a) *A vertex*.

There are 1782 of these, forming a single orbit.
The stabilizer of one is G_{2}(4).2 with vertex orbit sizes
1+416+1365.
The subgraph induced on the first subconstituent is the
G_{2}(4) graph, which is
strongly regular with parameters (v,k,λ,μ) = (416,100,36,20).

b) *A U _{4}(3) graph on 162 vertices*.

Each element g of Atlas type 3A of Suz fixes 162 vertices of Γ and Γ induces a U

The graph on these 22880 subgraphs, adjacent when disjoint, that is, when the corresponding elements g commute, is the unique distance-regular graph with intersection array {280,243,144,10; 1,8,90,280}, known as the Patterson graph. (And distance 0, 1, 2, 3, 4 in the Patterson graph corresponds to intersection size 162, 0, 18, 12, 42 for these 162-sets.)

The 2080 U

c) *Nice splits 6 ^{297}*.

Γ has a unique orbit (of size 32760) of partitions of the vertex set into 297 maximal 6-cocliques such that the union of any two parts induces a regular subgraph of Γ. (There are many other partitions into 297 maximal 6-cocliques.) The stabilizer of one such partition is U

The subgroup U

d) *A 2-coclique extension of the Schläfli graph*.

There are 135135 of these, forming a single orbit. The stabilizer of one is
2^{1+6}_{−}.U_{4}(2).2
with vertex orbit sizes 54+1728.
The graph induced on the orbit of size 54 is the
2-coclique extension of the Schläfli graph.
This graph has valency 32. Its local graph is the 2-coclique extension
of the Clebsch graph.
The mu-graphs of Γ consist of three copies of this graph on 32 vertices.

e) *A 3-edge-coloring*.

The subgroup 3^{5}:(M_{11}×2) of G
(of index 232960) is transitive on the 1782 vertices, but
has three orbits on the edges, giving a red subgraph of valency 20,
the disjoint union of 22
Brouwer-Haemers graphs,
and a bipartite green subgraph of valency 36,
and a blue subgraph of valency 360.

The 22 Brouwer-Haemers graphs are permuted by M_{11}×2
acting rank 3 with suborbits 1+1+20. The 1+1 induces a U_{4}(3)
graph, as under b). Thus, Γ has partitions into 11 U_{4}(3)
graphs.

f) *An edge*.

There are 370656 of these, forming a single orbit.
The stabilizer of one is J_{2}:2 × 2 with vertex orbit sizes
2+100+630+1050.

g) *A maximal 6-coclique*.

There are 405405 of these, forming a single orbit. The stabilizer of one is
2^{4+6}:3S_{6} with vertex orbit sizes 6+240+1536.
The vertices in the 240-orbit have 4 neighbours in the 6-coclique,
those in the 1536-orbit have 1.
Each maximal 6-coclique lies in 24 splits as under c).
Each maximal 6-coclique lies in 64 U_{4}(3) subgraphs,
and the maximal 6-cocliques of each U_{4}(3) subgraph
remain maximal in Γ.

Each vertex lies in 1365 maximal 6-cocliques. Each nonedge lies in 5.
Each non-edge xy lies in 1044 3-cocliques, on which the stabilizer
has orbits of sizes 20+1024, see below under i). Each triple xyz with
z in the orbit of size 20 lies in a unique maximal 6-coclique.
No triple xyz with z in the orbit of size 1024 lies in a maximal 6-coclique.

There are no smaller maximal cocliques. The largest cocliques have size 66.

h) *A nonincidence graph of PG(2,4)*.

There are 926640 of these, forming a single orbit.
The stabilizer of one is
(A_{4} × L_{3}(4):2_{3}):2
with vertex orbit sizes 42+480+1260.
The graph induced on the orbit of size 42 is the bipartite
point-line nonincidence graph of PG(2,4).

i) *A non-edge*.

There are 1216215 of these, forming a single orbit.
The stabilizer of one is
2^{2+8}:(S_{5} × S_{3})
with vertex orbit sizes 2+20+96+640+1024.
The orbit of size 20 is a K_{5×4} subgraph.
The orbit of size 96 is the mu-graph of Γ.

j) *A 792+990 split*.

The subgroup M_{12}.2 × 2 of G (of index 2358720)
has vertex orbit sizes 792+990.

k) *A 324+1458 split*.

The subgroup 3^{2+4}:2(S_{4} × D_{8})
of G (of index 3203200) has vertex orbit sizes 324+1458.

l) *A K _{6,6}*.

There are 10378368 of these, forming a single orbit. The stabilizer of one is (A

m) *A 72+270+1440 split*.

The stabilizer of one is ((3^{2}:8) × A_{6}).2
(of index 17297280) with vertex orbit sizes 72+270+1440
and valencies 26, 56 and 338, respectively.
Each 72 has a partition into 12 6-cliques.
For details, see Cliques, below.

n) *A 26K _{2}*.

The subgroup L

o) *A sub-Hoffman-Singleton graph*.

The subgroup Sym(7) of G (of index 177914880) has vertex orbit sizes
42+120+210+210+360+420+420.
The graph induced on the orbit of size 42 is the 2nd subconstituent
of the Hoffman-Singleton graph,
distance-regular with intersection array {6,5,1; 1,1,6}.

The largest cocliques have size 66 and form a single orbit
(Kuzuta, cf. Brouwer et al., 2009). The stabilizer of one
is U_{3}(4):4 with vertex orbit sizes 1+65+416+1300.
The 1716 points outside a 66-coclique all have 16 neighbours inside,
and we find a 3-(66,16,21) design.

If xyz is a special 3-coclique, then 48 vertices are adjacent to all three
and the subgraph induced on this orbit of size 48 is
3K_{4×4}.

There are 3+768 vertices (other than xyz) adjacent to none of xyz.
If uvw is this orbit of size 3, then uvwxyz is a maximal 6-coclique,
see above under g). All triples in uvwxyz are special.
The common neighbours of xyzw together with xyzw themselves
form a K_{5×4}, see above under i).

If xyz is a nonspecial 3-coclique, then 21 vertices are adjacent to all three
and the subgraph induced is 6K_{1}+3K_{5}.
The 6K_{1} part of this has 6 common neighbours,
and we find a K_{6,6}, see above under l).

A. E. Brouwer, A. Jurišić & J. H. Koolen,
*Characterization of the Patterson graph*,
J. Algebra **320** (2008) 1878-1886.

A. E. Brouwer, N. Horiguchi, M. Kitazume & H. Nakasora,
*A construction of the sporadic Suzuki graph from U _{3}(4)*,
J. Comb. Th. (A)

D. Pasechnik,
*Geometric characterization of graphs from the Suzuki chain*,
Europ. J. Combin. **14** (1993) 491-499.

L. H. Soicher,
*Three new distance-regular graphs*,
Europ. J. Combin. **14** (1993) 501-505.

M. Suzuki,
*A simple group of order 448,345,497,600*,
pp. 113-119 in: Theory of finite groups, R. Bauer & C.-S. Sah (eds.),
Benjamin, New York, 1969.