<!-- %W o8p2s3_o8p5s3.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 two.
This question arose in <Cite Key=
"BGS11"/>.
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=
"sect:overview_o8p2s3">
<Heading>Overview</Heading>
Let <M>S</M> denote the simple group
<M>O_8^+(5) \cong </M> P<M>\Omega^+(8,5)</M>,
that is, the nonabelian simple group that occurs as a composition factor
of the general orthogonal group GO<M>{}^+(8,5)</M> of <M>8 \times 8</M>
matrices over the field with five elements.
<P/>
The outer automorphism group of <M>S</M> is isomorphic to the symmetric group
on four points.
Let <M>G</M> be an automorphic extension of <M>S</M> by the symmetric group
on three points.
By <Cite Key=
"Kle87"/>,
the group <M>S</M> contains a maximal subgroup <M>M</M> of the
type
<M>O_8^+(2)</M> such that the normalizer <M>H</M>, say,
of <M>M</M> in <M>G</M> is an automorphic extension of <M>M</M> by
a symmetric group on three points.
(In fact, <M>H</M> is isomorphic to the full automorphism group of
<M>O_8^+(2)</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> and <M>3</M>, respectively.
Analogously, let <M>M.2 = H \cap S.2</M> and <M>M.3 = H \cap S.3</M>.
<P/>
In Section <Ref Subsect=
"sect:sect2"/>,
we use the following approach to construct representations of <M>M.2</M> and
<M>S.2</M>.
By <Cite Key=
"CCN85" Where=
"p. 85"/>,
the Weyl group <M>W</M> of
type <M>E_8</M> is a double cover of <M>M.2</M>,
and the reduction of its rational <M>8</M>-dimensional representation modulo
<M>5</M> embeds into the general orthogonal group GO<M>{}^+(8,5)</M>,
which has the structure <M>2.O_8^+(5).2^2</M>.
Then the actions of GO<M>{}^+(8,5)</M> and of an isomorphic image of <M>W</M>
in GO<M>{}^+(8,5)</M> on <M>1</M>-spaces in the
natural module of GO<M>{}^+(8,5)</M> yield <M>M.2</M> as a subgroup of
(a supergroup of) <M>S.2</M>,
where both groups are represented as permutation groups on <M>N = 19\,656</M>
points.
<P/>
In Section <Ref Sect=
"sect:sect3"/>,
first we use &GAP; to compute the automorphism group of <M>M</M>.
Then we take an outer automorphism <M>\alpha</M> of <M>M</M>, of order three,
and extend <M>\alpha</M> to an automorphism of <M>S</M>.
Concretely, we compute the images of generating sets of <M>S</M> and <M>M</M>
under <M>\alpha</M> and <M>\alpha^2</M>.
This yields permutation representations of <M>S.3</M> and its subgroup
<M>M.3</M> on <M>3 N = 58\,968</M> points.
<P/>
In Section <Ref Sect=
"sect:sect4"/>,
we put the above information together,
in order to construct permutation representations of <M>G</M> and <M>M</M>,
on <M>3 N</M> points.
<P/>
As an application, it is shown in Section <Ref Sect=
"sect:appl"/>
that the permutation representation of <M>G</M> on the cosets of <M>H</M>
has a base of length two;
this question arose in <Cite Key=
"BGS11"/>.
<P/>
In two appendices, it is discussed how to derive a part of this result
from the permutation character <M>(1_H^G)_H</M>
(see Section <Ref Subsect=
"sect:appendix_permchar"/>),
and a file containing the data used in the earlier sections is described
(see Section <Ref Subsect=
"sect:appendix_data"/>).
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=
"sect:sect2">
<Heading>Constructing Representations of <M>M.2</M> and <M>S.2</M></Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label=
"subsect:repres_W_E8">
<Heading>A Matrix Representation of the Weyl Group of
Type <M>E_8</M></Heading>
Following the recipe listed
in <Cite Key=
"CCN85" Where=
"p. 85, Section Weyl"/>,
we can generate the Weyl group <M>W</M> of
type <M>E_8</M> as a group of
rational <M>8 \times 8</M> matrices generated by the reflections in the
vectors
<
Display Mode=
"M">
\left(\pm 1/2, \pm 1/2, 0, 0, 0, 0, 0, 0\right)
</
Display>
plus the vectors obtained from these by permuting the coordinates,
plus those those vectors of the form
<
Display Mode=
"M">
\left( \pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2,
\pm 1/2, \pm 1/2, \pm 1/2, \pm 1/2 \right)
</
Display>
that have an even number of negative signs.
