[GAP Forum] Orbits

[Max Horn]

Dear Thekiso,

On 08.01.2014, at 13:06, "Thekiso Seretlo" <Thekiso.Seretlo at nwu.ac.za> wrote:

> Dear Forum Collegues
> It is still new year and my mind is still at rest but I am desperately trying to get the orbitlengths bwhen $M_11$ acts on the set of conjugates of $M_22$.

you told me in private communication that you are interested in these groups as subgroups of HS. Note that HS has two classes of maximal subgroups isomorphic to M11. 

If I understood your question correctly, one way to answer it is with the following GAP code:

gap> LoadPackage("atlasrep");
true
gap> g:= AtlasGroup( "HS" );
<permutation group of size 44352000 with 2 generators>

# Get the relevant maximal subgroups (ordered as in the atlas)
gap> m22:=AtlasSubgroup(g, 1);
<permutation group of size 443520 with 2 generators>
gap> m11a:=AtlasSubgroup(g, 8);
<permutation group of size 7920 with 2 generators>
gap> m11b:=AtlasSubgroup(g, 9);
<permutation group of size 7920 with 2 generators>

# Let's verify their isomorphism type
gap> StructureDescription(m22);
"M22"
gap> StructureDescription(m11a);
"M11"
gap> StructureDescription(m11b);
"M11"

# Finally, compute the orbit lengths of the two M11 acting on the conjugacy class of M22
gap> cc:=ConjugacyClassSubgroups(g, m22);;
gap> OrbitLengthsDomain(m11a, cc);
[ 66, 12, 22 ]
gap> OrbitLengthsDomain(m11b, cc);
[ 22, 12, 66 ]


Hope that helps,
Max