[GAP Forum] Maximal Subgroups for O(7,3)

Jack Schmidt jack at ms.uky.edu
Thu Nov 13 07:08:48 GMT 2008

You can use:

gap> g:=AtlasGroup("O7(3)",NrMovedPoints,1080);;
gap> a:=AutomorphismGroup(g);;
Time of last command: 16073 ms
gap> h:=Stabilizer(g,1);;
gap> f:=First(GeneratorsOfGroup(a),f->NrMovedPoints(Image(f,h))=1080);;
gap> k:=Image(f,h);;
gap> chi1:=PermutationCharacter(g,h);;
Time of last command: 3636 ms
gap> chi2:=PermutationCharacter(g,k);;
Time of last command: 12641 ms
gap> chi1=chi2;

The paper atlas mentions that these two maximals fuse in the  
automorphism group, so you know something like this should work.  The  
two conjugacy classes should occur equally often if you are sampling  
randomly, so your method should have worked, but this is a simple  
direct method.

On 2008-11-13, at 00:37, Joe Bohanon wrote:

> Sorry to those of you who get this twice.  I accidentally sent it to  
> the group pub forum first.
> I'm trying to get the maximal subgroups for O(7,3) and having some  
> trouble.  ATLAS 3.0 does not have them listed, but ATLAS 2.0 does  
> have the shape and there are 7 permutation representations that can  
> be called up by atlasrep.  For each of those seven, I did the  
> following with G set as the smallest permrep
> H:=Group(AtlasGenerators("O7(3)",i).generators);
> iso:=IsomorphismGroups(H,G);
> S:=Stabilizer(H,1);
> Then I simply ran Image(iso,S) to get the maximals corresponding to  
> the primitive permreps.  However for the two classes of G2(3), this  
> yields conjugate maximal subgroups.
> In addition, I also tried to take random elements of order 2 and 3  
> and try to generate a G2(3), and while I was able to create many of  
> them, none of them were out of this one conjugacy class.
> Am I missing something here?  I don't think there is a mistake  
> anywhere, as G2(3) is listed as having two classes in Kleidman's  
> tables.
> Thanks
> Joe
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