Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space a"e(3) are easier to feel by human's intuition. We give the maximum order of finite group actions on (a"e(3), I ) pound among all possible embedded closed/bordered surfaces with given geometric/algebraic genus greater than 1 in a"e(3). We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces.