The maximally symmetric surfaces in the 3-torus

Suppose an orientation-preserving action of a finite group G on the closed surface Σ _g of genus g> 1 extends over the 3-torus T³ for some embedding Σ _g⊂ T³. Then | G| ≤ 12 (g- 1) , and this upper bound 12 (g- 1) can be achieved for g= n²+ 1 , 3 n²+ 1 , 2 n³+ 1 , 4 n³+ 1 , 8 n³+ 1 , n∈ Z₊. The surfaces in T³ realizing a maximal symmetry can be either unknotted or knotted. Similar problems in the non-orientable category are also discussed. The connection with minimal surfaces in T³ is addressed and the situation when the maximally symmetric surfaces above can be realized by minimal surfaces is identified.