# The Hurwitz Subgroups of $E_6(2)$

## TL;DR

This paper proves that the group $E_6(2)$ is not a Hurwitz group and classifies all its Hurwitz subgroups, including those isomorphic to $L_2(8)$ and $L_3(2)$, up to conjugacy.

## Contribution

It establishes that $E_6(2)$ is not a Hurwitz group and completes the classification of its Hurwitz subgroups, providing new insights into their structure.

## Key findings

- $E_6(2)$ is not a Hurwitz group
- Complete classification of Hurwitz subgroups of $E_6(2)$
- Identification of subgroups isomorphic to $L_2(8)$ and $L_3(2)$

## Abstract

We prove that the exceptional group $E_6(2)$ is not a Hurwitz group. In the course of proving this, we complete the classification up to conjugacy of all Hurwitz subgroups of $E_6(2)$, in particular, those isomorphic to $L_2(8)$ and $L_3(2)$.