\hline
    \mathrm{CI}     & {\mathbb{Z}}& {\mathbb{Z}}^3             & 0             & 0                 & {\mathbb{Z}}^5     & {\mathbb{Z}}_2             & {\mathbb{Z}}& 0 
    \end{array}$$

$$\begin{array}{l||c||c|c|c|c|c||}
    \multicolumn{2}{l||}{} &
    \parbox{\fgw}{\includegraphics[height=\fgb]{C3_3D.png}} & 
    \parbox{\fgw}{\includegraphics[height=\fgb]{S6_3D.png}} & 
    \parbox{\fgw}{\includegraphics[height=\fgb]{D3_3D.png}} & 
    \parbox{\fgw}{\includegraphics[height=\fgb]{C3v_3D.png}} & 
    \parbox{\fgw}{\includegraphics[height=\fgb]{D3d_3D.png}}
    \\ \hline\hline
    \mathrm{Sch\ddot{o}n.}  & C_1 & C_3 & C_{3i},S_6 & D_3 & C_{3v} & D_{3d}
    \\ \hline
    \mathrm{HM} & 1 & 3 & \bar 3 & 32 & 3\mathrm{m} & \bar{3}\mathrm{m}
    \\ \hline\hline
    & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^3 & {\mathcal{C}}_{q}^3 & {\mathcal{C}}_{q+1}^3 & {\mathcal{C}}_{q}\times{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^3
    \\ \hline\hline
    \mathrm{A}      & 0     & 0                 & {\mathbb{Z}}^3             & 0         & {\mathbb{Z}}& {\mathbb{Z}}^3
    \\ \hline
    \mathrm{AIII}   & {\mathbb{Z}}& {\mathbb{Z}}^3             & 0                 & {\mathbb{Z}}^3     & {\mathbb{Z}}& 0
    \\ \hline\hline
    & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}\times{\mathcal{C}}_{q+1} & {\mathcal{R}}_{q-4}\times{\mathcal{C}}_{q} & {\mathcal{R}}_{q-3}^3 & {\mathcal{R}}_{q-4}\times{\mathcal{R}}_{q-5} & {\mathcal{R}}_{q-4}^3
    \\ \hline\hline
    \mathrm{AI}     & 0     & 0                 & {\mathbb{Z}}^2             & 0         & {\mathbb{Z}}& {\mathbb{Z}}^3
    \\ \hline
    \mathrm{BDI}    & 0     & {\mathbb{Z}}& 0                 & 0         & {\mathbb{Z}}& 0
    \\ \hline
    \mathrm{D}      & 0     & 0                 & {\mathbb{Z}}& 0         & 0                 & 0
    \\ \hline
    \mathrm{DIII}   & {\mathbb{Z}}& {\mathbb{Z}}^2             & 0                 & {\mathbb{Z}}^3     & 0                 & 0
    \\ \hline
    \mathrm{AII}    & {\mathbb{Z}}_2 & {\mathbb{Z}}_2             & {\mathbb{Z}}^2             & {\mathbb{Z}}_2^3   & {\mathbb{Z}}& {\mathbb{Z}}^3 
    \\ \hline
    \mathrm{CII}    & {\mathbb{Z}}_2 & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}_2             & {\mathbb{Z}}_2^3   & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}_2^3       
    \\ \hline
    \mathrm{C}      