We design a supersymmetric $SU(5)$ grand unified theory model using $\Delta (54)$, a finite non-Abelian subgroup of $SU(3{)}_f$. Heavy right-handed neutrinos are introduced which transform as a three-dimensional representation of our chosen family group. The model successfully reproduces the hierarchical structures of the standard model and the Cabibbo-Kobayashi-Maskawa mixing matrix. It then provides predictions for the light neutrinos with a normal hierarchy and masses such that $m_{\nu ,1}\approx 5\times {10}^{-3}\mathrm{eV}$, $m_{\nu ,2}\approx 1\times {10}^{-2}\mathrm{eV}$, and $m_{\nu ,3}\approx 5\times {10}^{-2}\mathrm{eV}$. We also provide predictions for masses of the heavy neutrinos, and corrections to the tribimaximal matrix that fit within experimental limits, e.g., a reactor angle of $-7.31°$. A simple modification to our model is introduced at the end and is shown to also produce predictions that fall well within those limits.

• Received 7 May 2011

DOI:https://doi.org/10.1103/PhysRevD.84.073009