## Abstract

We consider the finite exceptional group of Lie type G=E_{6} ^{ε}(q) (universal version) with 3|q−ε, where E_{6} ^{+1}(q)=E_{6}(q) and E_{6} ^{−1}(q)=^{2}E_{6}(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

Original language | English |
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Pages (from-to) | 4020-4039 |

Number of pages | 20 |

Journal | Journal of Pure and Applied Algebra |

Volume | 222 |

Issue number | 12 |

DOIs | |

Publication status | Published - 1 Dec 2018 |