Three generations of Higgses and the cyclic groups

Multi Higgs doublet models are interesting extensions of the Standard Model that can be related to flavor. The reason is that most flavor models usually involve the presence of several additional scalar fields. In this work we present an analysis that shows that for renormalizable flavor models based on the cyclic group of order $N$, if there is one flavored SU(2) double Higgs per generation, the smallest $N$ that can be used to reproduce the Nearest-Neighbor-Interaction texture for the quark mass matrices is N=5. Results for the Higgs spectrum and consistency under $K - \bar{K}$ mixing in a specific model with $Z_5$ are also presented.