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For g 2, let Mod (S₆) be the mapping class group of the closed orientable surface S₆ of genus g. In this paper, we provide necessary and sufficient conditions for a pair of elements in Mod (S₆) to generate an infinite metacyclic subgroup. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of Mod (S₆) with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of Mod (S₆) isomorphic to Z₌, Z₍, and Z. Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of Mod (S₆) that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class F is either F or F i, where i is a hyperelliptic involution.
Kapari et al. (Tue,) studied this question.