# We describe a connection between the new near octagon of order (2,10) and # Soicher’s first graph. # We import from another file a permutation representation of the group 3*Suz:2 # on 5346 points. This permutation group is denoted by 3Suz2. Read("Desktop/SuzGroup.g"); # We show that with respect to a suitable suborbit, the group 3Suz2 defines a # distance-transitive graph on the set [1..5346]. This graph is moreover # distance-regular with parameters {416,315,64,1;1,32,315,416}. G := Stabilizer(3Suz2,1); Orbs := Orbits(G,[1..5346]); Dist0:=Set(Filtered(Orbs,x->Size(x)=1)[1]); Dist1:=Set(Filtered(Orbs,x->Size(x)=416)[1]); Dist2:=Set(Filtered(Orbs,x->Size(x)=4095)[1]); Dist3:=Set(Filtered(Orbs,x->Size(x)=832)[1]); Dist4:=Set(Filtered(Orbs,x->Size(x)=2)[1]); # The action of G on [1..5346] is transitive, and there are four suborbits. # The suborbits with respect to 1 are named Dist0, Dist1, Dist2, Dist3 and # Dist4. We show that the graph defined by the suborbit Dist1 is # distance-regular with the above-mentioned parameters. If this is the case, # then Check1 will be true. # We first define the perp function in the graph. Perp:=function(x) return OnSets(Dist1,RepresentativeAction(3Suz2,1,x)); end; B0:=Size(Intersection(Dist1,Perp(Dist0[1]))); B1:=Size(Intersection(Dist2,Perp(Dist1[1]))); B2:=Size(Intersection(Dist3,Perp(Dist2[1]))); B3:=Size(Intersection(Dist4,Perp(Dist3[1]))); B4:=0; C0:=0; C1:=Size(Intersection(Dist0,Perp(Dist1[1]))); C2:=Size(Intersection(Dist1,Perp(Dist2[1]))); C3:=Size(Intersection(Dist2,Perp(Dist3[1]))); C4:=Size(Intersection(Dist3,Perp(Dist4[1]))); A0:=Size(Intersection(Dist0,Perp(Dist0[1]))); A1:=Size(Intersection(Dist1,Perp(Dist1[1]))); A2:=Size(Intersection(Dist2,Perp(Dist2[1]))); A3:=Size(Intersection(Dist3,Perp(Dist3[1]))); A4:=Size(Intersection(Dist4,Perp(Dist4[1]))); Check1 := (A0+B0+C0=B0) and (A1+B1+C1=B0) and (A2+B2+C2=B0) and (A3+B3+C3=B0) and (A4+B4+C4=B0); Check1:= Check1 and B0=416 and B1=315 and B2=64 and B3=1 and C1=1 and C2=32 and C3=315 and C4=416; # We construct the distance function in this graph distance:=function(x,y) local r,z; r:=RepresentativeAction(3Suz2,x,1); z:=y^r; if z in Dist0 then return 0; fi; if z in Dist1 then return 1; fi; if z in Dist2 then return 2; fi; if z in Dist3 then return 3; fi; return 4; end; # We show that the elementwise stabilizer of Dist1 in G is trivial. If this is # the case, then Check2 will be true. Check2:=IsTrivial(Stabilizer(G,Dist1,OnTuples)); # We now show that with respect to a suitable suborbit, G defines a strongly # regular graph on Dist1 with parameters (v,k,lambda,mu)=(416,100,36,20). # If this is the case, then Check3 will be true. H := Image(ActionHomomorphism(G,Dist1)); orbs := Orbits(Stabilizer(H,1),[1..416]); dist0:=Set(Filtered(orbs,x->Size(x)=1)[1]); dist1:=Set(Filtered(orbs,x->Size(x)=100)[1]); dist2:=Set(Filtered(orbs,x->Size(x)=315)[1]); perp:=function(x) return OnSets(dist1,RepresentativeAction(H,1,x)); end; b0:=Size(Intersection(dist1,perp(dist0[1]))); b1:=Size(Intersection(dist2,perp(dist1[1]))); b2:=0; c0:=0; c1:=Size(Intersection(dist0,perp(dist1[1]))); c2:=Size(Intersection(dist1,perp(dist2[1]))); a0:=Size(Intersection(dist0,perp(dist0[1]))); a1:=Size(Intersection(dist1,perp(dist1[1]))); a2:=Size(Intersection(dist2,perp(dist2[1]))); Check3 := (a0+b0+c0=b0) and (a1+b1+c1=b0) and (a2+b2+c2=b0); Check3:= Check3 and b0=100 and b1=63 and c1=1 and c2=20; # The action of H on [1..416] is distance-transitive and thus also # primitive. By GAP’s library of primitive groups, we know that there # is, up to equivalence, only one primitive action of a group of size # |H|=|G2(4):2| on a set of 416 elements. We can then conclude that the # above strongly regular graph is the G2(4) graph. By Pasechnik we then # know that the original distance-regular graph should be the first graph # constructed by Soicher. # We prove that there exists a natural bijective correspondence between # Dist2 and the set of central involutions of the group G = G2(4):2. In # view of this, the G2(4) near octagon can be completely described in terms # of the vertices of Dist2. # We first construct the central involutions of the group G, and collect these # in the set 2A InvClasses := function(U) return Filtered(List(ConjugacyClasses(U),Representative), x->Order(x)=2); end; IsCentralInv := function(U,c) local i,n; i :=1; n:=Size(U); while n mod 2 = 0 do i := 2*i; n := n/2; od; return Size(Centralizer(U,c)) mod i = 0; end; 2A := List(ConjugacyClass(G,Filtered(InvClasses(G), x -> IsCentralInv(G,x))[1]));; # We show that there exists a natural bijective correspondence between the # central involutions of G and the special subsets of the local graph at the # vertex 1. If this is the case then Check4 will be true. g := 2A[1]; S := Set(Filtered(Dist1,x -> x^g = x)); Check4 := Index(G,Stabilizer(G,S,OnSets))=4095 and Size(Stabilizer(G,S,OnTuples))=2; # We show that there exists a natural bijective correspondence between the set # Dist2 and the set of special subsets of the local graph at the vertex 1. If # this is the case then Check5 will be true. T:=Intersection(Dist1,Perp(Dist2[1])); Check5 := RepresentativeAction(G,S,T,OnSets) <> fail; # With the following GAP code we compute the information that allows to # distinguish the various suborbits (for the action of G on Dist2) by means # of graph-theoretical properties. SubOrbs := Orbits(Stabilizer(G,Dist2[1]),Dist2); x0:=Filtered(SubOrbs,x->Size(x)=1)[1][1]; y:=Filtered(SubOrbs,x->Size(x)=1)[1][1]; Info0:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))]; y:=Filtered(SubOrbs,x->Size(x)=2)[1][1]; Info1a:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))]; y:=Filtered(SubOrbs,x->Size(x)=20)[1][1]; Info1b:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))]; y:=Filtered(SubOrbs,x->Size(x)=40)[1][1]; Info2a:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))]; y:=Filtered(SubOrbs,x->Size(x)=320)[1][1]; Info2b:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))]; y:=Filtered(SubOrbs,x->Size(x)=640)[1][1]; Info3a:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))]; y:=Filtered(SubOrbs,x->Size(x)=1024)[1][1]; Info3b:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))]; y:=Filtered(SubOrbs,x->Size(x)=2048)[1][1]; Info4:=[distance(x0,y),Size(Intersection(Perp(y),Perp(x0),Dist1)), Size(Intersection(Perp(y),Perp(x0),Dist2)), Size(Intersection(Perp(y),Perp(x0),Dist3))];