Characters and Brauer trees of the covering group of ${}^2{E}_6(2)$

Abstract

Let $G$ be the finite simple Chevalley group of type ${}^2{E}_6(2)$. It has a Schur multiplier of type $C_2^2\times C_3$. We determine the ordinary character tables of the central extensions $3.G$, $6.G$, $(2^2\times 3).G$ of $G$ and their extensions by an automorphism of order $2$, that is $3.G.2$, $6.G.2$ and $(2^2\times 3).G.2$. Furthermore we determine all Brauer trees of all groups of type $Z.G.A$ (where $Z$ is central in $Z.G⊲Z.G.A$ and $A\cong Z.G.A/Z.G$) for which the ordinary character table is known.