There are two known $\mathrm{GQ}(5,5)$s, namely $\mathrm{O_5}(5)$ and $\mathrm{Sp}_4(5)$. Each one is the dual of the other.
| Number of vertices: | $156$ |
| Diameter: | $2$ |
| Intersection array: | $\{30,25;1,6\}$ |
| Spectrum: | $30^1 4^{90} (-6)^{65}$ |
| Automorphism group: | $\mathrm{PS_p}(4,5).2$ |
| Distance-transitive: | Yes |
| Primitive |