& 4\mathrm{mm} & \bar{4}2\mathrm{m} & 4/\mathrm{mmm}
    \\ \hline\hline
    & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q+1}^4 & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_q^4 & {\mathcal{C}}_{q+1}^5 & {\mathcal{C}}_{q}^2\times{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^2\times{\mathcal{C}}_{q+1} & {\mathcal{C}}_{q}^5
    \\ \hline\hline
    \mathrm{A}      & 0     & 0                 & {\mathbb{Z}}^2             & {\mathbb{Z}}^4             & 0         & {\mathbb{Z}}^2             & {\mathbb{Z}}^2             & {\mathbb{Z}}^5
    \\ \hline
    \mathrm{AIII}   & {\mathbb{Z}}& {\mathbb{Z}}^4             & 0                 & 0                 & {\mathbb{Z}}^5     & {\mathbb{Z}}& {\mathbb{Z}}& 0
    \\ \hline\hline
    & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-3}^2\times{\mathcal{C}}_{q+1} & {\mathcal{R}}_{q-4}\times{\mathcal{R}}_{q-2} & {\mathcal{R}}_{q-4}^2\times{\mathcal{C}}_q & {\mathcal{R}}_{q-3}^5 & {\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-5} & {\mathcal{R}}_{q-4}^2\times{\mathcal{R}}_{q-3}    & {\mathcal{R}}_{q-4}^5
    \\ \hline\hline
    \mathrm{AI}     & 0     & 0                 & {\mathbb{Z}}& {\mathbb{Z}}^3             & 0         & {\mathbb{Z}}^2             & {\mathbb{Z}}^2             & {\mathbb{Z}}^5
    \\ \hline
    \mathrm{BDI}    & 0     & {\mathbb{Z}}& 0                 & 0                 & 0         & {\mathbb{Z}}& 0                 & 0
    \\ \hline
    \mathrm{D}      & 0     & 0                 & {\mathbb{Z}}& {\mathbb{Z}}& 0         & 0                 & 0                 & 0
    \\ \hline
    \mathrm{DIII}   & {\mathbb{Z}}& {\mathbb{Z}}^3             & {\mathbb{Z}}_2             & 0                 & {\mathbb{Z}}^5     & 0                 & {\mathbb{Z}}& 0
    \\ \hline
    \mathrm{AII}    & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2           & {\mathbb{Z}}\times{\mathbb{Z}}_2    & {\mathbb{Z}}^3             & {\mathbb{Z}}_2^5   & {\mathbb{Z}}^2             & {\mathbb{Z}}^2\times{\mathbb{Z}}_2  & {\mathbb{Z}}^5 
    \\ \hline
    \mathrm{CII}    & {\mathbb{Z}}_2 & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2             & {\mathbb{Z}}_2^2           & {\mathbb{Z}}_2^5   & {\mathbb{Z}}_2^2\times{\mathbb{Z}}& {\mathbb{Z}}_2^3           & {\mathbb{Z}}_2^5       
    \\ \hline
    \mathrm{C}      & 0     & 0                 & {\mathbb{Z}}_2\times{\mathbb{Z}}& {\mathbb{Z}}_2^2\times{\mathbb{Z}}& 0         & {\mathbb{Z}}_2^3           & {\mathbb{Z}}_2^2           & {\mathbb{Z}}_2^5
    \\ 