   & 0     & 0     & {\mathbb{Z}}& {\mathbb{Z}}& 0     & 0         & 0     & 0     & {\mathbb{Z}}^0     & {\mathbb{Z}}^0     & 0     & 0      \end{array}$$  $$\begin{array}{l||c||c|c|c|c|c|c|c} \mathrm{Sch\ddot{o}n.}  & C_0 & C_0 & S_0 & C_{0h} & D_0 & C_{0v} & D_{0d} & D_{0h} \\ \hline\hline & {\mathcal{C}}_{q}   & {\mathcal{C}}_{q}^0 & {\mathcal{C}}_{q}^0 &{\mathcal{C}}_{q}^0 & {\mathcal{C}}_{q+0}^0\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q+0}^0\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q+0}^0\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q}^0 \\ \hline\hline \mathrm{A}      & {\mathbb{Z}}& {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & {\mathbb{Z}}^0 & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}^0 \\ \hline                            \mathrm{AIII}   & 0     & 0                 & 0                 & 0     & {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & 0 \\ \hline\hline & {\mathcal{R}}_{q-0} & {\mathcal{R}}_{q-0}^0\times{\mathcal{C}}_{q} & {\mathcal{R}}_{q-0}^0\times{\mathcal{C}}_{q} &{\mathcal{C}}_{q}^0 & {\mathcal{R}}_{q-0}^0\times{\mathcal{R}}_{q-0} & {\mathcal{R}}_{q-0}^0\times{\mathcal{R}}_{q-0} & {\mathcal{R}}_{q-0}^0\times{\mathcal{R}}_{q-0} & {\mathcal{R}}_{q-0}^0 \\ \hline\hline \mathrm{AI}     & 0     & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}^0 & {\mathbb{Z}}& {\mathbb{Z}}& {\mathbb{Z}}& 0  \\ \hline \mathrm{BDI}    & 0     & 0                 & 0                 & 0     & 0                 & 0                 & 0                 & 0  \\ \hline \mathrm{D}      & {\mathbb{Z}}& {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & {\mathbb{Z}}^0 & 0                 & 0                 & 0                 & {\mathbb{Z}}^0 \\ \hline \mathrm{DIII}   & {\mathbb{Z}}_0 & {\mathbb{Z}}_0^0           & {\mathbb{Z}}_0^0           & 0     & {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & {\mathbb{Z}}_0^0 \\ \hline \mathrm{AII}    & {\mathbb{Z}}_0 & {\mathbb{Z}}_0^0\times{\mathbb{Z}}& {\mathbb{Z}}_0^0\times{\mathbb{Z}}& {\mathbb{Z}}^0 & {\mathbb{Z}}_0^0\times{\mathbb{Z}}& {\mathbb{Z}}_0^0\times{\mathbb{Z}}& {\mathbb{Z}}_0^0\times{\mathbb{Z}}& {\mathbb{Z}}_0^0 \\ \hline \mathrm{CII}    & 0     & 0                 & 0                 & 0     & {\mathbb{Z}}_0^0               & {\mathbb{Z}}_0^0           & {\mathbb{Z}}_0^0           & 0 \\ \hline \mathrm{C}      & {\mathbb{Z}}& {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & {\mathbb{Z}}^0 & {\mathbb{Z}}_0             & {\mathbb{Z}}_0             & {\mathbb{Z}}_0             & {\mathbb{Z}}^0 \\ \hline \mathrm{CI}     & 0     & 0                 & 0                 & 0     & {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & {\mathbb{Z}}^0             & 0 \end{array}$$  $$\begin{array}{l||c||c|c|c|c|c} \mathrm{Sch\ddot{o}n.}  & C_0 & C_0 & C_{0i},S_0 & D_0 & C_{0v} & D_{0d} \\ \hline\hline & {\mathcal{C}}_{q}   & {\mathcal{C}}_{q}^0 & {\mathcal{C}}_{q}^0 & {\mathcal{C}}_{q+0}\times{\mathcal{C}}_{q} & {\mathcal{C}}_{q+0}\times{\mathcal{C}}_{q} & {\mathcal{C}}_