Skipping the easy cases of genus 0 and 1, what groups can arise as the group of conformal transformations of a Riemann surfaces of genus, say, 2 or 3?

I'm frustrated because there are papers that supposedly answer this question (which are even open-access):

* Izumi Kuribayashi and Akikazu Kuribayashi, [Automorphism groups of compact Riemann surfaces of genera three and four](http://www.sciencedirect.com/science/article/pii/002240499090107S), _Journal of Pure and Applied Algebra_ **65** (1990), 277-292

* Izumi Kuribayashi, [On an algebraization of the Riemann-Hurwitz relation](https://projecteuclid.org/euclid.kmj/1138036909), _Kodai Math. J._ **7** (1984), 222-237.

Unfortunately, I don't understand the notation for groups used in this paper.  There are lots of groups with names like $G(60,120)$ and $H(5 \times 40)$.  These are actually specific subgroups of $\mathrm{GL(g,\mathbb{C})}$ where $g$ is the genus of the surface, obtained by looking at how automorphisms of a Riemann surface act on its space of [holomorphic 1-forms](https://en.wikipedia.org/wiki/Differential_of_the_first_kind).  But I don't understand how they are defined, and I don't know how to answer questions like this:

* Which 32-element groups show up as automorphism groups of Riemann surfaces of genus 3?

This last question is the one I really want answered right now, but in general I think it would be nice to more about automorphism groups of low-genus Riemann surfaces.  I get the feeling that when such a group is reasonably big, it preserves a regular tiling of the surface by regular polygons, making the Riemann surface into the quotient of the hyperbolic plane by some [Fuchsian group](https://en.wikipedia.org/wiki/Fuchsian_group).  

The classic example is of course [Klein's quartic curve](http://math.ucr.edu/home/baez/klein.html), the genus-3 surface tiled by 24 regular heptagons, whose automorphism group is $PSL(2,7)$, the largest allowed by the [Hurwitz automorphism theorem](https://en.wikipedia.org/wiki/Hurwitz's_automorphisms_theorem).