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gap> pg := ProjectiveSpace(2,4);
ProjectiveSpace(2, 4)
gap> points := Points(pg);
<points of ProjectiveSpace(2, 4)>
gap> pointslist := AsList(points);
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)> ]
gap> Display(pointslist[1]);
[1..]
gap> frame := [[1,0,0],[0,1,0],[0,0,1],[1,1,1]]*Z(2)^0;
[ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0, Z(2)^0 ] ]
gap> frame := List(frame,x -> VectorSpaceToElement(pg,x));
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ]
gap> frame := StandardFrame( pg );
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ]
gap> pairs := Combinations(frame,2);
[ [ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ],
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ],
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ],
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ],
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ],
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ] ]
gap> secants := List(pairs,p -> Span(p[1],p[2]));
[ <a line in ProjectiveSpace(2, 4)>, <a line in ProjectiveSpace(2, 4)>,
<a line in ProjectiveSpace(2, 4)>, <a line in ProjectiveSpace(2, 4)>,
<a line in ProjectiveSpace(2, 4)>, <a line in ProjectiveSpace(2, 4)> ]
gap> leftover := Filtered(pointslist,t->not ForAny(secants,s->t in s));
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ]
gap> hyperoval := Union(frame,leftover);
[ <a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)>,
<a point in ProjectiveSpace(2, 4)>, <a point in ProjectiveSpace(2, 4)> ]
gap> g := CollineationGroup(pg);
The FinInG collineation group PGammaL(3,4)
gap> stab := Stabilizer(g,Set(hyperoval),OnSets);
<projective collineation group of size 720 with 3 generators>
gap> StructureDescription(stab);
"S6"
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