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.