We will describe two combinatorial problems in the theory of automorphism groups of compact Riemann surfaces of genus two or greater: enumerating the topological actions of a finite group on surfaces and determining the set of genera of surfaces admitting such a group action, called the genus spectrum. We will illustrate results in the important case of quasiplatonic cyclic group actions. Specifically, using formulas of Benim and Wootton (2014), we show that the number of quasiplatonic cyclic group actions is roughly one-sixth the number of regular dessins d'enfants with a cyclic group of symmetries. By optimizing the Riemann-Hurwitz formula under certain conditions, we discuss the second-smallest genus of surfaces admitting a quasiplatonic cyclic group action, which appears to resemble the known minimal genus action due to Harvey (1966). We determine the second-smallest genus action when the smallest prime divisor of a positive integer has exponent one.