(Clearly it is sufficient to consider only one vector form a pair
<M>\pm v</M>.)
<P/>
<Example><![
CDATA[
gap> rootvectors:= [];;
gap> for i in Combinations( [ 1 .. 8 ], 2 ) do
> v:= 0 * [ 1 .. 8 ];
> v{i}:= [ 1, 1 ];
> Add( rootvectors, v );
> v:= 0 * [ 1 .. 8 ];
> v{i}:= [ 1, -1 ];
> Add( rootvectors, v );
> od;
gap> Append( rootvectors,
> 1/2 * Filtered( Tuples( [ -1, 1 ], 8 ),
> x -> x[1] = 1 and Number( x, y -> y = 1 ) mod 2 = 0 ) );
gap> we8:= Group( List( rootvectors, ReflectionMat ) );
<matrix group with 120 generators>
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label=
"subsect:embedding_W_E8">
<Heading>Embedding the Weyl group of
Type <M>E_8</M> into GO<M>{}^+(8,5)</M></Heading>
The elements in the group constructed above respect the symmetric bilinear
form that is given by the identity matrix.
<P/>
<Example><![
CDATA[
gap> I:= IdentityMat( 8 );;
gap> ForAll( GeneratorsOfGroup( we8 ), x -> x * TransposedMat(x) = I );
true
]]></Example>
<P/>
So the reduction of the matrices modulo <M>5</M> yields a group <M>W^{\ast}</M>
of orthogonal matrices w. r. t. the identity matrix.
The group GO<M>{}^+(8,5)</M> returned by the &GAP; function
<Ref Func=
"GO" BookName=
"ref"/> leaves
a different bilinear form invariant.
<P/>
<Example><![
CDATA[
gap> largegroup:= GO(1,8,5);;
gap>
Display( InvariantBilinearForm( largegroup ).matrix );
. 1 . . . . . .
1 . . . . . . .
. . 2 . . . . .
. . . 2 . . . .
. . . . 2 . . .
. . . . . 2 . .
. . . . . . 2 .
. . . . . . . 2
]]></Example>
<P/>
In order to conjugate <M>W^{\ast}</M> into this group,
we need a <M>2 \times 2</M> matrix <M>T</M> over the field with five elements
with the property that <M>T T^{tr}</M>
is half of the upper left <M>2 \times 2</M> matrix in the above matrix.
<P/>
<Example><![
CDATA[
gap> T:= [ [ 1, 2 ], [ 4, 2 ] ] * One( GF(5) );;
gap>
Display( 2 * T * TransposedMat( T ) );
. 1
1 .
gap> I:= IdentityMat( 8, GF(5) );;
gap> I{ [ 1, 2 ] }{ [ 1, 2 ] }:= T;;
gap> conj:= List( GeneratorsOfGroup( we8 ), x -> I * x * I^-1 );;
gap> IsSubset( largegroup, conj );
true
]]></Example>
</Subsection>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label=
"subsect:120pt">
<Heading>Compatible Generators of <M>M</M>, <M>M.2</M>, <M>S</M>, and <M>S.2</M></Heading>
For the next computations,
we switch from the natural matrix representation of GO<M>{}^+(8,5)</M>
to a permutation representation of PGO<M>{}^+(8,5)</M>,
of degree <M>N = 19\,656</M>,
which is given by the action of GO<M>{}^+(8,5)</M> on the smallest orbit
of <M>1</M>-spaces in its natural module.
<P/>
<Example><![
CDATA[
gap> orbs:= OrbitsDomain( largegroup, NormedRowVectors( GF(5)^8 ),
> OnLines );;
gap> List( orbs, Length );
[ 39000, 39000, 19656 ]
gap> N:= Length( orbs[3] );
19656
gap> orbN:= SortedList( orbs[3] );;
gap> largepermgroup:= Action( largegroup, orbN, OnLines );;
]]></Example>
<P/>
In the same way,
permutation representations of the subgroup <M>M.2 \cong </M>SO<M>{}^+(8,2)</M>
and of its derived subgroup <M>M</M> are obtained.
