Teichmüller isometries induced by certain irreducible periodic mapping classes

Let $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ be the mapping class group of the closed orientable surface of genus $g\ge 2$, and let $\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{h}(S_g)$ be the Teichmüller space of $S_g$. In this paper, we provide an algorithm for determining the isometries induced in $\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{h}(S_g)$ by certain irreducible periodic mapping classes in Fenchel-Nielsen coordinates. As a demonstration of this method, we provide a description of the isometry induced by periodic mapping class of order $4g+2$ and also derive some of its geometric properties.