Let $p=2$ or 3, and let $k$ be an algebraically closed field of char. $p.$ Let $E$ be the supersingular elliptic curve over $k$ (with $j=0$). Let $G$ be the automorphism group of $E,$ which has order 12 (resp. 24) when $p=3$ (resp. 2). Then the $\ell$-adic cohomology $H^1(E,\mathbb Q_{\ell})$ is a 2-dimensional representation of $G.$ Is it irreducible?

Since we know the group structure of $G$ (cf. Silverman's AEC, Appendix A, exercise..), does anyone have a reference for its character table?