But first we compute a smaller generating set of the simple group <M>M</M>,
using a permutation representation on <M>120</M> points.
<P/>
<Example><![
CDATA[
gap> orbwe8:= SortedList( Orbit( we8, rootvectors[1], OnLines ) );;
gap> Length( orbwe8 );
120
gap> we8_to_m2:= ActionHomomorphism( we8, orbwe8, OnLines );;
gap> m2_120:= Image( we8_to_m2 );;
gap> m_120:= DerivedSubgroup( m2_120 );;
gap> sml:= SmallGeneratingSet( m_120 );; Length( sml );
2
gap> gens_m:= List( sml, x -> PreImagesRepresentative( we8_to_m2, x ) );;
]]></Example>
<P/>
Now we compute the actions of <M>M</M> and <M>M.2</M>
on the above orbit of length <M>N</M>.
For generating <M>M.2</M>, we choose an
element <M>b_N \in M.2 \setminus M</M>,
which is obtained from the action of a matrix <M>b \in 2.M.2 \setminus 2.M</M>.
<P/>
<Example><![
CDATA[
gap> gens_m_N:= List( gens_m,
> x -> Permutation( I * x * I^-1, orbN, OnLines ) );;
gap> m_N:= Group( gens_m_N );;
gap> b:= I * we8.1 * I^-1;;
gap> DeterminantMat( b );
Z(5)^2
gap> b_N:= Permutation( b, orbN, OnLines );;
gap> m2_N:= ClosureGroup( m_N, b_N );;
]]></Example>
<P/>
(Note that <M>M.2</M> is not contained in PSO<M>{}^+(8,5)</M>,
since the determinant of <M>b</M> is <M>-1</M> in the field with five elements.)
<P/>
The group <M>S</M> is the derived subgroup of PSO<M>{}^+(8,5)</M>,
and <M>S.2</M> is generated by <M>S</M> together with <M>b_N</M>.
<P/>
<Example><![
CDATA[
gap> s_N:= DerivedSubgroup( largepermgroup );;
gap> s2_N:= ClosureGroup( s_N, b_N );;
]]></Example>
</Subsection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=
"sect:sect3">
<Heading>Constructing Representations of <M>M.3</M> and <M>S.3</M></Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Subsection Label=
"sect:action_M.3_M">
<Heading>The Action of <M>M.3</M> on <M>M</M></Heading>
Let <M>\alpha</M> be an automorphism of <M>M</M>, of order three.
Then a representation of the semidirect product <M>M.3</M> of <M>M</M> by
<M>\langle \alpha \rangle</M> can be constructed as follows.
<P/>
If <M>M</M> is given by a matrix representation then we map <M>g \in M</M>
to the block diagonal matrix
<Alt Only=
"LaTeX">
\[
\left[ \begin{array}{ccc}
g & & \\
& g^\alpha & \\
& & g^{(\alpha^2)}
\end{array} \right] ,
\]
</Alt>
<Alt Only=
"Text">
<Verb>
⌈ g ⌉
| g^α |
⌊ g^(α^2) ⌋,
</Verb>
</Alt>
<Alt Only=
"HTML"><![
CDATA[
<div class=
"pcenter">
<table class=
"GAPDocTable">
<tr>
<td class=
"tdright"><span class=
"SimpleMath">g</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">g^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">g^(α^2)</span></td>
</tr>
</table>
</div>
]]></Alt>
and we represent <M>\alpha</M> by the block permutation matrix
<Alt Only=
"LaTeX">
\[
\left[ \begin{array}{ccc}
& & I \\
I & & \\
& I &
\end{array} \right] ,
\]
</Alt>
<Alt Only=
"Text">
<Verb>
⌈ I ⌉
| I |
⌊ I ⌋,
</Verb>
</Alt>
<Alt Only=
"HTML"><![
CDATA[
<div class=
"pcenter">
<table class=
"GAPDocTable">
<tr>
<td class=
"tdright"><span class=
"SimpleMath"> </span></td>
<td class=
"tdright"><span class=
"SimpleMath"> </span></td>
<td class=
"tdright"><span class=
"SimpleMath">I</span></td>
</tr>
<tr>
<td class=
"tdright"><span class=
"SimpleMath">I</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">I</span></td>
<td class=
"tdright"><span class=
"SimpleMath"> </span></td>
</tr>
</table>
</div>
]]></Alt>
where <M>I</M> is the identity
element in <M>M</M>.
