Quotients of the Hermitian curve from subgroups of ${\rm PGU}(3,q)$ without fixed points or triangles

Abstract: In this paper we deal with the problem of classifying the genera of quotient curves $\mathcal{H}q/G$, where $\mathcal{H}_q$ is the $\mathbb{F}{q^2}$-maximal Hermitian curve and $G$ is an automorphism group of $\mathcal{H}_q$. The groups $G$ considered in the literature fix either a point or a triangle in the plane ${\rm PG}(2,q^6)$. In this paper, we give a complete list of genera of quotients $\mathcal{H}_q/G$, when $G \leq {\rm Aut}(\mathcal{H}_q) \cong {\rm PGU}(3,q)$ does not leave invariant any point or triangle in the plane. As a result, the classification of subgroups $G$ of ${\rm PGU}(3,q)$ satisfying this property is given up to isomorphism.