Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})$

See pp. 449--457 of Peter Scott's article The geometries of 3-manifolds for a complete description of all 3-manifolds with finite fundamental group. The article is available on his website. There don't seem to be any with dihedral fundamental groups (see Allen Hatcher's comment below), but the fundamental groups of the prism manifolds are the binary dihedral groups, i.e. non-split central extensions of $D_{2n}\cong \mathbb{Z}/n\rtimes\mathbb{Z}/2$ by $\mathbb{Z}/2$.

A connected sum of the appropriate lens space and $\mathbb{R}P^3$ will have fundamental group $\mathbb{Z}_m \ast \mathbb{Z}_2.$ Otherwise, the only abelian fundamental groups of $3$-manifolds are $\mathbb{Z},$ $\mathbb{Z}^3$ and $\mathbb{Z}/n \mathbb{Z}$ - see Stefan Friedl's notes (introduction to 3-manifolds and their fundamental group), so that rules out interesting direct products.