& 0                                     
    \\ \hline
    2 & {\mathbb{R}}\hookrightarrow{\mathbb{C}}& {\mathcal{R}}_2=O(2n)/U(n)                              & \mathrm{D}    & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}& 0                         
    \\ \hline
    3 & {\mathbb{C}}\hookrightarrow{\mathbb{H}}& {\mathcal{R}}_3=U(2n)/Sp(n)                             & \mathrm{DIII} & 0     & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 & {\mathbb{Z}}\\ \hline
    4 & {\mathbb{H}}\hookrightarrow{\mathbb{H}}\oplus{\mathbb{H}}& {\mathcal{R}}_4=\prod_{k+m=n} Sp(n)/(Sp(k)\times Sp(m)) & \mathrm{AII}  & {\mathbb{Z}}& 0     & {\mathbb{Z}}_2 & {\mathbb{Z}}_2 
    \\ \hline
    5 & {\mathbb{H}}\oplus{\mathbb{H}}\hookrightarrow M_2({\mathbb{H}})    & {\mathcal{R}}_5=Sp(n)                                   & \mathrm{CII}  & 0     & {\mathbb{Z}}& 0     & {\mathbb{Z}}_2                                     
    \\ \hline
    6 & {\mathbb{H}}\hookrightarrow M_2({\mathbb{C}})             & {\mathcal{R}}_6=Sp(n)/U(n)                              & \mathrm{C}    & 0     & 0     & {\mathbb{Z}}& 0                     
    \\ \hline
    7 & {\mathbb{C}}\hookrightarrow M_2({\mathbb{R}})             & {\mathcal{R}}_7=U(n)/O(n)                               & \mathrm{CI}   & 0     & 0     & 0     & {\mathbb{Z}}\end{array}$$

The results are brought for all point group symmetries in all crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. All groups are brought both in Schönflies (Schön.) notation and in Hermann–Mauguin (HM) notation, they are accompanied by a solid displaying the crystalline symmetry group, taken with permission from Ashcroft and Mermin [@ashcroft1976solid]. Bulk topological invariants for all AZ symmetry classes are presented in Tables \[tab:p1\]-\[tab:m1\]. All classifying spaces for all three-dimensional (3D) point group symmetries in all crystal systems are compactly presented in Table \[tab:class\]. In Appendix \[app:layer\] we use our techniques to calculate all classifying spaces 