@witten2015three] of topological insulators and superconductors, which follows from the Atiyah-Bott-Shapiro construction [@kitaev2009periodic; @atiyah1964clifford]; see Table \[tab:per\]. This classification of Dirac Hamiltonians captures the bulk topological invariants corresponding to strong topological phases [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @franz2013topological; @witten2015three; @abramovici2012clifford], and it is these bulk strong topological invariants we herein generalize to incorporate spatial crystalline symmetry. Possible future analyses of weak [@FuKaneMele2007Topological; @Moore2007Topological; @Varjas2017Space] and fragile [@Po2018Fragile; @bouhon2018wilson; @bradlyn2018disconnected] crystalline topological phases are discussed in Sec. \[sec:discussion\].

$$\begin{array}{l||c|c||c|c|c||c|c|c||}
    &
    \parbox{\fgw}{\includegraphics[width=\fgw]{C1_3D.png}} & 
    \parbox{\fgw}{\includegraphics[width=\fgw]{Ci_3D.png}} & 
    \parbox{\fgw}{\includegraphics[width=\fgw]{C2_3D.png}} & 
    \parbox{\fgw}{\includegraphics[width=\fgw]{Cs_3D.png}} & \parbox{\fgw}{\includegraphics[width=\fgw]{C2h_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D2_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{C2v_3D.png}} & \parbox{\fgw}{\includegraphics[height=\fgb]{D2h_3D.png}}
    \\ \hline \hline
    \mathrm{Sch\ddot{o}n.}  & C_1 & C_i,S_2 & C_2,D_1 & C_s,C_{1h},C_{1v} & C_{2h},D_{1d} & D_2 & C_{2v},D_{1h} & D_{2h}
    \\ \hline
    \mathrm{HM} & 1 & \bar 1 & 2 & \mathrm{m} & 2/\mathrm{m} & 222 & \mathrm{mm}2 & \mathrm{mmm}
    \\ \hline\hline
    & {\mathcal{C}}_{q+1} & {\mathcal{C}}_{q} & {\mathcal{C}}_{q+1}^2 & {\mathcal{C}}_{q} & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q+1}^4 & {\mathcal{C}}_{q}^2 & {\mathcal{C}}_{q}^4
    \\ \hline\hline
    \mathrm{A}      & 0     & {\mathbb{Z}}& 0         & {\mathbb{Z}}& {\mathbb{Z}}^2     & 0         & {\mathbb{Z}}^2     & {\mathbb{Z}}^4                         
    \\ \hline
    \mathrm{AIII}   & {\mathbb{Z}}& 0     & {\mathbb{Z}}^2     & 0     & 0         & {\mathbb{Z}}^4     & 0         & 0                                 
    \\ \hline\hline
    & {\mathcal{R}}_{q-3} & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-3}^2 & {\mathcal{R}}_{q-4} & {\mathcal{R}}_{q-4}^2 & {\mathcal{R}}_{q-3}^4 & 