Γ is the second subconstituent of the McL graph Λ on 275 vertices.

Γ can also be found inside the
O^{–}_{6}(3) graph
on 112 vertices. That graph contains 648 Gewirtz subgraphs
forming a single orbit.
The intersection sizes are 0 or 56 (1x), 16 or 40 (42x),
20 or 36 (56x), 24 or 32 (105x), 28 (240x), and meeting in
0, 20, 24, 32, 36 or 56 vertices is an equivalence relation
with two equivalence classes of size 324.
Pick one equivalence class. It consists of 162 complementary pairs (`splits').
These are the vertices of Γ. Call two splits adjacent when their parts
meet in 20 or 36 vertices, and nonadjacent when they meet in 24 or 32.
This yields the graph Γ.
(In fact one sees inside the McL graph Λ that it is possible
to find 162 representatives of the 162 splits such that any two
representatives meet in either 20 or 32 vertices.)

a) *Splits 81+81*.
Γ has 112 splits into two
VO^{–}_{4}(3)
subgraphs, forming a single orbit.
The stabilizer of one is 3^{4}:(2 × M_{10}).
(How does one find all VO^{–}_{4}(3) subgraphs?
By searching the 21-dimensional eigenspace for eigenvectors
that are 1 on the subgraph and –1 on the complement. The result
is that these 112 are the only splits of Γ into two
subgraphs of valency 20.)

b) *Vertices*.
There are 162 of these, forming a single orbit.
The stabilizer of one is L_{3}(4):2^{2}
with vertex orbit sizes 1+56+105.
The 56 induce a Gewirtz graph.
The 105 are the flags of PG(2,4). where two flags (p,L) and (q,M)
are adjacent when p,q are distinct and L,M are distinct,
and p is on M or q on L. The induced subgraph is strongly regular
with parameters (105,32,4,12) and group L_{3}(4).D_{12}
acting rank 4.

c) *Pairs of 21-cocliques*.
There are 162 of these pairs of 21-cocliques, forming a single orbit.
The stabilizer of one is L_{3}(4):2^{2}
with vertex orbit sizes 42+120.
Γ has 324 21-cocliques forming a single orbit.
The stabilizer of one is L_{3}(4):2 with vertex orbit sizes
21+120+21, where both 21-sets are 21-cocliques, forming a pair.
Their union of is distance-regular with parameters
{16,15,5;1,12,16}, the nonincidence graph of points and lines of PG(2,4).
The 120 can be identified as one orbit of Fano subplanes of this PG(2,4),
adjacent when they meet in 1 point (and 1 line).

d) *Splits 6 ^{27}*.
Γ has one orbit (of size 252) of partitions of the vertex set
into 27 maximal 6-cocliques. The stabilizer of one is
U

e) *Splits 54+54+54*.
Γ has 280 splits into three copies of the same graph on 54 vertices,
forming a single orbit. The stabilizer of one is
3_{+}^{1+4}.2_{–}^{1+4}.S_{3},
transitive on the 162 vertices.
(There are 15120 triangles, forming a single orbit. Each triangle
determines a unique subgraph 9K_{3}, and there are 1680
such subgraphs, forming a single orbit. Each subgraph 9K_{3}
has a unique partner, disjoint from the first, such that each of its
vertices has two neighbours in each of the nine K_{3}'s of
the first, and we find 840 54-point subgraphs of valency 20.
Each of these 54-point subgraphs is disjoint from precisely two others
and there is an element of order 3 that permutes the three.)

f) *U _{3}(3) graphs*.
Γ has 540 subgraphs isomorphic to the
U

g) *Maximal 6-cocliques*.
There are 1134 of these, forming a single orbit.
The stabilizer of one is
(2^{4}:A_{6}).2 = 2^{4}.S_{6}.
with vertex orbit sizes 6+60+96.

h) *2-coclique extension of K _{3}×K_{3}*.
There are 2835 subgraphs isomorphic to the 2-coclique extension of
K

i) *Edges*.
There are 4536 of these, forming a single orbit.
The stabilizer of one is A_{6}.2^{2} × 2
with vertex orbit sizes 2+10+60+90. The orbit of size 10 is the
first of the orbits of 10-cocliques.

j) *K _{6,6}'s of the second kind*.
Γ has two orbits of K

The two groups found under i) and j) are conjugate in U

k) *Splits 72+90*.
There are 4536 splits of Γ into two subgraphs of sizes
72 and 90, with valencies 26 and 32.
*** check: not more such splits? ***
They form a single orbit.
The stabilizer of one is A_{6}.2^{2} × 2
with vertex orbit sizes 72+90.
For A_{6}, the orbit of size 72 splits into 36+36,
where each 36 carries a Sylvester graph.
For A_{6}, the orbit of size 90 splits into 45+45,
where each 45 carries the distance-2 graph of the incidence graph
of GQ(2,2).

l) *Maximal 15-cocliques of the first kind*.
Γ has two orbits of maximal 15-cocliques.
The first orbit has size 5184 and stabilizer A_{7}
with vertex orbit sizes 15+42+105.
(The second orbit has size 38880 and stabilizer 2 × L(3,2)
with vertex orbit sizes 1+14 on the 15-coclique, and
1+7+7+14+14+21+42+56 on all 162 points.)

m) *Nonedges*.
There are 8505 of these, forming a single orbit.
The stabilizer of one is (4^{2} × 2)(2 × S_{4})
with vertex orbit sizes 2+8+24+64+64. The orbit of size 8 induces a
K_{4,4}.

6: 1134 9: 498960 = 90720 + 408240 10: 7987896 = 4536 + 362880 + 1088640 + 3265920 + 3265920 12: 2063880 = 22680 + 408240 + 544320 + 1088640 15: 44064 = 5184 + 38880 21: 324

P.J. Cameron, J.-M. Goethals & J.J. Seidel,
*Strongly regular graphs having strongly regular subconstituents*,
J. Algebra **55** (1978) 257-280.

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