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$G(3,m,n)$ is the group presented by $\langle a,b\mid a^5={\left(ab\right)}^2=b^{m+3}{a}^{-n}{b}^m{a}^{-n}=1\rangle$. In this paper, we study the structure of $G(3,m,n)$. We also give a new efficient presentation for the Projective Special Linear group $PSL(2,5)$ and in particular we prove that $PSL(2,5)$ is isomorphic to $G(3,m,n)$ under certain conditions.