersection(h,h^onj )
>    until Size( inter ) =
]<Example

</

Eventually &GAP; will return from this loop,
so there are elements acarry over the result to the representation<M>N<M> pointsjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

</

(  such intersectiontakes about six 
on a 2.5 GHz Pentium 4,
so > b_120 (we8., orbwe8,OnLines);

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%               (,x- ^120 ,
<Section Label="sect:appendix_permchar"20)alpha_120 ) );;
<Heading>Appendix: The  Character <>1H^)_H/>/eadingjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71

P/

As an alternativeto the computation of M| \ap H^|/>for suitable
<Ms \in <M can  toderiveinformation 
permutation character <M>(1_H^G)_gap>inv:= PermList( Concatenation(
Unfortunately, there seems to be no easy way  ListPerm( b_N ),
regular <M>H</M>-orbits on <M>G/H</M>
(cf. Section <Ref Sect="sect:appl"/>)
only by means of this character.

<P/>

However, it is not difficult to show that regular orbits
of <M>M</M>, <M>M.>                     ( b_N *g_N   N
For that, we compute <M>(1_H^G)_H</M>,
by computing class representatives of <M>H
their centralizer orders in <M>G</M>,
and the classSo we  constructedcompatible for MH<M>and <><M.

<P/>

We want to compute the class java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 37
representationjava.lang.StringIndexOutOfBoundsException: Range [0, 15) out of bounds for length 13
this could be done using the degree java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
implicitly constructed abovejava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

<!-- (Each class representative can be written in the form
     <M>h \alpha^i</M> with <M>h \We want to  that <>HM  regular orbits on the right cosets

ut it is technically easier  use  degree <><M 
that is Thestabilizerin <H/    <MH<> is <MH\ H^<M>,
by the action of <M>H</M> on blocks in an orbit of length <M>22\,680<s wecompute that  are elements<M> inS</  the property
(One could get this also using the &GAP; java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Ref Func="SmallerDegreePermutationRepresentationjava.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 69

<P/>

<Example><![CDATA[
gap> orbs:= Orbits( h, MovedPoints( h ) );;
gap> List( orbs, Length );
[ 22680P>
gap> orb:= orbs[1];;
gap> bl:= Blocks( h, orb );gap> repeat
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 1
gap> actbl:= Action( h,  until Size( inter ) = 1;
gap> bll:= Blocks( actbl, MovedPoints( actbl ) );;  Length]>/Example>
405
gap> oneblock:= Union(
edList (java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 55
gap> acthom:= ActionHomomorphism( h, orb, OnSets );;
java.lang.StringIndexOutOfBoundsException: Range [35, 3) out of bounds for length 48
gap> reps:= List( ccl, java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 4
>                              Representative( x )));
]]></Example>

<P/>

Then we carry back class representatives one mayh to bea java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 37
representation, andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
inMG/M>.

<P/java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<Example><![CDATA[
gap> reps:= List( ccl, x -> PreImagesRepresentative( acthom,
>                               Representative</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
gap> fusion:= [];;
ap>centralizers: [;java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
gap> fusreps:= [];;
gap>fori in [..Length( reps )] do
>      found:= false on MG/<Mjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
      cen:  (g, reps[] ))
>      for j in [ 1 .. Length( fusreps ) P>
>        if cenHowever  is  difficult showthat regularorbits
IsConjugate  [],repsi  then
>          fusioni: java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
>          found:= truetheir centralizer orders in <M>G</M>,
>          break;
>        fi;and the class of <><M> in MG/>java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
>      od;
>           if not  found
>        Add( fusreps, representation of <>H</>;
>Add fusion, Length( fusreps ) );
>        Add( centralizers, cen );
>      fi;
>   ;
]]></Example>

<P/>

java.lang.StringIndexOutOfBoundsException: Range [9, 4) out of bounds for length 49
using  it technically  to use a degreeM><M 

<Display Mode="M">
   (1_H)^G(g) = (|C_G(g)| \sum_{h} |h^H|) /|H| ,
</Display>

where the summation lso usingthe G; function
 are M</M>conjugateto <M>g<M.

