A family of $(p,n)$-gonal Riemann surfaces with several $(p,n)$-gonal groups

Let $p⩾3$ be a prime number and let $n⩾0$ be an integer such that $p-1$ divides $n\mathrm{.}$ In this short note we construct a family of $(p,n)$-gonal Riemann surfaces of maximal genus $2np+(p-1{)}^2$ with more than one $(p,n)$-gonal group.