These suborbits can be described geometrically as follows: Given one basis {a,b,c}, the suborbit of size 15 consists of the bases that have an element in common with {a,b,c}. The first suborbit of size 150 consists of the bases that are disjoint from {a,b,c} but contain a point orthogonal to one of a,b,c. Associated with a basis {a,b,c} is the triangle consisting of the 15 isotropic points on the three lines ab, ac, and bc. The suborbits of sizes 1, 15, 100, 150, 150 correspond to the bases with triangles having 15, 5, 3, 2, 5 points in common. Thus, Γ can be described as the graph on the 416 triangles, adjacent when they have 3 points in common ([1]).
a) A vertex.
There are 416 of these, forming a single orbit.
The stabilizer of one is HJ.2 with vertex orbit sizes 1+100+315.
The subgraph induced on the first subconstituent is the
Hall-Janko graph, which is
strongly regular with parameters (v,k,λ,μ) = (100,36,14,12).
b) Three copies of the 2-coclique extension of the Clebsch graph.
The 65520 nonedges of Γ form a single orbit. The stabilizer of one
has vertex orbit sizes 2+10+20+64+160+160.
These 65520 nonedges fall into 1365 sets of 48, where each set
induces a subgraph of size 96 that is the disjoint union of three
copies of the 2-coclique extension of the Clebsch graph (the halved 5-cube),
and the 48 nonedges are the 2-cocliques.
These 1365 subgraphs of size 96 can be seen as μ-graphs in the Suzuki graph: The rank 3 Suzuki graph Σ on 1782 = 1 + 416 + 1365 vertices is strongly regular with parameters v = 1782, k = 416, λ = 100, μ = 96. We can take Γ to be the first subconstituent of Σ. Now the 1365 vertices of the second subconstituent are seen in Γ as 1365 subgraphs of size 96, regular of valency 20. These 1365 subgraphs form a single orbit. The stabilizer of one is 22+8:(3 × A5):2 with vertex orbit sizes 96+320.
The 4095 subgraphs isomorphic to the 2-coclique extension of the halved 5-cube form a single orbit. The stabilizer of one is 22+8:A5:2 with vertex orbit sizes 32+64+320.
c) Unions of 16 16-cocliques.
Γ contains a unique orbit of 21840 16-cocliques
(there are four more orbits of 16-cocliques, see below
in the Cocliques section) each stabilized by a group
(A4 x (24 : A5)) : 2
of order 23040 with vertex orbit sizes 16+240+160.
The union of the orbits of sizes 16+240 gives a 256-point subgraph
of valency 60 stabilized by a subgroup H of shape
24+6:(A5×3):2 of order 368640
with vertex orbit sizes 256+160. There are 1365 such 256-point subgraphs.
Such a 256-point subgraph Δ contains many 16-cocliques, but there is
only one orbit of size 16 of H on 16-cocliques in Δ (giving a partition
of the vertex set of Δ).
The 1365 subgraphs of size 96 found under b) and the 1365 subgraphs of size 256 found here form the points and lines of a generalized hexagon GH(4,4) (with 5 points/line and 5 lines/point) if we let disjoint subgraphs of different types be incident.
d) Sets of 208 6-cocliques.
The maximal subgroup U3(4):4 was used above to construct
the graph Γ. It is transitive on the 416 vertices.
Each of the 208 hyperbolic lines in PG(2,16) gives 6 bases forming
a 6-coclique in Γ. This yields a family of 208 6-cocliques
(3 on each vertex) invariant for U3(4):4.
e) A Gewirtz subgraph.
Each element of Atlas type 3A of G2(4) fixes 56 vertices
of Γ, and Γ induces a
Gewirtz graph on these 56-sets.
There are 2080 of these, forming a single orbit.
The stabilizer of one is 3.L3(4).22
with vertex orbit sizes 56+360. A vertex outside a Gewirtz subgraph
has 14 neighbours inside, inducing a
co-Heawood subgraph.
The 280 Gewirtz subgraphs on a vertex p induce the 280 10-cocliques
in the Hall-Janko subgraph on the neighbours of p. If C is one of
these 10-cocliques, then each vertex nonadjacent to p has two neighbours in C.
f) An edge.
There are 20800 of these, forming a single orbit.
The stabilizer of one is U3(3):2×2 with
vertex orbit sizes 2+36+126+252.
g) A 6-coclique or a 50-point subgraph P#5.
Γ has 698880 5-cliques, forming a single orbit.
The stabilizer of one is Sym(5)×Sym(3) with vertex
orbit sizes 5+6+15+30+60+60+120+120. The orbit of size 6 here
is the 6-coclique meant - it is not a maximal coclique, but
its stabilizer is (A5×A5):2
with vertex orbit sizes 6+50+60+300.
Let P be the Petersen graph on 10 vertices, and let P#5 denote the graph obtained from P by replacing each vertex x by the five (mutually adjacent) vertices (x,i) (i=1,2,3,4,5), and each edge x~y by the twenty edges (x,i)~(y,j) for i notequal j. Now P#5 has 50 vertices and is regular of degree 16. It has 5 10-cocliques and 10+450=460 5-cliques. The group of P#5 is Sym(5) × Sym(5).
Each 5-clique L determines 3 others Li (i=1,2,3) (forming a single orbit in the stabilizer of L) such that the union of L and Li induces K2#4. The union of the orbits with sizes 5+15+30 of the stabilizer of L has shape P#5. There are 69888 subgraphs P#5 forming a single orbit. The stabilizer of one is (A5×A5):2 with vertex orbit sizes 6+50+60+300.
The 6-coclique consists of the 6 vertices without neighbours in the P#5 (so that it is contained in 16-cocliques).
h) 78+91+91+156.
L(2,13) is a maximal subgroup of G with vertex orbit sizes 78+91+91+156,
where the two representations on 91 points are nonisomorphic. The point
stabilizers are D14, A4, D12, C7.
The four subgraphs have valencies 28, 24, 24, 36.
7: 499200 9: 628992000 10: 111370604800 11: 408839558400 12: 67554614400 13: 3133132800 16: 3808896These cocliques fall into 1397 orbits, namely 1, 2, 291, 867, 199, 32, 5 orbits of maximal cocliques of size 7, 9, 10, 11, 12, 13, 16, respectively.
The unique orbit of maximal 7-cocliques is that of the parts of the bipartition of the coHeawood graphs found as common neighbours of three mutually adjacent points (so that there are 416*100*36*2/6 = 499200 of them).
The Hoffman bound for cocliques is 16, so that for each 16-coclique each point outside has 4 neighbours inside.
The 16-cocliques in the 5 orbits have stabilizers of order 384, 384, 600, 1536, 23040, respectively. Let C be a 16-coclique of the last type, and S its stabilizer. Then S has vertex orbit sizes 16+240+160. The 16+240 induces a subgraph of size 256 that is the union of 16 16-cocliques, and its stabilizer has order 368640. This is the subgraph studied above under c).