The maximal subgroups of the exceptional groups $F_{4}(q)$, $E_{6}(q)$ and $^{2}\!E_{6}(q)$ and related almost simple groups
David Craven
Abstract
Abstract This article produces a complete list of all maximal subgroups of the finite simple groups of type $F_{4}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> , $E_{6}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> and twisted $E_{6}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> over all finite fields. Along the way, we determine the collection of Lie primitive almost simple subgroups of the corresponding algebraic groups. We give the stabilizers under the actions of outer automorphisms, from which one can obtain complete information about the maximal subgroups of all almost simple groups with socle one of these groups. We also provide a new maximal subgroup of $^{2}\!F_{4}(8)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> <mml:none/> <mml:mprescripts/> <mml:none/> <mml:mn>2</mml:mn> </mml:mmultiscripts> <mml:mo>(</mml:mo> <mml:mn>8</mml:mn> <mml:mo>)</mml:mo> </mml:math> , correcting the maximal subgroups for that group from the list of Malle. This provides the first new exceptional groups of Lie type to have their maximal subgroups enumerated for three decades. The techniques are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilinear form on the 27-dimensional minimal module for $E_{6}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> . We provide a collection of supplementary Magma files that prove the author’s computational claims, yielding existence and the number of conjugacy classes of all maximal subgroups mentioned in the text.