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gap> q := 4;
4
gap> conic := ParabolicQuadric(2,q);
Q(2, 4)
gap> nucleus := NucleusOfParabolicQuadric(conic);
<a point in ProjectiveSpace(2, 4)>
gap> hyperoval := Union(List(Points(conic)),[nucleus]);
[ <a point in ProjectiveSpace(2, 4)>, <a point in Q(2, 4)>,
<a point in Q(2, 4)>, <a point in Q(2, 4)>, <a point in Q(2, 4)>,
<a point in Q(2, 4)> ]
gap> pg := PG(3,q);
ProjectiveSpace(3, 4)
gap> hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);
<a plane in ProjectiveSpace(3, 4)>
gap> em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);
<geometry morphism from <All elements of ProjectiveSpace(2,
4)> to <All elements of ProjectiveSpace(3, 4)>>
gap> O := List(hyperoval,x->x^em);
[ <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
<a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
<a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)> ]
gap> points := Set(Filtered(Points(pg),x->not x in hyp));;
gap> lines := Union(List(O,x->Filtered(Lines(x),y->not y in hyp)));;
gap> inc := \*;
<Operation "*">
gap> gp := GeneralisedPolygonByElements(points,lines,inc);
<generalised quadrangle of order [ 3, 5 ]>
gap> coll := CollineationGroup(gp);
<permutation group of size 138240 with 8 generators>
gap> act := CollineationAction(coll);
function( el, g ) ... end
gap> g := Random(coll);
(1,37,45,63,27,19)(2,53,13,64,11,51)(3,33,38,61,31,28)(4,49,6,62,15,60)(5,46,
47,59,20,17)(7,42,40,57,24,26)(8,58)(9,55)(10,39,41,56,25,23)(12,35,34,54,29,
32)(14,48,43,52,18,21)(16,44,36,50,22,30)(65,132,90,157,89,105)(66,68,131,143,
119,103)(67,135,76,123,130,106)(69,133,112,100,81,107)(70,134,150,88,155,
104)(71,99,79,144,93,149)(72,153,95,120,73,122)(74,125,115,128,140,87)(75,121,
136,117,113,91)(77,124,98,83,147,146)(78,145,84,118,85,142)(80,92,137,141,108,
97)(82,86,116,111,138,101)(94,127,126,102,109,96)(110,152,151,154,156,
129)(114,160,139,158,148,159)
gap> l := Random(Lines(gp));
<a line in <generalised quadrangle of order [ 3, 5 ]>>
gap> act(l,g);
<a line in <generalised quadrangle of order [ 3, 5 ]>>
gap> p := Random(Points(gp));
<a point in <generalised quadrangle of order [ 3, 5 ]>>
gap> act(p,g);
<a point in <generalised quadrangle of order [ 3, 5 ]>>
gap> stab := Stabilizer(coll,p,act);
<permutation group of size 2160 with 3 generators>
gap> List(Orbits(stab,List(Points(gp)),act),x->Length(x));
[ 45, 18, 1 ]
gap> List(Orbits(stab,List(Lines(gp)),act),x->Length(x));
[ 90, 6 ]
[ Dauer der Verarbeitung: 0.18 Sekunden
(vorverarbeitet)
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