three generations of ${\bf
  10}$-plets, $T$ and $T_3$, transform as ${\bf 2} + {\bf 1}$ under $S_4$, while the ${\bf \bar{5}}$-plets $F$ and the three gauge singlets $N$ form triplets ${\bf 3}$ under $S_4$. At LO, two flavon doublets $\Phi^u_2$ and $\wt\Phi^u_2$ are responsible for up quark masses, while three flavon multiplets $\Phi^d_2 \sim {\bf 2}$, $\Phi^d_3 \sim {\bf 3}$, $\wt\Phi^d_3 \sim
{\bf 3}$ are necessary for generating down quark and charged lepton masses as well as the quark mixing angles. Right-handed neutrinos become massive when the three flavon multiplets $\Phi^\nu_1 \sim {\bf 1}$, $\Phi^\nu_2 \sim {\bf 2}$ and $\Phi^\nu_{3'} \sim {\bf 3'}$ acquire non-vanishing vacuum expectation values (VEVs). The specific alignment of the latter preserves a $Z_2 \times Z_2$ subgroup of the group $S_4$ and leads to the prediction of TB mixing in the neutrino sector. The model contains three $SU(5)$ Higgs fields $H_{5}$, $H_{\ol{5}}$ and $H_{\ol{45}}$ being singlets of $S_4$. With the correct vacuum, the latter is responsible for the GJ relations. Although the model is formulated at the scale of grand unification, we shall not be concerned with the details of $SU(5)$ breaking which we assume to be unrelated to flavour physics. The particle content and its 