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.



Name for extension of the symplectic group
I think it is sometimes written $\operatorname{GSp}\_{2g}(\mathbb{Z})$, and called the group of symplectic similitudes


