[GAP Forum] Listing the class representatives of PGL(2, Z/pZ) and PSL(2, Z/pZ)

[Alexander Hulpke]

Dear Forum,

On Oct 1, 2012, at 10/1/12 10:31, Kashyap Rajeevsarathy <kashyap at iiserb.ac.in> wrote:

> Dear Forum,
> 
> Is there a way to make GAP list the elements the elements of the class
> representatives of class representatives of PGL(2,Z/pZ) and PGL(2,Z/pZ) as
> 2x2 matrices?

PGL as a permutation group is obtained from GL by acting on 1-dimensional spaces, a corresponding homomorphism is stored in PGL as 
G!.actionHomomorphism

Thus for example:
gap> g:=PGL(3,3);
Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ])
gap> hom:=g!.actionHomomorphism;
<action epimorphism>
gap> reps:=List(ConjugacyClasses(g),Representative);;
gap> List(reps,x->PreImagesRepresentative(hom,x));

will give you a list of matrices representing the classes of PGL.

Best,

   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
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