The same construction in other words: Take the 23 points and 253 blocks of S(4,7,23), with points forming a coclique, p ~ B when p is not in B, and B ~ B' when B,B' meet in 1 point. This yields a graph on 276 vertices that is member of the switching class of a regular two-graph. Now switch one point isolated (and delete it) to get Γ. See also h) below.

a) *Vertices*.
There are 275 of these, forming a single orbit.
The stabilizer of one (in the full automorphism group)
is U_{4}(3):2 with vertex orbit sizes 1+112+162.
The graphs induced on the
first and second
subconstituent are described separately.

b) *22-cocliques*.
There are 4050 of these, forming a single orbit.
The stabilizer of one is M_{22}
with vertex orbit sizes 22+77+176.
The graph induced on the 77 is the M_{22}-graph.

c) *Pairs of Hoffman-Singleton graphs*.
Γ has an orbit of splits 50+50+175 where the 50's induce
the Hoffman-Singleton graph
and the 175 induces the graph on the edges of the Hoffman-Singleton graph,
adjacent when they are disjoint and lie in the same pentagon.
This latter graph is strongly regular with parameters
(v,k,λ,μ) = (175,72,20,36).
If one takes the union of the two 50's and switches on this split
one obtains the Higman-Sims graph.
The stabilizer of a split 50+50+175 is U_{3}(5):2, where
the outer 2 interchanges the two 50's.
There are 7128 such splits.
*** Other HoSi's in Γ? ***

d) *5-cliques*.
There are 15400 of these, forming a single orbit.
The stabilizer of one is 3_{+}^{1+4}:4S_{5}
with vertex orbit sizes 5+270.

e) *Edges*.
There are 15400 of these, forming a single orbit.
The stabilizer of one is 3^{4}:(M_{10} × 2)
with vertex orbit sizes 2+30+81+162.
The graph induced on the 81 is the
VO^{–}_{4}(3) graph.

f) *Non-edges*.
There are 22275 of these, forming a single orbit.
The stabilizer of one is L_{3}(4):2^{2}
with vertex orbit sizes 2+56+105+112.
The graph induced on the 56 (the common neighbours) is the
Gewirtz graph.

g) *Odd graphs*.
Γ contains an orbit of 22275
Odd graphs O_{4}.
The stabilizer of one is 2.S_{8}
with vertex orbit sizes 35+240.
There are other subgraphs O_{4} in Γ
since the local graph also contains such graphs.
*** Any further Odd subgraphs? ***

h) *Maximal 7-cocliques*.
There are 44550 of these, forming a single orbit.
The stabilizer of one is 2^{4}:A_{7}
with vertex orbit sizes 7+16+112+140.
The 16 here is a maximal 16-coclique.
(This is what one gets from the regular two-graph construction
after switching a block isolated and deleting it.)

i) *Maximal 11-cocliques of the first kind*.
There are 113400 of these, forming a single orbit.
Γ has three orbits of maximal 11-cocliques.
The first has stabilizer 2 × M_{11},
with vertex orbit sizes 11+22+110+132.
The 11 is a maximal 11-coclique, the 22 a (nice) 11K_{2}.
(There is a unique nice 11K_{2} on each 2K_{2},
but there are many others as well.)
The other two orbits of maximal 11-cocliques have
stabilizers of order 12 and 36, not transtive on the coclique.

j) *Splits 125+150*.
Γ has an orbit of splits 125+150 with stabilizer
5_{+}^{1+2}:3:8:2
with vertex orbit sizes 125+150.
The 125 induces a rank 5 strongly regular graph
with parameters (v,k,λ,μ) = (125,52,15,26).

k) *3-Cocliques*.
There are 779625 of these, forming a single orbit.
The stabilizer of one is
2^{2+4}:(S_{3}×S_{3})
with vertex orbit sizes 3+8+32+64+72+96.
The orbit of size 8 induces a K_{4,4}.

The largest cocliques have size 22. Numbers of maximal cocliques of given size (with orbit sizes):

7: 44550 10: 13721400 = 1247400 + 12474000 11: 199697400 = 113400 + 49896000 + 149688000 13: 43659000 = 6237000 + 37422000 16: 757350 = 44550 + 712800 22: 4050

The largest cocliques in the line graph have size 11 (and there are 254016000 of these, 181440 on each edge, 2240 on each pair of disjoint edges).

pp 85-122 in: Enumeration and Design - Proc. Silver Jubilee Conf. on Combinatorics, Waterloo, 1982, D.M. Jackson & S.A. Vanstone (eds.), Academic Press, Toronto, 1984.

J.-M. Goethals & J. J. Seidel,

*The regular two-graph on 276 vertices*,

Discr. Math. **12** (1975) 143-158.