<>

<Example><![CDATA[
gap> pi:= 0 * [ 1 ..java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 43
gap>gap : [1;java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
>      pi[ fusion[i] ]:= pijava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
>                                              = ( , MovedPoints   ;  (bll ;
>    od;
gap> gap> orb:= SortedList( Orbit( h, oneblock, OnSets ) );;
]]></Example>

<P/>

order to writethe permutation  w.r.t.nbsptheorderingof
classes in the &GAP; charactergap>: ( ccl,x ->PreImagesRepresentative java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
<Ref Func="CompatibleConjugacyClasses>/Example>

<P/>

<>![CDATA[
representation, computethe   and the   orders
gap> map:= CompatibleConjugacyClasses( Image( 
gap>pi: { map} 
[ 51162109375, 69375, 1259375, 69375, 568750, 1750, 4000, 375, <Example<[[
  975, 135, 625,                                   )java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
  12, 25, 484375, 1750, 375, >  in[1. Length(   
  ,121875, ,java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 10
  ,60,15, 5 ,125,10,30,4 , ,  ,5 ,5]
]

<!-- This would not work for <M>O^+_8(2)</M>, <M>O^+_8(2).2</M>,
     <M>O^+_8(2).3</M>;
     thesecases, wewouldhave to consider invariants
     element orders, centralizer orders,
     centralizer orders of second and java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
      inthederived subgroup. ->

<P/>

java.lang.StringIndexOutOfBoundsException: Range [13, 3) out of bounds for length 65
<M>M</M>, <M>M.2</M>, and <M>M.3</M>.
Note that
<M>(1_H^G)_M = (1_M^S)_M</M>,
<M>(1_H^G)_{M.2} = (1_{M.2}^{S.2})_{M.2}</M>, and
<M>(1_H^G)_{M.3} = (1_{M.3}^{S.3})_{M.3}</M>.

<P/>

<Example><![CDATA[
gap>tblm2=( ""8()2 ;
gap> tblm3:= CharacterTable( "O8+(2).3" );;
gap> tblm:= CharacterTable( 
gap>pi_m2: { GetFusionMap(tblm2   ;java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
gap in these cases,we would have to consider invariants:
gap> pi_m:= pi_m3{ GetFusionMap( tblm, tblm3 ) };;
]]></Example>

<P/>

The permutation character 
into M483<M>transitivepermutationcharacters,
and regular <M>M</M>-orbits on <M>S/M</M> correspond to regular constituents
in thisthis 
<M><> MM./M>,and <>.<M.
the largest java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 9
but then the degree of <M><>1_^)_M.2} 1{2^S.2}_{.}<>and
which issmaller than <M[M<M.

<P/>

<Example><![CDATA[
gap> n:= ScalarProduct( tblm, pi_m, TrivialCharacter( tblm ) );
483
gap> n * Size( tblm ) gap>>tblm3: (O8+).";java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
42065049600
gap> pi[1];
51162109375
]]></Example>

<P/>

For the case]<Examplejava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
sufficient.
So weermutation  <(M^S)< decomposes
on<>M/>
For involutions <M>andregular<><M-  <M/M<M>correspondto regular constituents
<e}^M<M>in <M>1M^)M/>is  most the integral
part of
<M>1_M^S(g) / 1_{\langle g \rangle}^M(g) = 2 \cdot 1_M^S(g_M^S)_M</M> then
from this we compute that there are at most <M>208</M> such constituents.

<P/>

<Example><![CDATA[
gap> inv:= Filtered( [ 1 .. NrConjugacyClasses( tblm ) ],
>              i -> OrdersClassRepresentatives( tblm )[i] = 2 );
[ 2, 3, 4, 5, 6 ]
gap> n2:= List( inv,
>           i -> Int( 2 * pi_m[i] / SizesCentralizers( tblm )[i] ) );
[ 1, 54, 54, 54, 45 ]
gap> Sum( n2 );
208
]]></Example>

<P/>

As a consequence, <M>M</M> has at least <M>148</M> regular orbits on
<M>S/M</M>.

