AbstractThe topic of this paper is the construction of a finite set of generators for a subgroup of finite index in the unit group u(ℤG) of the integral group ring of a finite group G. The present paper is a continuation of earlier research by Bass and Milnor, Jespers and Leal, and Ritter and Sehgal who constructed such generators provided that the group G does not have a non-abelian fixed-point free epimorphic image and the rational group algebra ℚG does not have simple epimorphic images that are two-by-two matrices over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra. In this paper we allow simple images of the type M₂(ℚ). We will do so by introducing new additional generators using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of PSL₂(ℤ). Furthermore, for each simple Wedderburn component M₂(ℚ) of ℚG, the new generators give a free subgroup that is embedded in M₂(ℤ).