<P/>
We need the action of <M>\alpha</M> on <M>M</M>.
More precisely, we need images of the chosen generators of <M>M</M>
under <M>\alpha</M> and <M>\alpha^2</M>.
<P/>
The group <M>M</M> is small enough for asking &GAP; to compute
its automorphism group, which is isomorphic with <M>O^+_8(2).S_3</M>;
for that, we use the degree <M>120</M> permutation representation
constructed in Section <Ref Subsect=
"subsect:120pt"/>.
<P/>
<Example><![
CDATA[
gap> aut_m:= AutomorphismGroup( m_120 );;
]]></Example>
<P/>
We pick an outer automorphism <M>\alpha</M> of order three.
<P/>
<Example><![
CDATA[
gap> nice_aut_m:= NiceMonomorphism( aut_m );;
gap> der:= DerivedSubgroup( Image( nice_aut_m ) );;
gap> der2:= DerivedSubgroup( der );;
gap> repeat x:= Random( der );
> ord:= Order( x );
> until ord mod 3 = 0 and ord mod 9 <> 0 and not x in der2;
gap> x:= x^( ord / 3 );;
gap> alpha_120:= PreImagesRepresentative( nice_aut_m, x );;
]]></Example>
<P/>
Next we compute the images of the generators <C>sml</C> under <M>\alpha</M>
and <M>\alpha^2</M>,
and the corresponding elements in the action of <M>M</M> on <M>N</M> points.
<P/>
<Example><![
CDATA[
gap> sml_alpha:= List( sml, x -> Image( alpha_120, x ) );;
gap> sml_alpha_2:= List( sml_alpha, x -> Image( alpha_120, x ) );;
gap> gens_m_alpha:= List( sml_alpha,
> x -> PreImagesRepresentative( we8_to_m2, x ) );;
gap> gens_m_alpha_2:= List( sml_alpha_2,
> x -> PreImagesRepresentative( we8_to_m2, x ) );;
gap> gens_m_N_alpha:= List( gens_m_alpha,
> x -> Permutation( I * x * I^-1, orbN, OnLines ) );;
gap> gens_m_N_alpha_2:= List( gens_m_alpha_2,
> x -> Permutation( I * x * I^-1, orbN, OnLines ) );;
]]></Example>
<P/>
Finally, we use the construction descibed in the beginning of this section,
and obtain a 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 group that is not contained in <M>M</M>.
Then we choose <M>s \in C_S(x) \setminus M</M>,
and using that <M>s^\alpha</M> must lie in <M>C_S(C_M(s)^\alpha)</M>,
we then check which elements of this small subgroup can be the desired
image.
<P/>
Each
element <M>x</M> of order nine in <M>M</M> has a root <M>s</M> of order
<M>63</M> in <M>S</M>,
and <M>C_S(x)</M> has order <M>189</M>.
For suitable such <M>x</M>,
exactly one
element <M>y \in C_S(C_M(s)^\alpha)</M> has order <M>63</M> and
satisfies the necessary conditions
that the orders of the products of <M>s</M> and the generators of <M>M</M>
are equal to the orders of the product of <M>y</M> and the images of these
generators under <M>\alpha</M>.
In other words, we have <M>s^\alpha = y</M>.
