Jordan's theorem for the diffeomorphism group of some manifolds

Let $M$ be a compact connected $n$-dimensional smooth manifold admitting an unramified covering $\widetilde{M}\to M$ with cohomology classes $\alpha_1,\dots,\alpha_n \in H^1(\widetilde{M};\mathbb{Z})$ satisfying $\alpha_1\cup\dots\cup\alpha_n\neq 0$. We prove that there exists some number $c$ such that: (1) any finite group of diffeomorphisms of $M$ contains an abelian subgroup of index at most $c$; (2) if $\chi(M)\neq 0$, then any finite group of diffeomorphisms of $M$ has at most $c$ elements. We also give a new and short proof of Jordan's classical theorem for finite subgroups of $\mathrm{GL}(n,\mathbb{C})$, of which our result is an analogue for $\mathrm{Diff}(M)$.