On the extension of $\mathrm {G}_{2}(3^{2n+1})$ by the exceptional graph automorphism

Abstract

The main aim of this paper is to compute the character table of $\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$, where $\sigma$ is the graph automorphism of $\mathrm{G}_{2}(3^{2n+1})$ such that the fixed-point subgroup $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$ is the Ree group of type $\mathrm{G}_{2}$. As a consequence we explicitly construct a perfect isometry between the principal $p$-blocks of $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$ and $\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$ for prime numbers dividing $q^2-q+1$.