nX-Complementary Generations of the Chevalley Group G2(3)

A finite non-abelian group G is said to be (l, m, n)-generated if it can be generated by two elements x and y such that o(x) = l, o(y) = m and o(xy) = n. Also, G is said to be nX-complementary generated if given an arbitrary non-identity element x ∈ G, there exists an element y ∈ nX such that G = hx, yi. We studied the (p, q, r)-generation for the Chevalley group G2(3), where p, q and r are all the primes dividing the order of G2(3). In the current paper, we classify all the non-trivial conjugacy classes of G2(3) whether they are complementary generators or not. To achieve this, we mainly used the structure constant method together with other results applied to establish generation and non-generation of the group G2(3) by the (p, q, r) triples. Some particular algorithms, as well as the (Gap) programming tool, and the Atlas of finite groups have been exploited in our computations.