A quasiplatonic surface is a compact Riemann surface of genus at least two admitting a highly symmetric, embedded bipartite graph, i.e., a regular dessin d'enfant. Just as there are only five Platonic solids on the sphere, it turns out there are only finitely many regular dessins, and hence quasiplatonic surfaces, in each genus. The exact number of quasiplatonic surfaces in every genus, however, is currently unknown. Towards this count, we enumerate a subclass of these surfaces: quasiplatonic surfaces which have a cyclic group of a fixed order as a group of automorphisms. We accomplish this count by considering cyclic group actions. We will also demonstrate the connections between counting group actions, distinguishing quasiplatonic surfaces, and enumerating regular dessins d'enfants embedded on these surfaces.