Exceptional supersymmetric standard models with non-abelian discrete family symmetry

We introduce a non-Abelian discrete Δ27 family symmetry into the recently proposed classes of Exceptional Supersymmetric Standard Model (E6SSM) based on a broken E6 Grand Unified Theory (GUT) in order to solve the flavour problem in these models and in particular to account for tri-bimaximal neutrino mixing. We consider both the minimal version of the model (the ME6SSM) with gauge coupling unification at the string scale and the E6SSM broken via the Pati-Salam chain with gauge coupling unification at the conventional GUT scale. In both models there are low energy exotic colour triplets with couplings suppressed by the symmetries of the model, including the family symmetry. This leads to suppressed proton decay and long lived TeV mass colour triplet states with striking signatures at the LHC.