Embedding periodic maps of surfaces into those of spheres with minimal dimensions

It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $n\ge 3g$. We show that for each integer $k>1$ there exist infinitely many periodic maps such that the smallest possible $m$ is equal to $k$.