transformation properties under $SU(5) \times S_4 \times U(1)$ are summarised in table \[particles\] together with the new field $\eta$ which we assume to transform as a trivial singlet under $S_4$. Note that we fix the $U(1)$ charges in the present note to be $(x,y,z)=(5,4,1)$, since it has been shown in [@HKL10] that a model with such charges does not suffer from operators which significantly perturb the LO results of fermion masses and mixing and/or the vacuum alignment.

$$\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||c|}\hline
\text{Field}\!\!\phantom{\Big|} & T_3 & T & F & N & H^{}_{5} & H_{\ol{5}} & H_{\ol{45}} &  \Phi^u_2 & \wt\Phi^u_2 & \Phi^d_3 & \wt\Phi^d_3 & \Phi^d_2  & \Phi^\nu_{3'} & \Phi^\nu_2 & \Phi^\nu_1 & \eta\\\hline
\!SU(5)\!\!\!\phantom{\Big|} & \bf 10 & \bf 10 & \bf \ol 5 & \bf 1 &\bf  5 &\bf \ol 5 &\bf \ol{45}
&\bf 1&\bf 1&\bf 1&\bf 1&\bf 1&\bf 1&\bf 1&\bf 1& \bf 1\\\hline
S_4\!\!\phantom{\Big|} & \bf 1&\bf 2&\bf 3&\bf 3&\bf 1&\bf 1&\bf 1&\bf 2&\bf 2&\bf 3&\bf 3&\bf
2&\bf 3'&\bf 2&\bf 1 & \bf 1\\ \hline
U(1)\!\!\phantom{\Big|} & 0&5&4&-4&0&0&1&\!-10\!&0&\!-4\!&\!-11\! &1 &8&8&8& 7 \\\hline
\end{array}$$

The leading operators of the matter superpotential are (order one coefficients are suppressed with the exception of the terms involving right-handed neutrinos $N$) \[LOup\] && T\_3T\_3H\^\_[5]{} + 