 Projective special unitary group over a finite field

The projective quotient of determinant-one matrices preserving a nondegenerate Hermitian form over ${\mathbb{F}}_{q^2}$, with conjugation $x\mapsto x^q$. In dimension three its natural two-transitive action has $q^3+1$ isotropic points. For $q\ge 3$ its order is $q^3(q^3+1)(q^2-1)/\gcd (3,q+1)$.