<P/>
<Example><![
CDATA[
gap> alpha:= GroupHomomorphismByImagesNC( m_N, m_N,
> gens_m_N, gens_m_N_alpha );;
gap> CheapTestForHomomorphism:= function( gens, genimages, x, cand )
> return Order( x ) = Order( cand ) and
> ForAll( [ 1 .. Length( gens ) ],
> i -> Order( gens[i] * x ) = Order( genimages[i] * cand ) );
> end;;
gap> repeat
> repeat
> x:= Random( m_N );
> until Order( x ) = 9;
> c_s:= Centralizer( s_N, x );
> repeat
> s:= Random( c_s );
> until Order( s ) = 63;
> c_m_alpha:= Images( alpha, Centralizer( m_N, s ) );
> good:= Filtered( Elements( Centralizer( s_N, c_m_alpha ) ),
> x -> CheapTestForHomomorphism( gens_m_N,
> gens_m_N_alpha, s, x ) );
> until Length( good ) = 1;
gap> s_alpha:= good[1];;
gap> c_m_alpha_2:= Images( alpha, c_m_alpha );;
gap> good:= Filtered( Elements( Centralizer( s_N, c_m_alpha_2 ) ),
> x -> CheapTestForHomomorphism( gens_m_N_alpha, gens_m_N_alpha_2,
> s_alpha, x ) );;
gap> s_alpha_2:= good[1];;
]]></Example>
<P/>
Using the
notation of the previous section,
this means that the permutation representation of <M>M.3</M> on <M>3 N</M>
points can be extended to <M>S.3</M> by choosing the permutation corresponding
to the block diagonal matrix
<Alt Only=
"LaTeX">
\[
\left[ \begin{array}{ccc}
s & & \\
& s^\alpha & \\
& & s^{(\alpha^2)}
\end{array} \right] ,
\]
</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/>
<Example><![
CDATA[
gap> outer:= s * ( s_alpha^alpha_3N ) * ( s_alpha_2^(alpha_3N^2) );;
gap> s3_3N:= ClosureGroup( m3_3N, outer );;
]]></Example>
<P/>
(And of course we have <M>S = \langle M, s \rangle</M>,
which yields generators for <M>S</M> that are compatible with those of
<M>M</M>.)
<P/>
<Example><![
CDATA[
gap> s_3N:= ClosureGroup( m_3N, outer );;
]]></Example>
</Subsection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=
"sect:sect4">
<Heading>Constructing Compatible Generators of <M>H</M> and <M>G</M></Heading>
After having constructed compatible representations of <M>M.2</M> and
<M>G.2</M> on <M>N</M> points
(see Section <Ref Subsect=
"subsect:120pt"/>)
and of <M>M.3</M> and <M>S.3</M> on <M>3 N</M> points
(see Section <Ref Subsect=
"subsect:M.3andS.3"/>),
the last construction step is to find a permutation on <M>3 N</M> points
with the following properties:
<List>
<Item>
The induced automorphism <M>\beta</M> of <M>M</M> extends to <M>M.3</M>
such that the automorphism <M>\alpha</M> of <M>M</M> is inverted,
modulo inner automorphisms of <M>M</M>.
</Item>
<Item>
The action on the first <M>N</M> points coincides with that of the
element
<M>b_N \in M.2 \setminus M</M> that was constructed in
Section <Ref Subsect=
"subsect:120pt"/>.
</Item>
</List>
<P/>
Using the
notation of the previous sections,
we represent <M>\beta</M> by a block matrix
<Alt Only=
"LaTeX">
\[
\left[ \begin{array}{ccc}
b & & \\
& & b d \\
& b g &
\end{array} \right] ,
\]
</Alt>
<Alt 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>
<td class=
"tdright"><span class=
"SimpleMath"> </span></td>
<td class=
"tdright"><span class=
"SimpleMath">b d</span></td>
</tr>
<tr>
<td class=
"tdright"><span class=
"SimpleMath"> </span></td>
<td class=
"tdright"><span class=
"SimpleMath">b g</span></td>
<td class=
"tdright"><span class=
"SimpleMath"> </span></td>
</tr>
</table>
</div>
]]></Alt>
where <M>b</M> describes the action of <M>\beta</M> on <M>M</M>
(on <M>N</M> points),
<M>g</M> describes the inner automorphism <M>\gamma</M> of <M>M</M>
that is defined by the condition <M>\beta \alpha = \alpha^2 \beta \gamma</M>,
and <M>d</M> describes <M>\gamma \gamma^\alpha</M>.
<P/>
So we compute an
element in <M>M</M> that induces the conjugation automorphism
<M>\gamma</M>, and its image under <M>\alpha</M>.
We do this in the representation of <M>M</M> on <M>120</M> points,
and carry over the result to the representation on <M>N</M> points,
via the rational matrix representation;
this approach had been used already in
Section <Ref Subsect=
"subsect:120pt"/>.
