Let $V$ be the $p$ dimensional vector space of functions from $\mathbb{F}_p $ to the $\mathbb{C}$ and let $\hat{f}$ be the discrete fourier transform of $f$. There exists a representation, namely, the Weil Representation, $\rho: SL_2(\mathbb{F}_p) \to GL(V) $ where $\rho (\omega)(f) = \hat{f} $
where $\omega= \left( {\begin{array}{cc} 0 &1 \\ -1 & 0 \\ \end{array} } \right) \ $.
I was curious if there were other (interesting) group representations $V$ for which some element acted as the fourier transform.