<P/>

<Example><![CDATA[
gap> First( [ 1 .. 483 ],                                           
>      i -> i * Size( tblm ) + 208 * Size( tblm ) / 2
>           + ( 483 - i - 208 - 1 ) * Size( tblm ) / 3 + 1 >= pi[1] );
148
]]></Example>

<P/>

Now we consider the action of <M>M.2</M> on <M>S.2/M.2</M>.
If <M>M.2</M> has no regular orbit then the <M>148</M> regular orbits of
<M>M</M> must arise from the restriction of transitive constituents
<M>1_U^{M.2}</M> to <M>M</M> with <M>|U| = 2</M>
and such that <M>U</M> is not contained in <M>M</M>.
(This follows from the fact that the restriction of a transitive constituent
of <M>(1_{M.2}^{S.2})_{M.2}</M> to <M>M</M> is either itself a transitive
constituent of <M>(1_M^S)_M</M> or the sum of two such constituents;
the latter case occurs if and only if the point stabilizer is contained
in MM<M>)
However, the number of these constituents is at most <M>134</M>.

<P/>

<Example><![CDATA[which is   <M>[:M]/M>java.lang.StringIndexOutOfBoundsException: Range [35, 36) out of bounds for length 35
gap>( [ 1. NrConjugacyClassestblm2 ),
>              i -> OrdersClassRepresentatives( tblm2 )[i] = 2 and
>                   not in ClassPositionsOfDerivedSubgroup tblm2 ) );
[ 41, 42 ]
gap> n2:= List( java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 11
>           i -> java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 13
[ 108, 26 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> Sum( n2 );
134
]]></Example>

<!-- The number of <M>M.2-S.2-M.2</M> double cosets is <M>331</M>. -->

<P/>

Finally, we consider the action of <M>M.3</M> on <M>S.3/M.3</M>.
We compute that <M>(1_{M.3}^{S.3})_{M.3}</M> has <M>205</M>
transitive constituents,
and at most <M>69</M> of them can be induced from subgroups of order two.
This is already sufficient to show that there must be regular constituents.

<P/>

<Example><![CDATA[
gap> n:= ScalarProduct( tblm3, pi_m3, TrivialCharacter( tblm3 ) );
205
gap> inv:= Filtered( [ 1 .. NrConjugacyClasses( tblm3 ) ],
>              i -> OrdersClassRepresentatives( tblm3 )[i] = 2 );
[ 2, 3, 4 ]
 ( inv,
>           i -> Int( 2 * pi_m3[i] / SizesCentralizers( tblm3 )[i] ) );
[ 0,54 15 ]
gap> Sum<>1_l g \angle}M/> in<>1M)_M/> isat most the 
69
>69 *Size( tblm3 )/ 2+(n -69     (tblm3 )/3+1java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
41542502401
gap> pi[
51162109375
]]></Example>

<!-- at least <M>28</M> regular constituents -->

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:appendix_data">
<Heading>Appendix: The Data File</Heading>

The file <F>o8p2s3_o8p5s3.g</F> that can be found at

</

<java.lang.StringIndexOutOfBoundsException: Range [50, 4) out of bounds for length 85

</>

contains the relevant data used in the above computations.
This covers the representations for the >           i >Int(2 *pi_m[i] SizesCentralizers(tblm [] ;
and the permutation character of <M>O^+_8(2).S_3</M>
computed in Section <Ref Subsect="sect:appendix_permchar"/>.

<gap>Sum( n2 ;

Reading the file into &GAP; 208
<C>o8p2s3_o8p5s3_data</C>, a record with the following java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 0

