Abstract In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-Abelian group G for which every nonlinear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary Abelian 2-group. Key Words: Free and finite index subgroupsIntegral group ringUnit groupAMS Subject Classification: 16S3420C0516O6020E05 ACKNOWLEDGMENT Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek-Vlaanderen (Belgium) and Bilateral Scientific and Technological Cooperation BWS 05/07 (Flanders-Poland). A. Dooms is a postdoctoral Onderzoeker van het Fonds voor Wetenschappelijk Onderzoek-Vlaanderen (Belgium). M. Ruiz's research partially supported by the Fundación Séneca of Murcia and D.G.I. of Spain. Notes Communicated by M. Ferrero.