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 $$2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})$$ of $$\mathrm{GL}_{5}(\mathbb{F}_{2})$$ by its natural module of order $$2^5$$. The uniqueness of such a nonsplit extension was shown by, and the existence by , who showed using some computer calculations of that the Dempwolff group  is contained in the compact Lie group $$E_{8}$$ as the subgroup fixing a certain lattice  in the Lie algebra of $$E_{8}$$,  and is also contained in the  Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

showed that any extension of $$\mathrm{GL}_{n}(\mathbb{F}_{q})$$ by its natural module $$\mathbb{F}_{q}^{n}$$ splits if $$q>2$$. Note that this theorem does not necessarily apply to extensions of $$\mathrm{SL}_{n}(\mathbb{F}_{q})$$; for example, there is a non-split extension $$5^{3\,.}\mathrm{SL}_{n}(\mathbb{F}_{q})$$, which is a maximal subgroup of the Lyons group. showed that it also splits if $$n$$ is not 3, 4, or 5, and in each of these three cases there is just one non-split extension. These three nonsplit extensions can be constructed as follows:
 * The nonsplit extension $$2^{3\,.}\mathrm{GL}_{3}(\mathbb{F}_{2})$$ is a maximal subgroup of the Chevalley group $$G_{2}(\mathbb{F}_{3})$$.
 * The nonsplit extension $$2^{4\,.}\mathrm{GL}_{4}(\mathbb{F}_{2})$$ is a maximal subgroup of the sporadic Conway group Co3.
 * The nonsplit extension $$2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})$$ is a maximal subgroup of the Thompson sporadic group Th.