Why must such a group be dihedral? I tend to think dihedral groups are easy to recognize, but I don't quite see why if _G_ is a quotient of $$U = \langle x, y, z : x^2 = y^2 = z^2 = 1, yx=xy, zy=yz \rangle$$ and _G_ has order 4 mod 8 (so, 4, 12, 20, etc.) then _G_ must in fact be a dihedral group. This is related to my previous question on coset graphs having 4-cycles, and nearly confirms my suspicion about dihedral groups.

We think of $U$ as $U = y \times (x * z) \cong C_2 \times (C_2 * C_2)$.

In any finite factorgroup $\langle x,z\rangle$ is isomorphic to some dihedral group, say $D_{2n}$. Either $y\in\langle x,z\rangle$ in the factorgroup and we are done, or the quotient is isomorphic to $D_{2n}\times C_2$, which is isomorphic to $D_{4n}$ whenever $n$ is odd.