It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle W2 (known as the Cremona–Richmond configuration).

Each vertex corresponds to an edge or a perfect matching, and connected vertices represent the incidence structure between edges and matchings.

Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a symmetric graph; it has a group of 1440 automorphisms, which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b).

In addition, the outer automorphisms of the group of permutations swap one side of the bipartition for the other.

This graph is the spherical building associated to the symplectic group

Concretely, the Tutte-Coxeter graph can be defined from a 4-dimensional symplectic vector space