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 formula_1 of formula_2 by its natural module of order formula_3. 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 formula_4 as the subgroup fixing a certain lattice in the Lie algebra of formula_4, 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 formula_6 by its natural module formula_7 splits if formula_8, and showed that it also splits if formula_9 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: 