symplectic groups in GAP

Question

For the symplectic groups in GAP : grp:=SymplecticGroup(n,q), what is the underlying metric or bilinear form ($M$)? For example if I use A:=Random(grp) to get a random symplectic matrix this matrix should satisfy $A^\dagger A=1$ where $A^\dagger=M A^t M^{-1}$; what is $M$? I tried $[[0,-I],[I,0]]$ but that didn't work.

Answer 1

As explained in the manual (https://www.gap-system.org/Manuals/doc/ref/chap50.html#X8674AAA578FE4AEE), the form is stored in the InvariantBilinearForm attribute (as entry .matrix):

gap> grp:=SP(4,5);;
gap> M:=InvariantBilinearForm(grp).matrix;;
gap> Display(M);
 . . . 1
 . . 1 .
 . 4 . .
 4 . . .
gap> List(GeneratorsOfGroup(grp),x->TransposedMat(x)*M*x=M);
[ true, true ]