# We implement a computer model of the generalized hexagon H(4)^D with point set # {1,2,…,1365}, line set lines and automorphism group G. v:=1365; Points:=[1..v]; G:=AllPrimitiveGroups(DegreeOperation,v)[4]; Stab:=Stabilizer(G,1); Orbs:=Orbits(Stab,[1..v]); Dist0:=[1]; Dist1:=Filtered(Orbs,x->Size(x)=20)[1]; Dist2:=Filtered(Orbs,x->Size(x)=320)[1]; Dist3:=Filtered(Orbs,x->Size(x)=1024)[1]; Perp1:=Union(Dist0,Dist1); Perp2:=OnSets(Perp1,RepresentativeAction(G,1,Dist1[1])); line:=Intersection(Perp1,Perp2); lines:=Orbit(G,line,OnSets); # If x is a point of H(4)^D, then Perp(x) denotes the set of points at distance # 1 from x. Perp:=function(x) return OnSets(Perp1,RepresentativeAction(G,1,x)); end; # We now find a subhexagon of order (4,1) of H(4)^D. We denote this subhexagon # by hex. a:=1; b:=Dist3[1]; c1:=Intersection(Perp(b),Dist2)[1]; c2:=Intersection(Perp(b),Dist2)[2]; Complete:=function(X) local Y,i; Y:=ShallowCopy(X); for i in [1..1365] do if Size(Intersection(Perp(i),Y))>1 then AddSet(Y,i); fi; od; return Y; end; hex:=Set([a,b,c1,c2]); while Size(hex)<>105 do hex:=Complete(hex); od; # We denote by H1 the setwise stabilizer of hex and by H2 the elementwise # stabilizer of hex. # We verify that |H1|=241920, |H2|=3 and that H2 is the center of the derived # subgroup of H1. If these claims hold, then Check1 will be true. Otherwise, # Check1 will be false. H1:=Stabilizer(G,hex,OnSets); H2:=Stabilizer(G,hex,OnTuples); H:=H1; Check1 := Size(H1)=241920 and Size(H2)=3; Check1 := Check1 and H2=Center(DerivedSubgroup(H1)); # We determine the conjugacy class of G that consists of the 4095 central # involutions (Type 2A using Atlas notation) and the conjugacy class of H # that consists of 315 involutions. 2AG:=Set(Filtered(ConjugacyClasses(G),x->Size(x)=4095)[1]); 2AH:=Set(Filtered(ConjugacyClasses(H),x->Size(x)=315)[1]); # We verify the various claims made in Theorem 2.4 of the paper. If these are # true, then Check2 till Check 9 will be true. Check2:= 2AH = Intersection(2AG,H); centrG:=Centralizer(G,2AH[1]); suborbsG:=OrbitsDomain(centrG,2AG); Sub1AG:=Filtered(suborbsG,x->Size(x)=2)[1]; Sub1BG:=Filtered(suborbsG,x->Size(x)=20)[1]; Sub2AG:=Filtered(suborbsG,x->Size(x)=40)[1]; Sub2BG:=Filtered(suborbsG,x->Size(x)=320)[1]; Sub3AG:=Filtered(suborbsG,x->Size(x)=640)[1]; Sub3BG:=Filtered(suborbsG,x->Size(x)=1024)[1]; Sub4G:=Filtered(suborbsG,x->Size(x)=2048)[1]; centrH:=Centralizer(H,2AH[1]); suborbsH:=OrbitsDomain(centrH,2AH); Sub1AH:=Filtered(suborbsH,x->Size(x)=2)[1]; Sub1BH:=Filtered(suborbsH,x->Size(x)=8)[1]; Sub2AH:=Filtered(suborbsH,x->Size(x)=16)[1]; Sub2BH:=Filtered(suborbsH,x->Size(x)=32)[1]; Sub3AH:=Filtered(suborbsH,x->Size(x)=64 and Order(2AH[1]*x[1])=4)[1]; Sub3BH:=Filtered(suborbsH,x->Size(x)=64 and Order(2AH[1]*x[1])=3)[1]; Sub4H:=Filtered(suborbsH,x->Size(x)=128)[1]; Check3 := IsSubset(Sub1AG,Sub1AH); Check4 := IsSubset(Sub1BG,Sub1BH); Check5 := IsSubset(Sub2AG,Sub2AH); Check6 := IsSubset(Sub2BG,Sub2BH); Check7 := Sub3AH<>[] and IsSubset(Sub3AG,Sub3AH); Check8 := Sub3AH<>[] and IsSubset(Sub3BG,Sub3BH); Check9 := IsSubset(Sub4G,Sub4H); # Additional verifications # We show that the group H1 is of type SL(3,4):2^2. With the command # StructureDescription, we can verify that the derived subgroup H3 of H1 # has type SL(3,4) # H3:=DerivedSubgroup(H1); # We construct a number of involutions of H1. # InvClasses := function(U) # return Filtered(List(ConjugacyClasses(U),Representative), x->Order(x)=2); # end; # invo:=InvClasses(H1); # x1:=Filtered(invo,x->Size(ConjugacyClass(H1,x))=315)[1]; # x2:=Filtered(invo,x->Size(ConjugacyClass(H1,x))=280)[1]; # x3:=Filtered(invo,x->Size(ConjugacyClass(H1,x))=360)[1]; # x4:=Filtered(invo,x->Size(ConjugacyClass(H1,x))=1008)[1]; # We construct a group U of type 2^2 # int:=Intersection(Centralizer(H1,x2),ConjugacyClass(H1,x3)); # U:=Group([x2,int[1]]); # We verify that U and H3 intersect trivially. If this is the case, # then Check10 will be true. # Check10 := Intersection(U,H3)=Group(());