Tutte-Coxeter graph

This article defines a particular undirected graph, i.e., the definition here determines the graph uniquely up to graph isomorphism.
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Definition

Definition in terms of duads and synthemes

Consider a fixed set of size 6. Define the duads as the subsets of size 2 of this set. There are ${\binom {6}{2}}=15$ duads. Define the synthemes as the partitions of the set into triples of subsets of size 2. There are ${\frac {1}{3!}}{\binom {6}{2,2,2}}=15$ synthemes.

The Tutte-Coxeter graph is defined as follows:

1. Its vertex set is the union of the set of duads and the set of synthemes
2. Its edges are defined as follows: two vertices are adjacent if one of them is a duad, the other is a syntheme, and the duad is one of the three pieces in the partition described by the syntheme.

Alternate definitions

The Tutte-Coxeter graph can also be defined in the following equivalent ways:

1. It is the Levi graph corresponding to the Cremona-Richmond configuration, which in turn is the generalized quadrangle with parameters (2,2).
2. It is the unique (up to graph isomorphism) (3,8)-cage.