gap> Read("fundamental_v2.g"); ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.6.1 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~leonard/). Homepage: http://www.maths.qmul.ac.uk/~leonard/grape/ ───────────────────────────────────────────────────────────────────────────── gap> autmcl:=OnePrimitiveGroup(DegreeOperation,275,Size,898128000*2); McL:2 gap> gamma:=EdgeOrbitsGraph(autmcl,[1,2]);; gap> if VertexDegrees(gamma)=[112] then > mclgraph:=gamma; > else > mclgraph:=ComplementGraph(gamma); > fi; gap> GlobalParameters(mclgraph); [ [ 0, 0, 112 ], [ 1, 30, 81 ], [ 56, 56, 0 ] ] gap> sigma:=DistanceSetInduced(mclgraph,2,1);; gap> OrderGraph(sigma); 162 gap> GlobalParameters(sigma); [ [ 0, 0, 56 ], [ 1, 10, 45 ], [ 24, 32, 0 ] ] gap> delta:=DistanceSetInduced(sigma,2,1);; gap> Unbind(delta.names); gap> OrderGraph(delta); 105 gap> GlobalParameters(delta); [ [ 0, 0, 32 ], [ 1, 4, 27 ], [ 12, 20, 0 ] ] gap> aut:=AutomorphismGroup(delta);; gap> Size(aut); 241920 gap> DisplayCompositionSeries(aut); G (6 gens, size 241920) | Z(2) S (6 gens, size 120960) | Z(2) S (3 gens, size 60480) | Z(3) S (2 gens, size 20160) | A(2,4) = L(3,4) 1 (0 gens, size 1) gap> delta:=NewGroupGraph(aut,delta);; gap> F:=FundamentalRecordSimplicialComplex(delta,"abelianized");; gap> T:=F.spanningTree;; gap> Collected(AbelianInvariants(F.group)); [ [ 2, 16 ], [ 3, 2 ] ] gap> # gap> # Now we investigate the connected double covers of delta. gap> # gap> r:=2; 2 gap> F:=FundamentalRecordSimplicialComplex(delta,"abelianized",r,T);; gap> Size(F.group); 65536 gap> M:=MaximalSubgroups(F.group);; gap> Length(M); 65535 gap> fibres:=List([1..delta.order],x->List([1..r],y->r*(x-1)+y));; gap> C:=[]; [ ] gap> Cf:=[]; [ ] gap> for H in M do > cov:=CoveringGraph(delta,F.group,F.edgeLabels,H); > covf:=GraphWithUnorderedVertexPartition(cov,fibres); > found:=false; > for cf in Cf do > if IsIsomorphicGraph(covf,cf,false) then > found:=true; > break; > fi; > od; > if not found then > # The r-cover covf is not isomorphic (as an r-cover of delta) > # to those found so far. > Add(C,cov); > Add(Cf,covf); > fi; > od; gap> Length(C); 13 gap> Collected(List(C,IsDistanceRegular)); [ [ false, 13 ] ] gap> Collected(List(C,x->[OrderGraph(x),GlobalParameters(x)])); [ [ [ 210, [ [ 0, 0, 32 ], [ 1, 4, 27 ], [ -1, -1, -1 ], [ -1, -1, -1 ], [ 32, 0, 0 ] ] ], 13 ] ] gap> F:=List(C,x->FundamentalRecordSimplicialComplex(x,"abelianized"));; gap> Collected(List(F,x->Collected(AbelianInvariants(x.group)))); [ [ [ [ 2, 15 ], [ 3, 2 ] ], 13 ] ] gap> Runtime(); # total runtime so far in milliseconds 10086490 gap> # gap> # Now we investigate the connected triple covers of delta. gap> # gap> r:=3; 3 gap> F:=FundamentalRecordSimplicialComplex(delta,"abelianized",r,T);; gap> Size(F.group); 9 gap> M:=MaximalSubgroups(F.group);; gap> Length(M); 4 gap> fibres:=List([1..delta.order],x->List([1..r],y->r*(x-1)+y));; gap> C:=[]; [ ] gap> Cf:=[]; [ ] gap> for H in M do > cov:=CoveringGraph(delta,F.group,F.edgeLabels,H); > covf:=GraphWithUnorderedVertexPartition(cov,fibres); > Add(C,cov); > Add(Cf,covf); > od; gap> Length(C); 4 gap> List(C,IsDistanceRegular); [ true, true, true, false ] gap> List(C,x->[OrderGraph(x),GlobalParameters(x)]); [ [ 315, [ [ 0, 0, 32 ], [ 1, 4, 27 ], [ 4, 20, 8 ], [ 27, 4, 1 ], [ 32, 0, 0 ] ] ], [ 315, [ [ 0, 0, 32 ], [ 1, 4, 27 ], [ 4, 20, 8 ], [ 27, 4, 1 ], [ 32, 0, 0 ] ] ], [ 315, [ [ 0, 0, 32 ], [ 1, 4, 27 ], [ 4, 20, 8 ], [ 27, 4, 1 ], [ 32, 0, 0 ] ] ], [ 315, [ [ 0, 0, 32 ], [ 1, 4, 27 ], [ -1, -1, -1 ], [ -1, -1, -1 ], [ 32, 0, 0 ] ] ] ] gap> F:=List(C,x->FundamentalRecordSimplicialComplex(x,"abelianized"));; gap> List(F,x->Collected(AbelianInvariants(x.group))); [ [ [ 2, 18 ], [ 3, 2 ] ], [ [ 2, 18 ], [ 3, 2 ] ], [ [ 2, 18 ], [ 3, 2 ] ], [ [ 2, 16 ], [ 3, 2 ] ] ] gap> # gap> # Now determine the number of pairwise nonisomorphic (as 3-covers of delta) gap> # elements of Cf. gap> # gap> Length(GraphIsomorphismClassRepresentatives(Cf)); 2 gap> Runtime(); # total runtime so far in milliseconds 10111761 gap> # gap> # Now we look at the connected double covers of the distance-regular gap> # triple covers of delta. gap> # gap> s:=2; 2 gap> idgrpA4:=IdGroup(AlternatingGroup(4)); [ 12, 3 ] gap> forest:=CoverOfSpanningTree(T,r);; gap> fibres:=List([1..delta.order],x->List([1..4],y->4*(x-1)+y));; gap> CA4:=[]; [ ] gap> CfA4:=[]; [ ] gap> for cov in C do > if IsDistanceRegular(cov) then > F:=FundamentalRecordSimplicialComplex(cov,"abelianized",s,forest); > Print("\n",Collected(AbelianInvariants(F.group))); > L:=[]; > M:=MaximalSubgroups(F.group); > for H in M do > Ccov:=CoveringGraph(cov,F.group,F.edgeLabels,H); > labels:=EdgeLabelsFromCover(Ccov,r*s); > G:=Group(Set(labels),()); > idgrp:=IdGroup(G); > Add(L,[idgrp,IsDistanceRegular(Ccov)]); > if idgrp=idgrpA4 then > # G is isomorphic to A_4. > covA4deg4:=CoveringGraph(delta,G,labels,SylowSubgroup(G,3)); > Add(CA4,covA4deg4); > Add(CfA4,GraphWithUnorderedVertexPartition(covA4deg4,fibres)); > fi; > od; > Print("\n",Collected(L),"\nRuntime in milliseconds=",Runtime(),"\n"); > fi; > od; [ [ 2, 18 ] ] [ [ [ [ 6, 2 ], false ], 65535 ], [ [ [ 12, 3 ], false ], 3 ], [ [ [ 24, 13 ], false ], 196605 ] ] Runtime in milliseconds=29028605 [ [ 2, 18 ] ] [ [ [ [ 6, 2 ], false ], 65535 ], [ [ [ 12, 3 ], false ], 3 ], [ [ [ 24, 13 ], false ], 196605 ] ] Runtime in milliseconds=48086661 [ [ 2, 18 ] ] [ [ [ [ 6, 2 ], false ], 65535 ], [ [ [ 12, 3 ], false ], 3 ], [ [ [ 24, 13 ], false ], 196605 ] ] Runtime in milliseconds=67751889 gap> # gap> # Now determine the number of pairwise nonisomorphic (as 4-covers of delta) gap> # elements of CfA4. gap> # gap> Length(GraphIsomorphismClassRepresentatives(CfA4)); 1 gap> Size(AutomorphismGroup(CfA4[1])); 80640 gap> aut:=AutomorphismGroup(CA4[1]);; gap> Size(aut); 80640 gap> A:=NewGroupGraph(aut,CA4[1]);; gap> GlobalParameters(A); [ [ 0, 0, 32 ], [ 1, 4, 27 ], [ -1, 20, -1 ], [ 27, 4, 1 ], [ 32, 0, 0 ] ] gap> quit;