Hurwitz Groups and G₂(q)

Finite factor groups of are called Hurwitz groups. Here we prove that apart from 2 G 2 (3), G 2 (2), G 2 (3) and G 2 (4), all the groups 2 G 2 (3 2n+1 ) and G 2 (q), q = p n , p € P, are Hurwitz groups. For the proof, (2, 3, 7) structure constants are calculated from the character tables [2], [7], and then with the lists of maximal subgroups [8], [5], the existence of generating triples is deduced.