<P/>
<Example><![
CDATA[
gap> b_120:= Permutation( we8.1, orbwe8, OnLines );;
gap> g_120:= RepresentativeAction( m_120,
> List( sml_alpha_2, x -> x^b_120 ),
> List( sml, x -> (x^b_120)^alpha_120 ), OnTuples );;
gap> g_120_alpha:= g_120^alpha_120;;
gap> g_N:= Permutation( I * PreImagesRepresentative( we8_to_m2, g_120 )
> * I^-1, orbN, OnLines );;
gap> g_N_alpha:= Permutation( I * PreImagesRepresentative( we8_to_m2,
> g_120_alpha ) * I^-1, orbN, OnLines );;
gap> inv:= PermList( Concatenation(
> ListPerm( b_N ),
> ListPerm( b_N * g_N ) + 2*N,
> ListPerm( b_N * g_N * g_N_alpha ) + N ) );;
]]></Example>
<P/>
So we have constructed compatible generators for <M>H</M> and <M>G</M>.
<P/>
<Example><![
CDATA[
gap> h:= ClosureGroup( m3_3N, inv );;
gap> g:= ClosureGroup( s3_3N, inv );;
]]></Example>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=
"sect:appl">
<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
<M>G/H</M>.
The stabilizer in <M>H</M> of the coset <M>H g</M> is <M>H \cap H^g</M>,
so we compute that there are elements <M>s \in S</M> with the property
<M>|H \cap H^s| = 1</M>.
<P/>
(Of course this implies that also in the permutation representations of
the subgroups <M>S</M>, <M>S.2</M>, and <M>S.3</M> of <M>G</M> on the
cosets of the intersection with <M>H</M>,
the point stabilizers have regular orbits.)
<P/>
<Example><![
CDATA[
gap> repeat
> conj:= Random( s_3N );
> inter:= Intersection( h, h^conj );
> until Size( inter ) = 1;
]]></Example>
<P/>
Eventually &GAP; will return from this loop,
so there are elements <M>s</M> with the required property.
<P/>
(Computing one such intersection takes about six minutes
on a 2.5 GHz Pentium 4,
so one may have to be a bit patient.)
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=
"sect:appendix_permchar">
<Heading>Appendix: The Permutation Character <M>(1_H^G)_H</M></Heading>
<P/>
As an alternative to the computation of <M>|H \cap H^s|</M> for suitable
<M>s \in S</M>, we can try to derive information from the
permutation character <M>(1_H^G)_H</M>.
Unfortunately, there seems to be no easy way to prove the existence of
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.2</M>, and <M>M.3</M> exist.
For that, we compute <M>(1_H^G)_H</M>,
by computing class representatives of <M>H</M>,
their centralizer orders in <M>G</M>,
and the class fusion of <M>H</M>-classes in <M>G</M>.
<P/>
We want to compute the class representatives in a small permutation
representation of <M>H</M>;
this could be done using the degree <M>360</M> representation that was
implicitly constructed above,
<!-- (Each class representative can be written in the form
<M>h \alpha^i</M> with <M>h \in M.2</M>.) -->
but it is technically easier to use a degree <M>405</M> representation
that is obtained from the degree <M>58\,968</M> representation
by the action of <M>H</M> on blocks in an orbit of length <M>22\,680</M>.
(One could get this also using the &GAP; function
<Ref Func=
"SmallerDegreePermutationRepresentation" BookName=
"ref"/>.)
<P/>
<Example><![
CDATA[
gap> orbs:= Orbits( h, MovedPoints( h ) );;
gap> List( orbs, Length );
[ 22680, 36288 ]
gap> orb:= orbs[1];;
gap> bl:= Blocks( h, orb );; Length( bl[1] );
2
gap> actbl:= Action( h, bl, OnSets );;
gap> bll:= Blocks( actbl, MovedPoints( actbl ) );; Length( bll );
405
gap> oneblock:= Union( bl{ bll[1] } );;
gap> orb:= SortedList( Orbit( h, oneblock, OnSets ) );;
gap> acthom:= ActionHomomorphism( h, orb, OnSets );;
gap> ccl:= ConjugacyClasses( Image( acthom ) );;
gap> reps:= List( ccl, x -> PreImagesRepresentative( acthom,
> Representative( x ) ) );;
]]></Example>
<P/>
Then we carry back class representatives to the degree <M>58\,968</M>
representation, and compute the class fusion and the centralizer orders
in <M>G</M>.
