The Sp(4,2) Generalized Quadrangle

There is a unique strongly regular graph Γ with parameters v = 15, k = 6, λ = 1, μ = 3, and spectrum 6¹ 1⁹ (−3)⁵. It is the complement of the Triangular graph T(6).

Generalized Quadrangle

Γ is the collinearity graph of the unique generalized quadrangle GQ(2,2), that can be defined as having as points the 15 pairs from a 6-set and as lines the 15 partitions of that 6-set into three pairs, with natural incidence.

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.

Group

The full group of automorphisms of Γ is Sym(6), acting rank 3 with point stabilizer S₄ × 2.

Subgraphs

Substructures belonging to the maximal subgroups of the automorphism group:

a) A 5-coclique. There are 6 of these, forming a single orbit. The stabilizer of one is S₅, 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 S₅, 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 3²:D₈, 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 S₄ × 2 with vertex orbit sizes 1+6+8. The subgraph induced on the 8 is the cube 2³.

e) A line. There are 15 of these, forming a single orbit. The stabilizer of one is S₄ × 2 with vertex orbit sizes 3+12.

The incidence graph

The point-line incidence graph Δ of this geometry is the unique distance-regular graph with intersection array {3,2,2,2;1,1,1,3}. It has spectrum 3¹ 2⁹ 0¹⁰ (−2)⁹ (−3)¹. It is known as Tutte's 8-cage.

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.

Group

The full group of automorphisms of Δ is Sym(6).2, acting distance transitively with point stabilizer S₄ × 2.

Subgraphs

Substructures belonging to the maximal subgroups of the automorphism group:

a) 6 points and 6 lines, pairwise nonincident. There are 10 of these, forming a single orbit. The stabilizer of one is 3²:D₈, 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 S₅, with vertex orbit sizes 5+10+15.

c) A vertex. There are 30 of these, forming a single orbit. The stabilizer of one is S₄ × 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 5K₂.) 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 2⁵ with vertex orbit sizes 2+4+8+16. The subgraph induced on the 16 is the union of two octagons.