// 2015-06-16 by Akihiro Munemasa

// See BCN, p.399
PDB:=PerfectGroupDatabase();
NumberOfGroups(PDB,3*Factorial(7)) eq 1;
GG:=PermutationGroup(PDB,3*Factorial(7),1);
Hs:=Subgroups(GG:OrderEqual:=#GG div 63);
#Hs eq 1;
G:=CosetImage(GG,Hs[1]`subgroup);
H:=Stabilizer(G,1);
o10s:=[o:o in Orbits(H)|#o eq 10];
o10s:=[o:o in o10s|[Random(o),1] in Orbit(G,[1,Random(o)])];
#o10s eq 1;
Gamma:=OrbitalGraph(G,1,{Representative(o10s[1])});
IntersectionArray(Gamma);
Delta:=sub<Gamma|Neighbors(Random(Vertices(Gamma)))>;
IsIsomorphic(Delta,OddGraph(3));

// See BCN, p.381
G:=PrimitiveGroup(65,1);
H:=Stabilizer(G,1);
o10s:=[o:o in Orbits(H)|#o eq 10];
#o10s eq 1;
Gamma:=OrbitalGraph(G,1,{Representative(o10s[1])});
IntersectionArray(Gamma) eq [10,6,4,1,2,5];
Delta:=sub<Gamma|Neighbors(Random(Vertices(Gamma)))>;
IsIsomorphic(Delta,OddGraph(3));