<P/>
<Example><![
CDATA[
gap> reps:= List( ccl, x -> PreImagesRepresentative( acthom,
> Representative( x ) ) );;
gap> fusion:= [];;
gap> centralizers:= [];;
gap> fusreps:= [];;
gap> for i in [ 1 .. Length( reps ) ] do
> found:= false;
> cen:= Size( Centralizer( g, reps[i] ) );
> for j in [ 1 .. Length( fusreps ) ] do
> if cen = centralizers[j] and
> IsConjugate( g, fusreps[j], reps[i] ) then
> fusion[i]:= j;
> found:= true;
> break;
> fi;
> od;
> if not found then
> Add( fusreps, reps[i] );
> Add( fusion, Length( fusreps ) );
> Add( centralizers, cen );
> fi;
> od;
]]></Example>
<P/>
Next we compute the permutation character values,
using the formula
<
Display Mode=
"M">
(1_H)^G(g) = (|C_G(g)| \sum_{h} |h^H|) /|H| ,
</
Display>
where the summation runs over class representatives <M>h \in H</M>
that are <M>G</M>-conjugate to <M>g</M>.
<P/>
<Example><![
CDATA[
gap> pi:= 0 * [ 1 .. Length( fusreps ) ];;
gap> for i in [ 1 .. Length( ccl ) ] do
> pi[ fusion[i] ]:= pi[ fusion[i] ] + centralizers[ fusion[i] ] *
> Size( ccl[i] );
> od;
gap> pi:= pi{ fusion } / Size( h );;
]]></Example>
<P/>
In order to write the permutation character w.r.t. the ordering of
classes in the &GAP; character table, we use the &GAP; function
<Ref Func=
"CompatibleConjugacyClasses" BookName=
"ref"/>.
<P/>
<Example><![
CDATA[
gap> tblh:= CharacterTable(
"O8+(2).S3" );;
gap> map:= CompatibleConjugacyClasses( Image( acthom ), ccl, tblh );;
gap> pi:= pi{ map };
[ 51162109375, 69375, 1259375, 69375, 568750, 1750, 4000, 375, 135,
975, 135, 625, 150, 650, 30, 72, 80, 72, 27, 27, 3, 7, 25, 30, 6,
12, 25, 484375, 1750, 375, 375, 30, 40, 15, 15, 15, 6, 6, 3, 3, 3,
157421875, 121875, 4875, 475, 75, 3875, 475, 13000, 1750, 300, 400,
30, 60, 15, 15, 15, 125, 10, 30, 4, 8, 6, 9, 7, 5, 6, 5 ]
]]></Example>
<!-- This would not work for <M>O^+_8(2)</M>, <M>O^+_8(2).2</M>,
<M>O^+_8(2).3</M>;
in these cases, we would have to consider invariants:
element orders, centralizer orders,
centralizer orders of second and third power,
containment in the derived subgroup. -->
<P/>
Now we consider the restrictions of this permutation character to
<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:= CharacterTable(
"O8+(2).2" );;
gap> tblm3:= CharacterTable(
"O8+(2).3" );;
gap> tblm:= CharacterTable(
"O8+(2)" );;
gap> pi_m2:= pi{ GetFusionMap( tblm2, tblh ) };;
gap> pi_m3:= pi{ GetFusionMap( tblm3, tblh ) };;
gap> pi_m:= pi_m3{ GetFusionMap( tblm, tblm3 ) };;
]]></Example>
<P/>
The permutation character <M>(1_M^S)_M</M> decomposes
into <M>483</M> transitive permutation characters,
and regular <M>M</M>-orbits on <M>S/M</M> correspond to regular constituents
in this decomposition.
If there is no regular transitive constituent in <M>(1_M^S)_M</M> then
the largest degree of a transitive constituent is <M>|M|/2</M>;
but then the degree of <M>1_M^S</M> is less than <M>483 |M|/2</M>,
which is smaller than <M>[S:M]</M>.
<P/>
<Example><![
CDATA[
gap> n:= ScalarProduct( tblm, pi_m, TrivialCharacter( tblm ) );
483
gap> n * Size( tblm ) / 2;
42065049600
gap> pi[1];
51162109375
]]></Example>
<P/>
For the case of <M>M.2 < S.2</M>, this argument turns out to be not
sufficient.
So we first compute a lower bound on the number of regular <M>M</M>-orbits
on <M>S/M</M>.
