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Motivated by the theory of Riemann surfaces, we classify all possibilitiesfor finite simple groups acting faithfully on a compact Riemann surface ofgenus at least 2 in such a way that all non-trivial elements have at most threefixed points on each non-regular orbit and at most four fixed points in total.In each case we also give information about the branching datum of the surface.There is a shorter version of this article (submitted for publication), and inthis extended version we give many more details about the GAP code that we usefor the calculations. We also explicitly include a lemma that we only quote inthe short version, so we can explain how exactly the GAP calculations and thelemma work together.

Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaces (extended version)