  For $g\geq 2$, let $\text{Mod}(S_g)$ be the mapping class group of the closed
orientable surface $S_g$ of genus $g$. In this paper, we provide necessary and
sufficient conditions for the existence of infinite metacyclic subgroups of
$\text{Mod}(S_g)$. In particular, we provide necessary and sufficient
conditions under which a pseudo-Anosov mapping class generates an infinite
metacyclic subgroup of $\text{Mod}(S_g)$ with a nontrivial periodic mapping
class. As applications of our main results, we establish the existence of
infinite metacyclic subgroups of $\text{Mod}(S_g)$ isomorphic to
$\mathbb{Z}\rtimes \mathbb{Z}_m, \mathbb{Z}_n \rtimes \mathbb{Z}$, and
$\mathbb{Z} \rtimes \mathbb{Z}$. Furthermore, we derive bounds on the order of
a nontrivial periodic generator of an infinite metacyclic subgroup of
$\text{Mod}(S_g)$ that are realized. Finally, we show that the centralizer of
an irreducible periodic mapping class $F$ is either $\langle F\rangle$ or
$\langle F\rangle \times \langle i\rangle$, where $i$ is a hyperelliptic
involution.
