 Solution

Write the dihedral group as $D_{2n}=⟨r,s:r^n=s^2=e,srs=r^{-1}⟩$. Every reflection has order two, so an element of order three must be a rotation. Such rotations exist exactly when $3\mid n$, and then they are $r^{n/3}$ and $r^{2n/3}$. The answer is therefore