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gap> pg1 := ProjectiveSpace(2,81);
ProjectiveSpace(2, 81)
gap> f2 := GF(9);
GF(3^2)
gap> em := NaturalEmbeddingByFieldReduction(pg1,f2);
<geometry morphism from <All elements of ProjectiveSpace(2,
81)> to <All elements of ProjectiveSpace(5, 9)>>
gap> f2 := GF(3);
GF(3)
gap> em := NaturalEmbeddingByFieldReduction(pg1,f2);
<geometry morphism from <All elements of ProjectiveSpace(2,
81)> to <All elements of ProjectiveSpace(11, 3)>>
gap> pg2 := ProjectiveSpace(11,3);
ProjectiveSpace(11, 3)
gap> em := NaturalEmbeddingByFieldReduction(pg1,pg2);
<geometry morphism from <All elements of ProjectiveSpace(2,
81)> to <All elements of ProjectiveSpace(11, 3)>>
gap> pg1 := PG(1,9);
ProjectiveSpace(1, 9)
gap> em := NaturalEmbeddingByFieldReduction(pg1,GF(3));
<geometry morphism from <All elements of ProjectiveSpace(1,
9)> to <All elements of ProjectiveSpace(3, 3)>>
gap> i := Intertwiner(em);
MappingByFunction( The FinInG projectivity group PGL(2,9), <projective colline
ation group of size 720 with
2 generators>, function( m ) ... end, function( m ) ... end )
gap> spread := List(Points(pg1),x->x^em);
[ <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)>,
  <a line in ProjectiveSpace(3, 3)>, <a line in ProjectiveSpace(3, 3)> ]
gap> stab := Stabilizer(CollineationGroup(PG(3,3)),Set(spread),OnSets);
<projective collineation group of size 5760 with 3 generators>
gap> hom := HomographyGroup(pg1);
The FinInG projectivity group PGL(2,9)
gap> gens := GeneratorsOfGroup(hom);;
gap> group := Group(List(gens,x->x^i));
<projective collineation group with 2 generators>
gap> Order(group);
2880
gap> IsSubgroup(stab,group);
true

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