We extend the investigation of the recently introduced class Sk of four-dimensional N=1 superconformal field theories by considering a large family of quiver gauge theories within it, which we denote Sk1. These theories admit a realization in terms of Zk orbifolds of Type IIA configurations of D4-branes stretched among relatively rotated sets of NS-branes. This fact permits a systematic investigation of the full family, which exhibits new features such as nontrivial anomalous dimensions differing from free field values and novel ways of gluing theories. We relate these ingredients to properties of compactification of the six-dimensional (1,0) superconformal TNk theories on spheres with different kinds of punctures. We describe the structure of dualities in this class of theories upon exchange of punctures, including transformations that correspond to Seiberg dualities, and exploit the computation of the superconformal index to check the invariance of the theories under them

The work of H. H. and A. U. is supported by the Spanish Ministry of Economy and Competitiveness under Grant No. FPA2012-32828, Consolider-CPAN (No. CSD2007-00042), Grant No. SEV-2012-0249 of the Centro de Excelencia Severo Ochoa Programme, SPLE Advanced Grant under Contract No. ERC-2012- ADG_20120216-320421 and Contract “UNILHC” No. PITN-GA-2009-237920 of the European Commission. The work of H. H. is supported also by the REA Grant Agreement No. PCIG10-GA-2011-304023 from the People Programme of FP7 (Marie Curie Action)