On some flag-transitive non-classical \(c. C_2\)-geometries. II (Q1368015)

The generalized quadrangle \(T^*_2 (O_4)\) of Tits type, which is constructed from the (essentially unique) hyperoval \(O_4\) in the projective plane \(PG (2,4)\), is remarkable, because its automorphism group \(2^6: (3S_6)\) is flag-transitive, but \(T^*_2 (O_4)\) is not a Moufang quadrangle [compare \textit{S. Yoshiara}, Eur. J. Comb. 14, 59-77 (1993; Zbl 0769.51003)]. In the paper under review, the author undertakes the laborious task to determine all flag-transitive automorphism groups of \(T^*_2 (O_4)\), and he obtains precisely 35 conjugacy classes of such groups. This classification is used to prove that there are no flag-transitive circular extensions of \(T^*_2 (O_4)\) or of its dual other than those constructed by the author in loc. cit. As a consequence, there exist (up to isomorphy) precisely two flag-transitive extensions of \(T^*_2 (O_4)\), and precisely three flag-transitive extensions of the dual of \(T^*_2 (O_4)\), and these extensions are known explicitly.