Some non-split extensions of the orthogonal group \(\Omega(7,3)\) (Q1970978)

The author constructs two non-split extensions of the orthogonal group \(\Omega(7,3)\). The first non-split extension is \(3^7\cdot\Omega(7,3):2\) which can be realized as a maximal subgroup of Fischer's largest 3-transposition group \(F_{24}\). The second non-split extension is \(3\cdot\Omega(7,3):2\) which is the full automorphism group of the \(27\times 2\)-dimensional Conway-Norton algebra for \(\Omega(7,3)\). For those constructions the author uses the Moufang loop of order 81 which has been introduced by \textit{R. L. Griess}, jun. [J. Algebra 131, No. 1, 281-293 (1990; Zbl 0718.17028)].