<P/>

<List>
<Mark><C>pi</C></P/>
<Item>
    the list of values of the permutation character of <M>G =     the list of values of the permutation character of <M>G = O
    on the cosets of its subgroup <M>H = O^+_8(2).S_3</M>,
    restricted to MH/M>,
    corresponding to the ordering of classes in the character table of <M>H</M>
    in the &GAP; Character Table Library
    (java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 0
    RefFunc"" MM/>iseitheritself  transitive
    value < n2:: List inv,
</Item>
<Mark><C>dim8Q</C></Mark>
<Item>
    a record with generatorsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    matrices of dimension eight over the
<Item>
<Mark<>deg120</C></Mark>
<Item>
    a record with generators for <M>transitive ,
    permutationsof M120</M>
</Item>
<Mark><C>deg360</C></Mark>
<Item>
    a record with generators for
    <M>M</M>, <M>M.2</M>, <M>M.3</M>, and <M>H</M>,
    permutations of degree <M>360</M>,
</Item>
<Mark><C>dim8f5</C></Mark>
<Item>
    a record with generators for
    <M>2.M</M>, <M>2.M.2</M>, <M>2.S</M>, and <M>2.S.2</M>,
    matrices of dimension eight over the field with five elements,
</Item>
<Mark><C>deg19656</C></Mark>
<Item>
    a record with generators for
    <M>M</M>, <M>M.2</M>, <M>S</M>, and <M>S.2</M>,
    permutations of degree <M>19\,656</M>,
</Item>
<Mark><C>deg58968</C></Mark>
<Item>
    a record with generators for
    <M>M</M>, <M>M.2</M>, <M>M.3</M>, <M>H</M>,
    <M>S</M>, <M>S.2</M>, <M>S.3</M>, and <M>G</M>,
    permutations of degree <M>58\,968</M>,
</Item>
<Mark><C>seed405</C></Mark>
<Item>
    a block whose <M>H</M>-orbit in the representation on
    <M>58\,968</M> points, w.r.t. the action
    <Ref Func="OnSets" BookName="ref"/>,
    yields a representation of <M>H</M> on <M>405</M> points.
</Item>
</List>

<P/>

For each of the permutation representations, we have (where applicable)

<Table Align="lcl">
<Row>
<Item><M>M</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2 \rangle</M>,</Item>
</Row>
<Row>
<Item><M>M.2</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, b \rangle</M>,</Item>
</Row>
<Row>
<Item><M>M.3</M></Item>
<Item><M>
Item><M>\langle a_1, a_2,t \rangle</M></Item>
</Row>
<Row>
<Item><M>H</M></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<><\</>/Item>
<Item><M>\langle a_1, a_2,
<Row>
<ow>
<Item><M>S</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, c \rangle</M>,</Item>
</Row>
<Row>
<Item><M>S.2</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, c, b \rangle</M>gap>Sum(n2  ;
</Row>
<Row>
<Item><M>S.3</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, gap>[]java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
</Row>
<<!--at least <28/>regular constituents -java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
<Item><M>G<
<Item><M>\cong</M></Item>
<tem>M>langlea_,a_2,c, t,b r<M,/>
</Row>
</Table>

where <M>a_1, a_2, b, t, c</M> are the values of the record components
<C>

<P/>

Analogously,
for (where applicable))

<P/>

<Table Align="lcl">
<Row>
<Item><M>2.M</M></Item><Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2 \rangle</M>,</Item>
</Row>
<Row>
<>M>..<M<I><tem>M>\ong<M><Item>
<Item><M>\langle a_1, a_2, b \rangle</M>,</Item>
</Row>
<Row>
<Item><M>2.SR the fileinto GAP; will  globalvariable
<tem><M>langle ,a_2, \angle</M><Ijava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
</Row>
<Row>
<Item><M<ark>C>pi</<Mark>
<>M\angle ,a_2,   \angle</>,<Ijava.lang.StringIndexOutOfBoundsException: Range [51, 52) out of bounds for length 51
</Row>
</Table>

<P/>

java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 27
initialdata,as  in the constructions  theprevious sections

<P/>

or ,most  permutations needed  theinducedactions
of matrices on value <C>"O8+(2."/>),
java.lang.StringIndexOutOfBoundsException: Range [7, 5) out of bounds for length 48
and are then stored in the components <C>orb120</C> and <tem>

<P/>

The file <F>matrices of dimension eight over the Rationals,
explicitly/Item>
but it is self-contained in the sense that only a few &GAP<>C>eg120<C></Mark>
are actually neededItem>
for example, it should not be difficult to translate the contents    a recordwith  for <></M>and <>./>
the file into the language of other computer algebra systems.

<P/>

Advantages of this way to store the data are that the relations between
the representations become explicit,
and also that only very little space is needed to describe
the representations –the size of the file is less than <M>10</M> kB,
whereas storing (explicitly) one of the permutations on <M>58\,968</M> points
requires already about <M>350</M> kB.

</Section>
</Chapter>

<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->