Let $S_g$ denote an ortientable surface of genus $g$. Let $Diff(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $Diff(S_g) \to Aut(H_1(S_g))=GL_{2n}(\mathbb Z)$? It is called symplectic group if we restrict to diffeomorphisms that preserve the orientation.

Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\operatorname{Diff}(S_g) \to \operatorname{Aut}(H_1(S_g))=GL_{2n}(\mathbb Z)$? It is called symplectic group if we restrict to diffeomorphisms that preserve the orientation.