Symmetry types of cyclic covers of the sphere

We examine all compact Riemann surfaces of genus greater than one which admit a cyclic group of automorphisms that yields a covering of the Riemann sphere with exactly three branch points. We determine the number of non-conjugate symmetries of each of these surfaces. For each symmetry, we determine the number of ovals it fixes and whether the orbit space under the symmetry is orientable or not. This yields the species of each symmetry and the symmetry type of each surface. Explicit defining equations of each surface and symmetry are given.