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 ℝ³ are easier to feel by human’s intuition. We give the maximum order of finite group actions on (ℝ³, Σ) among all possible embedded closed/bordered surfaces with given geometric/algebraic genus greater than 1 in ℝ³. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces.