There is a unique strongly regular graph Γ with parameters
*v* = 15, *k* = 6, λ = 1, μ = 3,
and spectrum 6^{1} 1^{9} (−3)^{5}.
It is the complement of the Triangular graph T(6).

This generalized quadrangle is the Sp(4,2) generalized quadrangle, consisting of the 15 projective points and 15 totally isotropic lines of a 4-dimensional vector space over GF(2) provided with nondegenerate symplectic form.

This generalized quadrangle is self-dual. It has six ovoids, namely the six sets (of size 5) consisting of all pairs containing a fixed symbol. Dually, it has six spreads.

a) *A 5-coclique*.
There are 6 of these, forming a single orbit.
The stabilizer of one is S_{5},
with vertex orbit sizes 5+10.
The subgraph induced on the 10 is the
Petersen graph.
These are the six ovoids.

b) *A partition of the points into lines*.
There are 6 of these, forming a single orbit.
The stabilizer of one is S_{5},
with vertex orbit size 15. These are the six spreads.

c) *A 3x3 grid*.
There are 10 of these, forming a single orbit.
The stabilizer of one is 3^{2}:D_{8},
with vertex orbit sizes 9+6.
The subgraph induced on the 9 is the 3x3 grid, that is,
the generalized quadrangle GQ(2,1).
The subgraph induced on the 6 is K_{3,3}, that is,
the generalized quadrangle GQ(1,2).

d) *A vertex*.
There are 15 of these, forming a single orbit.
The stabilizer of one is S_{4} × 2
with vertex orbit sizes 1+6+8.
The subgraph induced on the 8 is the cube 2^{3}.

e) *A line*.
There are 15 of these, forming a single orbit.
The stabilizer of one is S_{4} × 2
with vertex orbit sizes 3+12.

This graph is also found as the graph on the orthonormal bases in PG(2,9) with a conic, adjacent when they have a point in common.

a) *6 points and 6 lines, pairwise nonincident*.
There are 10 of these, forming a single orbit.
The stabilizer of one is 3^{2}:D_{8},
with vertex orbit sizes 6+9+9+6.

b) *5-sets with all distances 4*.
There are 12 of these, forming a single orbit.
The stabilizer of one is S_{5},
with vertex orbit sizes 5+10+15.

c) *A vertex*.
There are 30 of these, forming a single orbit.
The stabilizer of one is S_{4} × 2
with vertex orbit sizes 1+3+6+12+8.

d) *A pair of disjoint decagons*.
There are 36 of these, forming a single orbit.
The stabilizer of one is 10:4
with vertex orbit sizes 10+20.
(There are 72 pentagons in Γ, and 72 decagons in Δ
the vertices nonadjacent to a decagon form another decagon;
the remaining vertices, adjacent to both decagons, induce 5K_{2}.)
These are the pairs {ovoid,spread} of the geometry.
If we call two such pairs (O,S) and (O',S') adjacent when
the unique point in both O and O' lies on the unique line
in both S and S', then we obtain the
Sylvester graph.

e) *An edge*.
There are 45 of these, forming a single orbit.
The stabilizer of one has order 2^{5}
with vertex orbit sizes 2+4+8+16.
The subgraph induced on the 16 is the union of two octagons.