The locus of Riemann surfaces of genus $2(p-1)$ with $4p$ automorphisms

Abstract: Let $p \geqslant 5$ be a prime number. In this article we provide a complete and explicit description of the locus formed by the compact Riemann surfaces of genus $2(p-1)$ that are endowed with a group of automorphisms of order $4p$. In addition, we provide isogeny decompositions of the corresponding Jacobian varieties and study if the most symmetric ones admit complex and real multiplication.