A cyclic n-gonal surfaces S, is a compact Riemann surface with an automorphism h such that the surface S/C has genus 0, where C = <h>. In certain circumstances (weakly malnormal actions, described in the previous talk by Aaron Wootton) C is normal in the automorphism group A=Aut(S) if the genus g of S is sufficiently large. For small genus, when n is composite, unlike the case where n is prime, it is not feasible to determine the exceptional cases where C is not normal by hand and instead it is necessary to resort to computer classification. In this talk, we give an overview of the computational methods used to determine these exceptional cases, and a tabulation of results achieved to date.