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Symmetries of surfaces can be studied via topological group actions. For compact, connected, orientable surfaces of genus two or greater, the full group of (orientation-preserving) isometries is a finite group by the Hurwitz automorphism theorem. The most highly symmetric surfaces admit quasiplatonic group actions and are thus called quasiplatonic surfaces, a generalization of the Platonic solids. On the other hand, a finite group acts quasiplatonically (with a given signature) in only finitely many ways. We study the quasiplatonic actions of the cyclic group. Using formulas of Benim and Wootton (2014), we show that the total number of quasiplatonic cyclic group actions includes the number of regular dessins d'enfants (or bipartite map embeddings) with a cyclic group of automorphisms. This has relevance in studying the number of conjugacy classes of finite cyclic subgroups of the mapping class group.