[GAP Forum] Orbit calculation in GUAVA

[Alexander Hulpke]

Dear Forum, Dear Siddhartha Sarkar,

> I have just started with coding and trying to experiment with GUAVA.
> 
> I am trying to reproduce the following calculation briefed in Robert
> Wilson's book
> "The Finite Simple Groups" Page 184 on hexacodes.
> 
> Hexacode W, as defined, is a GF(4)-subspace of GF(4)^6 generated by
> the following
> vectors : (w, w^2, w^2, w, w^2, w), (w^2, w, w, w^2, w^2, w), (w^2, w,
> w^2, w, w, w^2)
> 
> where GF(4) = { 0, 1, w, w^2 }
> 
> 3 X S_4 seems to be a symmetry group : 3 by multiplying scalers from
> GF(4)* and S_4
> by the permutation subgroup of S_6 generated by (1,2)(3,4),
> (1,3,5)(2,4,6), (1,3)(2,4)
> 
> I wish to reproduce the orbit calculation of 3 X S_4 to W.

As the action of 3xS4 is linear, the easiest is to represent the group by matrices over GF(4):

gap> f:=GF(4);
GF(2^2)
gap> w:=PrimitiveElement(f); # It does not matter whether this is w or w^2. For other fields one might have to be
                                              # more careful in chosing
Z(2^2)

gap> S4:=Group((1,2)(3,4),(1,3,5)(2,4,6), (1,3)(2,4)); # the S4 you gave
Group([ (1,2)(3,4), (1,3,5)(2,4,6), (1,3)(2,4) ])

Form permutation matrices in dimension 6 over f
gap> permmats:=List(GeneratorsOfGroup(S4),
> x->PermutationMat(x,6,f));
[ [ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], ...

# for the group we need to add the scalar matrices generated by multiplication with the primitive element
gap> G:=Group(Concatenation(permmats,[w*IdentityMat(6,f)]));
<matrix group with 4 generators>

# the seed vectors and their space
gap> vecs:=[[w, w^2, w^2, w, w^2, w],[w^2, w, w, w^2, w^2, w],        
> [w^2, w,w^2, w, w, w^2]];
[ [ Z(2^2), Z(2^2)^2, Z(2^2)^2, Z(2^2), Z(2^2)^2, Z(2^2) ], ...

gap> W:=VectorSpace(f,vecs);  
<vector space over GF(2^2), with 3 generators>
gap> Size(W);
64

Now we can form orbits:

gap> orbs:=Orbits(G,Elements(W));;
gap> List(orbs,Length);
[ 1, 9, 36, 6, 12 ]

Hope this helps,

   Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke