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

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. Also, the classification of subgroups G of \(\mathrm{PGU}(3,q)\) satisfying this property is given up to isomorphism.