Quotients of the Hermitian curve from subgroups of $\mathrm{PGU}(3,q)$ 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 $\mathrm{PG}(2,q^6)$. In this paper, we give a complete list of genera of quotients ${\mathcal{H}}_q/G$, when $G\le \mathrm{Aut}({\mathcal{H}}_q)\cong \mathrm{PGU}(3,q)$ does not leave invariant any point or triangle in the plane. As a result, the classification of subgroups $G$ of $\mathrm{PGU}(3,q)$ satisfying this property is given up to isomorphism.