Geometric invariant theory (GIT) has been used to construct many moduli spaces and birational models of moduli spaces. Predictions by Hassett and Keel suggest that one should get interesting models of M_g (the moduli space of stable curves) by changing the parameters of Gieseker's GIT construction of M_g. Ian Morrison and I developed Grobner basis techniques to help us study these new GIT quotients. One trick we use relies heavily on the representation theory of automorphisms of curves. I'll explain how we used various software packages in the original project, and some of the tools I have implemented or would like to implement in Magma.