Consider the discrete symmetry $C_3$ (rotations by $120^\circ$ leave system invariant). I believe then the matrix describing this action is simply the rotation matrix (in 3d) (clockwise rotation): $$R(120^\circ)=\begin{pmatrix} \cos(120^\circ) & -\sin(120^\circ) & 0\\ \sin(120^\circ) & \cos(120^\circ) & 0\\ 0 & 0 & 1 \end{pmatrix}$$
Consider also the following Permutation matrix, which maps basis vectors $e_1\rightarrow e_2$, $e_2\rightarrow e_3$, $e_3\rightarrow e_1$: $$P=\begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ \end{pmatrix}.$$
I am confused about the following:
Don't these matrices fundamentally describe the same symmetry? Shouldn't they share the same eigenvectors?
Thank you!