This is the unique graph on 126 vertices with the properties that (i) any maximal clique has six vertices, and (ii) if $C$ is a maximal clique and $v$ is a vertex outside of $C$, then $v$ has exactly two neighbours in $C$. Uniqueness was shown by Blokhuis and Brouwer (1984). It is a strongly regular graph with parameters $(126,45,12,18)$, but there are other examples of SRGs with these parameters.
| Number of vertices: | $126$ |
| Diameter: | $2$ |
| Intersection array: | $\{45,32;1,18\}$ |
| Spectrum: | $45^1 3^{90} (-9)^{35}$ |
| Automorphism group: | $\mathrm{PSU}(4,3).2^2$ |
| Distance-transitive: | Yes |
| Primitive |