Belolipetsky and Jones classified those compact Riemann surfaces of genus
g admitting a large group of automorphisms of order
, for each
under the assumption that
is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting
and
automorphisms, with
a prime number. As a consequence,
... [Show full abstract] we obtain the classification of Riemann surfaces admitting a group of automorphisms of order , with a prime number. We also provide isogeny decompositions of their Jacobian varieties.