This paper continues previous work, based on systematic use of a formula of L. Scott, to detect Hurwitz groups. It closes the problem of determining the finite simple groups contained in $PGL_n(F)$ for $nleq 7$ which are Hurwitz, where $F$ is an algebraically closed field. For the groups $G_2(q)$, $qgeq 5$, and the Janko groups $J_1$ and $J_2$ it provides explicit $(2,3,7)$-generators.
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