Genus zero systems for primitive groups of affine type

LetMg be the moduli space of genus g curves. A Hurwitz locus inMg is a locus of points representing G-covers of fixed genus g with a given ramification type. The classification of Hurwitz loci of complex curves admitting G is by the computation of orbits of a suitable surface braid group acting on the generating tuples of G. When the genus of the curve is low, the braid orbits can be enumerated explicitly using GAP (Groups, Algorithm, Programming) computer algebra system and the BRAID package by Magaard, Shpectorov and Völklein. However, the length of the orbits dramatically increases with the size of G and genus of the curve. In order to handle larger orbits, we propose to break up the tuples into two or more shorter pieces which can be computed within reasonable time, and then recombine them together as direct products to form the braid orbits.