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gap> w := SymplecticSpace(3,5);
W(3, 5)
gap> lines:=AsList(Lines(w));;
gap> duality := NaturalDuality(w);
<geometry morphism from <Elements of W(3, 5)> to <Elements of Q(4, 5)>>
gap> l:=lines[1];
<a line in W(3, 5)>
gap> l^duality;
<a point in Q(4, 5)>
gap> PreImageElm(duality,last);
<a line in W(3, 5)>
gap> hom := Intertwiner(duality);
MappingByFunction( PGammaSp(4,5), <projective collineation group of size
9360000 with 4 generators>, function( g ) ... end, function( g ) ... end )
gap> q := 5;
5
gap> q5q := EllipticQuadric(5,q);
Q-(5, 5)
gap> mat := [[0,1,0,0],[1,0,0,0],[0,0,0,Z(q)],[0,0,Z(q),0]]*Z(q)^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*Z(5), 0*Z(5), Z(5), 0*Z(5) ] ]
gap> hform := HermitianFormByMatrix(mat,GF(q^2));
< hermitian form >
gap> herm := PolarSpace(hform);
<polar space in ProjectiveSpace(
3,GF(5^2)): x1^5*x2+x1*x2^5+Z(5)*x3^5*x4+Z(5)*x3*x4^5=0 >
gap> duality := NaturalDuality(q5q,herm,true);
<geometry morphism from <Elements of Q-(5, 5)> to <Elements of H(3,
5^2): x1^5*x2+x1*x2^5+Z(5)*x3^5*x4+Z(5)*x3*x4^5=0>>
gap> hom := Intertwiner(duality);
MappingByFunction( PDeltaO-(6,5), <projective collineation group of size
58968000000 with 3 generators>, function( g ) ... end, function( g ) ... end )
gap> g := Random(CollineationGroup(q5q));
< a collineation: <cmat 6x6 over GF(5,1)>, F^0>
gap> g^hom;
< a collineation: <cmat 4x4 over GF(5,2)>, F^5>
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