Let S be a closed Riemann surface, and let G be a finite subgroup of the automorphism group of S. It is known that there exists a smooth G-equivariant embedding from S to some Euclidean space E, where G acts orthogonally on E. We will discuss some results about the minimal possible dimension n of such E. For example, we will show that n is at most |G| if |G|>4. Also, for the Hurwitz action on the Klein quartic, we have n=8. This is a joint work with Zhongzi Wang.