Yes. The character table of SL(2,q) [is known][1]. If q is odd, the sum of the dimensions of the irreducible representations of SL(2,q) is q<sup>2</sup>+q. If q is even, this sum is q<sup>2</sup>. In either case, this is bigger than the dimension of $\mathbb{C}\{X\}$, so there must exist irreps not in $\mathbb{C}\{X\}$.


  [1]: https://www2.bc.edu/mark-reeder/SL(2,q).pdf