Irreducible (2,3,7)-subgroups of PGLn(F), n⩽7

Let F be an algebraically closed field of characteristic p⩾0. For n⩽7 we classify (up to multiplication by scalar matrices) the similarity invariants of the triples (x,y,xy)∈SLn3(F) such that x2, y3 and 7(xy) are scalar, and 〈x,y〉 is an irreducible subgroup of SLn(F). Moreover, for those triples which are rigid, we determine the isomorphism type of the projective image of 〈x,y〉, provided p>0. This leads to the discovery of new Hurwitz groups and gives a significant contribution to the classification of the Hurwitz subgroups of PGLn(F), n⩽7, which is our final goal.