 Irreducible complex representations of a finite dihedral group

For $D_{2n}=⟨r,s:r^n=s^2=1,srs=r^{-1}⟩$, every irreducible complex representation has degree at most two. If $v$ is an eigenvector of $r$ with eigenvalue $\lambda$, then $sv$ has eigenvalue ${\lambda }^{-1}$, so irreducibility makes the representation equal to $\mathrm{span}\{v,sv\}$.