A classification of triangular Riemann surfaces with $2p^2$ automorphisms

In this article we provide a classification and description of compact Riemann surfaces admitting a triangular action of a group of order $2p^2,$ where $p$ is an odd prime number. We obtain that all such Riemann surfaces are isomorphic to curves defined over the rational numbers. As a by-product, we derive a classification of orientably-regular hypermaps whose orientation-preserving automorphism group has order $2p^2\mathrm{.}$