Let S be a Riemann surface and G a large subgroup of Aut(S) (Aut(S) may be unknown). We are particularly interested in regular n-gonal surfaces, i.e., the quotient surface S/G (and hence S/Aut(S)) has genus zero. For various H the ramification information of the branched coverings S/K -> S/H may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of Aut(S). The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy classes of subgroups. The only required information is a generating vector for the action of G, and the subgroup structure. The latter may be computed using Magma or GAP. The signatures and generating vectors of the subgroups are not required.
Broughton, Sean A., "Branching matrices for the automorphism group lattice of a Riemann surface" (2018). Mathematical Sciences Technical Reports (MSTR). 167.