The dihedral group $C_{nv}$ is isomorphic to a subgroup of $\SO(3)$.

The matrices $$ C_n\colon=\left(\begin{array}{ccc} \cos(2\pi/n)&-\sin(2\pi/n)&0\\ \sin(2\pi/n)&\cos(2\pi/n)&0\\ 0&0&1 \end{array}\right) \quad\mbox{and}\quad \s\colon=\left(\begin{array}{ccc} 1&0&0\\ 0&-1&0\\ 0&0&-1 \end{array}\right) $$ satisfy: $C_n^n=\s^2=1$ and $\s C_n=C_n^{-1}\s$.