The (p, q, r)-generations of the Chevalley group $$G_{2}(3)$$

Abstract A finite group G is called ( l , m , n ) -generated , if it is a quotient group of the triangle group $$T(l,m, n) = \left.$$ T ( l , m , n ) = x , y , z | x l = y m = z n = x y z = 1 . In [23], Moori posed the question of finding all the ( p , q , r ) triples, where $$p,\ q$$ p , q and r are prime numbers, such that a non-abelian finite simple group G is a ( p , q , r )-generated. In answering this question, we establish all the ( p , q , r )-generations for the group $$G_{2}(3).$$ G 2 ( 3 ) . We mainly used the structure constant method together with other results to establish the generation and non-generation of the $$G_{2}(3)$$ G 2 ( 3 ) by the triples ( p , q , r ). The Groups, Algorithms and Programming, GAP [21] and the Atlas of finite group representations [27] are used in our computations.