Dempwolff group

In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 2¹⁵·3²·5·7·31, that is the unique nonsplit extension ${\displaystyle 2^{5\,.}\mathrm {GL} _{5}(\mathbb {F} _{2})}$ of ${\displaystyle \mathrm {GL} _{5}(\mathbb {F} _{2})}$ by its natural module of order ${\displaystyle 2^{5}}$. The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group ${\displaystyle E_{8}}$ as the subgroup fixing a certain lattice in the Lie algebra of ${\displaystyle E_{8}}$, and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

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