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

In this paper, we deal with the problem of classifying the genera of quotient curves Hq/ G, where Hq is the Fq2-maximal Hermitian curve and G is an automorphism group of Hq. The groups G considered in the literature fix either a point or a triangle in the plane PG (2 , q6). In this paper, we give a complete list of genera of quotients Hq/ G, when G≤ Aut (Hq) ≅ PGU (3 , q) does not leave invariant any point or triangle in the plane. Also, the classification of subgroups G of PGU (3 , q) satisfying this property is given up to isomorphism.