Some Non-split Extensions of the Orthogonal Group Ω(7, 3)

We will give an explicit construction of the non-split extension 37 • Ω(7, 3) : 2, which is a maximal subgroup of Fischerʹs largest 3-transposition group F24. In the proof of non-splitness, we use some Moufang loop of order 81. By using this loop, we can also construct another non-split extension 3 • Ω(7, 3) : 2.