Characters and Brauer trees of the covering group of $^2E_6(2)$

TL;DR

This paper computes the character tables of certain covering groups of the finite simple group $^2E_6(2)$ and determines Brauer trees for related groups, expanding understanding of their modular representation theory.

Contribution

It provides the first complete character tables for specific central extensions of $^2E_6(2)$ and classifies all Brauer trees for related groups with known character tables.

Findings

Character tables of 3.G, 6.G, (2^2×3).G and their automorphism extensions are determined.

All Brauer trees for groups Z.G.A with known character tables are classified.

Results enhance understanding of modular representations of these groups.

Abstract

Let $G$ be the finite simple Chevalley group of type $^2E_6(2)$. 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 $Z.G \lhd Z.G.A$ and $A \cong Z.G.A/Z.G$) for which the ordinary character table is known.