x:=Indeterminate(GF(2)); SetName(x,"x"); f:=x^23-1; Factors(f); f:=First(Factors(f),i->Degree(i)>1); LoadPackage("guava"); cod:=GeneratorPolCode(f,23,GF(2)); IsPerfectCode(cod); ext:=ExtendedCode(cod); WeightDistribution(ext); autext:=AutomorphismGroup(ext); gens:=SmallGeneratingSet(autext);; m24:=Group(gens); SetName(m24,"m24"); Size(m24); Transitivity(m24,[1..24]); st:=Stabilizer(m24,[23,24],OnSets); gens:=SmallGeneratingSet(st);; st2:=Group(gens); m22a:=Action(st2,[1..22]); SetName(m22a,"m22a"); Size(m22a); s:=SylowSubgroup(m22a,2);; pciso:=IsomorphismPcGroup(s); a:=Image(pciso); n:=Filtered(NormalSubgroups(a),i->Size(i)=16 and IsElementaryAbelian(i)); n:=List(n,i->PreImage(pciso,i));; e:=Filtered(n,i->IsRegular(i,MovedPoints(i)));;Length(e); e:=e[1];; h:=Normalizer(m22a,e);; ophom:=ActionHomomorphism(m22a,RightCosets(m22a,h),OnRight);; mop:=Image(ophom); DegreeAction(mop); dp:=DirectProduct(m22a,mop);; emb1:=Embedding(dp,1);; emb2:=Embedding(dp,2);; diag:=List(GeneratorsOfGroup(m22a), i->Image(emb1,i)*Image(emb2,Image(ophom,i)));; diag:=Group(diag,());; SetName(diag,"M22.2-99"); LoadPackage("grape"); gamma:=NullGraph(diag,100); AddEdgeOrbit(gamma,[1,100]); AddEdgeOrbit(gamma,[100,1]); hexad:=First(Orbits(h,[1..22]),i->Length(i)=6); for i in hexad do AddEdgeOrbit(gamma,[i,23]); AddEdgeOrbit(gamma,[23,i]); od; Adjacency(gamma,23); stab:=Stabilizer(diag,23);; orbs:=Orbits(stab,[24..99]);; orbreps:=List(orbs,i->i[1]); rep1:=orbreps[1]; Adjacency(gamma,rep1); Intersection(hexad,Adjacency(gamma,rep1)); rep2:=orbreps[2]; Adjacency(gamma,rep2); Intersection(hexad,Adjacency(gamma,rep2)); AddEdgeOrbit(gamma,[23,rep2]); AddEdgeOrbit(gamma,[rep2,23]); IsSimpleGraph(gamma); Adjacency(gamma,23); IsDistanceRegular(gamma); aug:=AutGroupGraph(gamma); Size(aug); small := SmallGeneratingSet(aug);; Length(small); aug2 := Group(small); DisplayCompositionSeries(aug2); hs:=DerivedSubgroup(aug2); LoadPackage("ctbllib"); ct:=CharacterTable("U3(5)"); ct2:=CharacterTable("U3(5).2"); cths:=CharacterTable("hs"); OrdersClassRepresentatives(ct); OrdersClassRepresentatives(ct2); ordhs:= OrdersClassRepresentatives(cths); pos:= Position(ordhs,12); SizesCentralizers(cths)[pos]; repeat e1:=Random(hs); until OrderPerm(e1)=12; e2:=e1^6;; SizesConjugacyClasses(ct2)[2]; PowerMap(cths,3)[PowerMap(cths,2)[21]]; SizesConjugacyClasses(cths)[2]; count:=0;; repeat u:=Subgroup(hs,[e1,e2^Random(hs)]); count:=count+1; until Index(hs,u)=176; count; ophom:=ActionHomomorphism(hs,RightTransversal(hs,u),OnRight);; hsop:=Image(ophom); IsPrimitive(hsop,[1..176]); dp:=DirectProduct(hs,hsop);; emb1:=Embedding(dp,1);; emb2:=Embedding(dp,2);; diag:=List(GeneratorsOfGroup(hs), i->Image(emb1,i)*Image(emb2,Image(ophom,i)));; diag:=Group(diag,());;SetName(diag,"hs-276"); Size(diag); adj:=Adjacency(gamma,5); t:=Stabilizer(diag,[5,adj[5]],OnSets);; sg := SmallGeneratingSet(t);; t := Subgroup(diag, sg); DisplayCompositionSeries(t); aus:=AutomorphismGroup(t);; Size(aus); inner:=InnerAutomorphismsAutomorphismGroup(aus); Index(aus,inner); rt:=RightTransversal(aus,inner);; automs:=Filtered(rt,i->i^2 in inner and not i in inner);; Length(automs); List(automs,Order); List(automs,IsConjugatorAutomorphism); i := Position( List(automs,IsConjugatorAutomorphism), true ); element:=ConjugatorInnerAutomorphism(automs[i]);; if SignPerm( element ) = -1 then ot:=Filtered( Orbits(t,[1..276]), orb -> Length(orb)=2)[1]; element:=element*(ot[1],ot[2]); fi; co3:=Group(Concatenation(GeneratorsOfGroup(diag),[element]),()); Size(co3); DisplayCompositionSeries(co3); quit;