gives rise to a decomposition of $S_4$ into three cosets. Specifically, as coset-representants we may choose the three permutations $$\begin{gathered}
\label{3per}
(c_0,c_1,c_2,c_3)\ \to \ (c_0,c_1,c_2,c_3),\   (c_0,c_1,c_3,c_2),\  (c_0,c_2,c_3,c_1).\end{gathered}$$

Now that we have clarified the character of the symmetry group by exploiting the new couplings $c$, we may and will transform back to $g$. In particular, for a given $g$ the three maps (\[3per\]) are $$\begin{gathered}
g\to g,g',\tilde{g},\end{gathered}$$ where $$\begin{gathered}
  \tilde{g}_0\equiv \frac{1}{2}(g_0+g_1+g_2+g_3-1),\qquad \tilde{g}_1\equiv \frac{1}{2}(g_0+g_1-g_2-g_3+1),
  \nonumber\\
 \tilde{g}_2\equiv \frac{1}{2}(-g_0+g_1-g_2+g_3+1),\qquad \tilde{g}_3\equiv \frac{1}{2}(g_0-g_1-g_2+g_3+1).$$ We now summarize and extend the above analysis in the following theorem.

\[theorem3.1\] The group $G$ generated by the maps $$\begin{gathered}
\label{g1}
(g_0,g_1,g_2,g_3)   \to (g_0,g_1,1-g_2,g_3),
\\
\label{g2}
(g_0,g_1,g_2,g_3)   \to (g_0,g_1,g_2,1-g_3),\end{gathered}$$ and $$\begin{gathered}
\label{g3}
g\to J_Ng,\end{gathered}$$ with $J_N$ given by , is isomorphic to $S_4$ and contains the mirror map $$\begin{gathered}
(g_0,g_1,g_2,g_3)   \to (g_1,g_0,g_3,g_2).\end{gathered}$$ Now assume $g$ belongs to the set $\Pi_G$, cf. . Then the $G$-orbit of $g$ belongs to $\Pi_G$ and one has $$\begin{gathered}
\label{sprel}
\sigma (H(g))= \sigma(H(w(g))),\qquad \forall \, w\in G.\end{gathered}$$

We have already shown the first assertion. Next, we recall that $\Pi_r$ is defined by the five linear constraints $$\begin{gathered}
2g_0>-1,\qquad 2g_1>-1,\qquad g_0+g_1+g_2+g_3>0,\qquad g_0+g_1-g_2+g_3>-1,\\
 g_0+g_1+g_2-g_3>-1.\end{gathered}$$ Together with the extra constraint defining $\Pi_G$, i.e., $$\begin{gathered}
g_0+g_1-g_2-g_3>-2,\end{gathered}$$ this yields six constraints that 