# Demonstration of permutation groups for GAC 2010 in Allahabad a := (1,3)(4,6); b := (1,2,4,5)(3,6); c := a*b; g := Group(a,b); MovedPoints(g); IsTransitive(g); Size(g); IsPrimitive(g); blocks := Blocks(g,[1..6]); hom := ActionHomomorphism(g,blocks,OnSets); h := Image(hom); Size(h); k := Kernel(hom); Size(k); n := NormalSubgroups(g); List(n,Size); syl3 := SylowSubgroup(g,3); Size(syl3); norm := Normalizer(g,syl3); norm = g; Size(norm); cent := Centralizer(g,syl3); Size(cent); sym6 := SymmetricGroup(6); x := PseudoRandom(sym6); x in g; y := PseudoRandom(g); y in g; m := MaximalSubgroups(g); List(m,Size); List(m,IdGroup); fi23 := AtlasGroup("Fi23"); LargestMovedPoint(fi23); Size(fi23); g := Group(GeneratorsOfGroup(fi23)); Size(g); time; syl5 := SylowSubgroup(g,5); time; Size(syl5); norm := Normalizer(g,syl5); time; Size(norm);