# On some classical groups with small permutation degree

The following question arises while I am reading a paper of B. N. Cooperstein.

In the Table 1 of his paper, two groups have smaller permutation degree compared to other members in their family: $\mathrm{PSp}_4(3)$ with permutation degree 27 and $\mathrm{PSU}_3(5)$ with permutation degree 50, while $\mathrm{PSp}_{2n}(q)$ generally has a permutation degree of $$\frac{q^{2n}-1}{q-1}$$ and $\mathrm{PSU}_3(q)$ has a permutation degree of $q^3+1$. It is already known to Camille Jordan that $\mathrm{PSp}_4(3)$ is a permutation group on 27 lines of a cubic surface. I am curious if there is a similar geometric configuration for $\mathrm{PSU}_3(5)$.

Question: Is there any neat geometric configuration (with 50 elements) that the group $\mathrm{PSU}_3(5)$ is a permutation group which acts faithfully on the configuration?

Edit: A table for permutation degree of simple groups can be found here.

• perhaps it is also worth mentioning that the degree 27 action of Sp(4,3) arises from its isomorphism to $U(4,2)$, for which the action of degree 27 is the action on totally isotropic lines in its natural 4-dimensional module over GF(4). In the case of degree 50 action of U(3,5), nothing like this happens. – Dima Pasechnik Mar 4 '17 at 19:51