a) A vertex.
There are 1782 of these, forming a single orbit.
The stabilizer of one is G2(4).2 with vertex orbit sizes
1+416+1365.
The subgraph induced on the first subconstituent is the
G2(4) graph, which is
strongly regular with parameters (v,k,λ,μ) = (416,100,36,20).
b) A U4(3) graph on 162 vertices.
Each element g of Atlas type 3A of Suz fixes 162 vertices of Γ
and Γ induces a U4(3) graph
on these 162-sets. These U4(3) graphs are strongly regular
with parameters (v,k,λ,μ) = (162,56,10,24).
There are 22880 of these (one for each subgroup <g> of G),
forming a single orbit. The stabilizer of one is
32.U4(3).22133
with vertex orbit sizes 162+1620.
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 U4(3) subgraphs on a vertex p induce the 2080
Gewirtz subgraphs in the G2(4) graph on the neighbours of p.
c) Nice splits 6297.
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 U5(2):2.
Let (just here) C6@2 denote the 2-coclique extension
of the hexagon. Then the union of any two parts of such a partition
induces either 6K2 or C6@2.
Each partition carries the structure of a rank 3 strongly regular graph
with parameters (v,k,λ,μ) = (297,40,7,5), the collinearity graph
of the generalized quadrangle GQ(8,4) that is the dual polar graph
for U5(2), if we call two maximal 6-cocliques adjacent
when their union induces C6@2.
The subgroup U5(2):2 of G (of index 32760) is transitive
on the 1782 vertices, but has two orbits on the edges,
giving a red subgraph of valency 160
and a green subgraph of valency 256.
And three orbits on the nonedges (with valencies 1280, 80, and 5),
the last of which gives our partition of the vertex set into 6-cocliques.
The local graph of the red graph is the disjoint union of five
2-coclique extensions of the Clebsch graph.
d) A 2-coclique extension of the Schläfli graph.
There are 135135 of these, forming a single orbit. The stabilizer of one is
21+6−.U4(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 35:(M11×2) 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 M11×2
acting rank 3 with suborbits 1+1+20. The 1+1 induces a U4(3)
graph, as under b). Thus, Γ has partitions into 11 U4(3)
graphs.
f) An edge.
There are 370656 of these, forming a single orbit.
The stabilizer of one is J2:2 × 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
24+6:3S6 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 U4(3) subgraphs,
and the maximal 6-cocliques of each U4(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
(A4 × L3(4):23):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
22+8:(S5 × S3)
with vertex orbit sizes 2+20+96+640+1024.
The orbit of size 20 is a K5×4 subgraph.
The orbit of size 96 is the mu-graph of Γ.
j) A 792+990 split.
The subgroup M12.2 × 2 of G (of index 2358720)
has vertex orbit sizes 792+990.
k) A 324+1458 split.
The subgroup 32+4:2(S4 × D8)
of G (of index 3203200) has vertex orbit sizes 324+1458.
l) A K6,6.
There are 10378368 of these, forming a single orbit.
The stabilizer of one is
(A6:22×A5).2
with vertex orbit sizes 12+150+720+900.
m) A 72+270+1440 split.
The stabilizer of one is ((32:8) × A6).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 26K2.
The subgroup L2(25).22 of G
(of index 57480192) has vertex orbit sizes 52+130+300+650+650.
The graph induced on the orbit of size 52 is 26K2.
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 U3(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 3K4×4.
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 K5×4, see above under i).
If xyz is a nonspecial 3-coclique, then 21 vertices are adjacent to all three and the subgraph induced is 6K1+3K5. The 6K1 part of this has 6 common neighbours, and we find a K6,6, 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 U3(4), J. Comb. Th. (A) 116 (2009) 1056-1062.
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.