For involutions <M>g \in M</M>, the number of transitive constituents
<M>1_{\langle g \rangle}^M</M> in <M>(1_M^S)_M</M> is at most the integral
part of
<M>1_M^S(g) / 1_{\langle g \rangle}^M(g) = 2 \cdot 1_M^S(g) / |C_M(g)|</M>;
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 <M>M</M>.)
However, the number of these constituents is at most <M>134</M>.
<P/>
<Example><![
CDATA[
gap> inv:= Filtered( [ 1 .. NrConjugacyClasses( tblm2 ) ],
> i -> OrdersClassRepresentatives( tblm2 )[i] = 2 and
> not i in ClassPositionsOfDerivedSubgroup( tblm2 ) );
[ 41, 42 ]
gap> n2:= List( inv,
> i -> Int( 2 * pi_m2[i] / SizesCentralizers( tblm2 )[i] ) );
[ 108, 26 ]
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 ]
gap> n2:= List( inv,
> i -> Int( 2 * pi_m3[i] / SizesCentralizers( tblm3 )[i] ) );
[ 0, 54, 15 ]
gap> Sum( n2 );
69
gap> 69 * Size( tblm3 ) / 2 + ( n - 69 - 1 ) * Size( tblm3 ) / 3 + 1;
41542502401
gap> pi[1];
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
<P/>
<
URL>
http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/data/o8p2s3_o8p5s3.g</
URL>
<P/>
contains the relevant data used in the above computations.
This covers the representations for the groups
and the permutation character of <M>O^+_8(2).S_3</M>
computed in Section <Ref Subsect=
"sect:appendix_permchar"/>.
<P/>
Reading the file into &GAP; will define a global variable
<C>o8p2s3_o8p5s3_data</C>, a record with the following components.
<P/>
<List>
<Mark><C>pi</C></Mark>
<Item>
the list of values of the permutation character of <M>G = O^+_8(5).S_3</M>
on the cosets of its subgroup <M>H = O^+_8(2).S_3</M>,
restricted to <M>H</M>,
corresponding to the ordering of classes in the character table of <M>H</M>
in the &GAP; Character Table Library
(this table has the
<Ref Func=
"Identifier" BookName=
"ref"/>
value <C>
"O8+(2).3.2"</C>),
</Item>
<Mark><C>dim8Q</C></Mark>
<Item>
a record with generators for <M>2.M</M> and <M>2.M.2</M>,
matrices of dimension eight over the Rationals,
</Item>
<Mark><C>deg120</C></Mark>
<Item>
a record with generators for <M>M</M> and <M>M.2</M>,
permutations of degree <M>120</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>\cong</M></Item>
<Item><M>\langle a_1, a_2, t \rangle</M>,</Item>
</Row>
<Row>
<Item><M>H</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, t, b \rangle</M>,</Item>
</Row>
<Row>
<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>,</Item>
</Row>
<Row>
<Item><M>S.3</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, c, t \rangle</M>,</Item>
</Row>
<Row>
<Item><M>G</M></Item>
<Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, c, t, b \rangle</M>,</Item>
</Row>
</Table>
where <M>a_1, a_2, b, t, c</M> are the values of the record components
<C>a1</C>, <C>a2</C>, <C>b</C>, <C>t</C>, and <C>c</C>.
<P/>
Analogously,
for the matrix representations, we have (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>
<Item><M>2.M.2</M></Item><Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, b \rangle</M>,</Item>
</Row>
<Row>
<Item><M>2.S</M></Item><Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, c \rangle</M>,</Item>
</Row>
<Row>
<Item><M>2.S.2</M></Item><Item><M>\cong</M></Item>
<Item><M>\langle a_1, a_2, c, b \rangle</M>,</Item>
</Row>
</Table>
<P/>
Additional components are used for deriving the representations from
initial data, as in the constructions in the previous sections.
<P/>
For example, most of the permutations needed arise as the induced actions
of matrices on orbits of vectors;
these orbits are computed when the file is read,
and are then stored in the components <C>orb120</C> and <C>orb19656</C>.
<P/>
The file <F>o8p2s3_o8p5s3.g</F> does not contain the generators
explicitly,
but it is self-contained in the sense that only a few &GAP; functions
are actually needed to produce the data;
for example, it should not be difficult to translate the contents of
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>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
