on two generation methods for the simple linear group psl(3,7)

A finite group g is said to be (l, m, n)-generated, if it is a quotient group of the triangle group t(l, m, n) = x, y, z|x^ l = y^ m = z^ n = xyz = 1  . in [j. moori, (p, q, r)-generations for the janko groups j1 and j2, nova j. algebra and geometry, 2 (1993), no. 3, 277–285], moori posed the question of finding all the (p, q, r) triples, where p, q and r are prime numbers, such that a non-abelian finite simple group g is (p, q, r)-generated. also for a finite simple group g and a con[1]jugacy class x of g, the rank of x in g is defined to be the minimal number of elements of x generating g. in this paper we investigate these two generational prob[1]lems for the group p sl(3, 7), where we will determine the (p, q, r)-generations and the ranks of the classes of p sl(3, 7). we approach these kind of generations using the structure constant method. gap [the gap group, gap – groups, algorithms, and programming, version 4.9.3; 2018. (http://www.gap-system.org)] is used in our computations.