  In this paper, we use permutation elements to record cylinder decompositions
of a square-tiled surface $X$. Collecting all such possible permutation
elements that record cylinder decompositions, we can enumerate the
$SL_2(\mathbb{Z})$ orbit of a given surface $X$ and give a method to determine
whether or not a matrix $A\in SL_2(\mathbb{Z})$ is the differential of an
affine diffeomorphism of $X$.
