  In this article we consider compact Riemann surfaces that are uniquely
determined by the property of possessing a group of automorphisms of a
prescribed order, strengthening uniqueness results proved by Nakagawa. More
precisely, we deal with the cases in which such an order is $3g$ and $3g+3,$
where $g$ is the genus. We prove that if $g$ is odd (respectively $g$ even and
$g \not \equiv 2 \mbox{ mod } 3$) then there exists a unique Riemann surface of
genus $g$ with a group of automorphisms of order $3g$ (respectively $3g+3$). A
similar conclusion can be derived in terms of orientably-regular hypermaps. In
addition, we determine the full automorphism group of such Riemann surfaces and
provide decompositions of their Jacobians.
