and AIII) from the classifying spaces of the real AZ classes.

### Point Group Symmetries {#sec:pgs}

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                                ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](Ci_3D.png "fig:"){height="\fgya"}   ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](C6_3D.png "fig:"){height="\fgya"}   ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](Cs_3D.png "fig:"){height="\fgya"}   ![Solids with either inversion, rotation, reflection, or rotoreflection symmetry. The generators of the symmetry are also specified.[]{data-label="fig:PiG"}](S6_3D.png "fig:"){height="\fgya"}
                                                                                                                           Inversion                                                                                                                                                                                         Rotation                                                                                                                                                                                         Reflection                                                                                                                                                                                      Rotoreflection
  $G$                                                                                                                     $I^2\!=\!1$                                                                                                                                                                                     $(c_n)^n\!=\!1$                                                                                                                                                                                  $\sigma^2\!=\!1$                                                                                                                                                                                $(s_{2n})^{2n}\!=\!1$
  $\Pi(G)\!\!\!\!\!\!\!\!\!$                                                                                           $\hat{I}^2\!=\!1$                                                                                                                                                                              $(\hat{c}_n)^n\!=\!-1$                                                                                                                                                                            $\hat{\sigma}^2\!=\!-1$                                                                                                                                                                     $(\hat{s}_{2n})^{2n}\!=\!-(-1)^{n}$
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The point groups are generated by: inversions $I$, rotations $c_n$ by $\frac{2\pi}{n}$, reflections ${\sigma=Ic_2}$, and rotoreflections ${s_{2n}=c_{2n}\sigma_h}$, where $\sigma_h,\sigma_v$ are horizontal and vertical reflections.

Importantly, one must take into account the fermionic nature of the electrons and hence the projective spinor representation $\Pi(G)$; see Eq. (\[spinor\]). This spinor representation is constructed from subgroups of the Pin group, which is a double cover of the orthogonal group \[just as the Spin group, ${\mathrm{Spin}(3)\cong\mathrm{SU}(2)}$, is a double cover of the special orthogonal group\]: $$\begin{array}{ccccc}
\{\pm1\} & \hookrightarrow & \mathrm{SU}(2) & 