In this note, we use a new technique involving invariant subspaces of certain vector space that can be used to compute the soluble series and in particular to find all curves covered by the maximal soluble automorphism groups of Riemann surfaces of genus not exceeding some given integer N.

Riemann surfaces, automorphism groups, new technique, invariant subspaces, vector spaces, soluble series, maximal soluble automorphism groups, genus of a curve, the action of groups, maximal order of automorphism groups, presentations of groups in generators and relations.