  Suppose an orientation preserving action of a finite group $G$ on the closed
surface $\Sigma_g$ of genus $g>1$ extends over the 3-torus $T^3$ for some
embedding $\Sigma_g\subset T^3$. Then $|G|\le 12(g-1)$, and this upper bound
$12(g-1)$ can be achieved for $g=n^2+1, 3n^2+1, 2n^3+1, 4n^3+1, 8n^3+1, n\in
\mathbb{Z}_+$. Those surfaces in $T^3$ realizing the maximum symmetries can be
either unknotted or knotted. Similar problems in non-orientable category is
also discussed.
  Connection with minimal surfaces in $T^3$ is addressed and when the maximum
symmetric surfaces above can be realized by minimal surfaces is identified.
