Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic isometries. Where can one find a nice explicit description of $\Gamma$?
EDIT: Sorry for being vague. Basically I just want to work out certain computations on Riemann surfaces and so am looking for nice, explicit groups of isometries of the upper half plane (so that I can compute equivariantly on the universal cover instead of on the surface). I don't care about finding the group corresponding to a given holomorphic structure, but am just looking for nice ones to work with.