non-spatial symmetry actions on the Hilbert space are given by ${B=\langle{{\mathrm{Cl}}}_{p+1,q},\Pi(G)\rangle}$, which denotes the algebra generated by the elements of the extended non-spatial symmetry actions, ${{\mathrm{Cl}}}_{p+1,q}$, as well as by the projective spatial symmetries, $\Pi(G)$.

We focus on the 32 possible point group symmetries of 3D crystals [@graph]. These are determined by an abstract group structure, $G$, and an action on 3D space, ${G\subset\mathrm{O}(3)}$. We identify the algebra extension structure of each point group symmetry within each AZ symmetry class and construct the appropriate classifying space and its bulk topological invariants which are manifested by model Hamiltonians.

In the absence of crystalline symmetry, such bulk topological invariants always manifest in edge modes due to the bulk-boundary correspondence [@franz2013topological; @Kane2005Topological; @Kane2005Quantum; @FuKaneMele2007Topological; @Moore2007Topological; @Hsieh2009Observation; @Roy2009Topological; @Fu2007Topological; @Konig2007Quantum; @xia2009observation; @Chen2009Experimental; @Khalaf2018Symmetry]. However, in previously treated crystalline symmetries, it was noted that such crystalline bulk invariants may also either host no edge modes [@Hughes2011Inversion; @Shiozaki2014Topology], or in fact host higher-order [@parameswaran2017topological; @Benalcazar2017Quantized; @Benalcazar2017Electric; @Song2017d; @Langbehn2017Reflection; @Schindler2018Higher; @schindler2018bismuth; @xu2017topological; @Shapourian2018Topological; @lin2017topological; @Ezawa2018Higher; @Khalaf2018Higher; @Geier2018Second; @trifunovic2018higher; @fang2017rotation] hinge or corner protected topological modes. Possible treatment of the higher-order topological insulators and superconductors which correspond to the bulk invariants presented in this paper are discussed in 