gap> mat := [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]*Z(5)^0;
[ [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ], [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5) ] ]
gap> phi := HermitianPolarityOfProjectiveSpace(mat,GF(25));
<polarity of PG(3, GF(5^2)), >
gap> mat2 := IdentityMat(4,GF(5));
[ [ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ] ]
gap> psi := PolarityOfProjectiveSpace(mat2,GF(25));
<polarity of PG(3, GF(5^2)), >
gap> phi*psi = psi*phi;
true
gap> g := CorrelationGroup(PG(3,25));
<projective group with Frobenius with proj. space isomorphism of size
3719082276000000000000 with 4 generators>
gap> h := CollineationGroup(PG(3,25));
PGammaL(4,25)
gap> hom := Embedding(h,g);
MappingByFunction( PGammaL(4,25), <projective group with Frobenius with proj.
space isomorphism of size 3719082276000000000000 with
4 generators>, function( y ) ... end )
gap> coll := PreImagesRepresentative(hom,phi*psi);
<projective semilinear element: [ [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ],
[ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ],
[ 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5) ] ], F^5>