Table 1 Isotropy subgroups of periodic solutions from ${\mathbf{S}}_4\times {\mathbf{Z}}_2(\rho )$ symmetry. We use the notation $e_{\theta }(t)=e(t+\theta T)$ where $e(t)$ is T-periodic. Moreover, the frequency of u is three times the frequency of a, $v\approx -3a$, and the frequency of c is twice the frequency of a

${\Sigma }_0$

$\left(\begin{array}{cccc}{a}_0& a_0& a_0& a_0\\ a_{1/2}& a_{1/2}& a_{1/2}& a_{1/2}\end{array}\right)$

Figure 8(a)

${\Sigma }_1$

$\left(\begin{array}{cccc}{a}_0& a_{1/2}& a_{1/2}& a_0\\ a_{1/2}& a_0& a_0& a_{1/2}\end{array}\right)$

Figures 10(a) or 10(b) or 10(c)

${\Sigma }_2$

$\left(\begin{array}{cccc}c& a_0& a_{1/2}& c\\ c& a_{1/2}& a_0& c\end{array}\right)$

Fusion

${\Sigma }_3$

$\left(\begin{array}{cccc}{a}_0& a_{1/4}& a_{2/4}& a_{3/4}\\ a_{2/4}& a_{3/4}& a_0& a_{1/4}\end{array}\right)$

Figure 11

${\Sigma }_4$

$\left(\begin{array}{cccc}{a}_0& a_{2/6}& a_{4/6}& u_0\\ a_{3/6}& a_{5/6}& a_{1/6}& u_{1/2}\end{array}\right)$

Complicated transitions

${\Sigma }_5$

$\left(\begin{array}{cccc}{a}_0& a_0& a_0& v_0\\ a_{1/2}& a_{1/2}& a_{1/2}& v_{1/2}\end{array}\right)$

Figure 12