Manifolds with a finite but not trivial fundamental group I came across this nice result:

Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial:
  $$H^1_{dR}(M)=0.$$

However, I don't know any example of manifold with a finite but not trivial fundamental group. What are the nice examples of such manifolds?
 A: Real projective space $\mathbb{RP}^n$ has fundamental group $\mathbb Z/2\mathbb Z$ for $n\geq 2$. This the quotient of the sphere $S^n$ by the antipodal action $x\sim -x$. In fact $S^n$ is a $2$-sheeted universal cover, which implies by covering space theory that its fundamental group is of order $2$.
A: The special orthogonal group $SO(n)$ for $n\ge 3$ has fundamental group $\mathbb Z/2\mathbb Z$.
A: The Poincare homology sphere has fundamental group of order $120$, the binary icosahedral group. Any homology sphere must have a perfect fundamental group, but it need not be trivial.
A: In analogy to the answer of Robert Israel: The special linear group $\operatorname{SL}_n(\mathbb{R})$ (For a proof it is a smooth manifold see here.) has fundamental group $\mathbb{Z}_2$ for $n\geq 3$.
I am not totally sure, whether the projective orthogonal groups are smooth manifolds (somebody would need to verify that), but we have (See here.) for $n\geq 1$:
$$\pi_0\operatorname{PO}(2n)
\cong\mathbb{Z}_2$$
$$\pi_0\operatorname{PO}(2n+1)
\cong 1$$
$$\pi_1\operatorname{PO}(4n)
\cong\mathbb{Z}_2^2$$
$$\pi_1\operatorname{PO}(4n+2)
\cong\mathbb{Z}_4$$
$$\pi_1\operatorname{PO}(2n+1)
\cong\mathbb{Z}_2.$$
