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 of by its natural module of order . 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 as the subgroup fixing a certain lattice in the Lie algebra of , 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 by its natural module splits if , and showed that it also splits if 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 is a maximal subgroup of the Chevalley group . The nonsplit extension is a maximal subgroup of the sporadic Conway group Co3. The nonsplit extension is a maximal subgroup of the Thompson sporadic group Th. References External links Dempwolff group at the atlas of groups. Finite groups