Maximal subgroups of ²E₆(2) and its automorphism groups

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We give a new computer-assisted proof of the classification of maximal subgroups of the simple group ²E₆(2) and its extensions by any subgroup of the outer automorphism group S₃. This is not a new result, but no earlier proof exists in the literature. A large part of the proof consists of a computational analysis of subgroups generated by an element of order 2 and an element of order 3. This method can be effectively automated, and via statistical analysis also provides a sanity check on results that may have been obtained by delicate theoretical arguments.