The $3\times3$ Fourier matrix squares to the index-reversal permutation

If

$$A=\frac1{\sqrt3}\begin{pmatrix}1&1&1\\1&\omega&\omega^2\\1&\omega^2&\omega^4\end{pmatrix},$$

where $\omega=e^{2\pi i/3}$, then $A^2=$