$nX$-Complementary Generations of the Chevalley Group $G_{2}(3)$

‎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 $x\in G$‎, ‎there exists an element $y \in nX$ such that $G=\langle x,y\rangle$‎. ‎We studied the $(p,q,r)$-generation for the Chevalley group $G_{2}(3)$‎, ‎where $p$‎, ‎$q$ and $r$ are all the primes dividing the order of $G_{2}(3)$‎. ‎In the current paper‎, ‎we classify all the non-trivial conjugacy classes of $G_{2}(3)$ 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 $G_{2}(3)$ 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.