Horizontal Symmetries $\Delta(150)$ and $\Delta(600)$

Using group theory of mixing to examine all finite subgroups of SU(3) with an order less than 512, we found recently that only the group $\Delta(150)$ can give rise to a correct reactor angle $\th_{13}$ of neutrino mixing without any free parameter. It predicts $\sin^22\th_{13}=0.11$ and a sub-maximal atmospheric angle with $\sin^22\th_{23}=0.94$, in good agreement with experiment. The solar angle $\th_{12}$, the CP phase $\d$, and the neutrino masses $m_i$ are left as free parameters. In this article we provide more details of this case, discuss possible gain and loss by introducing right-handed symmetries, and/or valons to construct dynamical models. A simple model is discussed where the solar angle agrees with experiment, and all its mixing parameters can be obtained from the group $\Delta(600)$ by symmetry alone. The promotion of $\Delta(150)$ to $\Delta(600)$ is on the one hand analogous to the promotion of $S_3$ to $S_4$ in the presence of tribimaximal mixing, and on the other hand similar to the extension from $A_4$ to $S_4$ in that case.