@slager2013space; @Chiu2013Classification; @Benalcazar2014Classification; @Varjas2015Bulk; @Shiozaki2015Z2; @Cho2015Topological; @Yang2015Topological; @wang2016hourglass; @Ezawa2016Hourglass; @Varjas2017Space; @Yang2017Topological; @Wieder2018Wallpaper; @Bouhon2017Global; @bouhon2017bulk; @Kruthoff2017Topological; @kruthoff2017topology]. Many of these proposed topological phases have been measured in various experiments [@hsieh2012topological; @dziawa2012topological; @tanaka2012experimental; @xu2012observation] in materials such as $\mathrm{PbTe}$, $\mathrm{Pb_{1-x}Sn_xTe}$, and $\mathrm{Pb_{1-x}Sn_xSe}$. Other crystalline topological phases host exotic surface behaviour such as hinge and corner states [@parameswaran2017topological; @Benalcazar2017Quantized; @Benalcazar2017Electric; @Song2017d; @Langbehn2017Reflection; @Schindler2018Higher; @schindler2018bismuth; @xu2017topological; @Shapourian2018Topological; @lin2017topological; @Ezawa2018Higher; @Khalaf2018Higher; @Geier2018Second; @trifunovic2018higher; @fang2017rotation; @okuma2018topological]. Such higher-order topological insulators and superconductors have been also recently experimentally observed in bismuth by Schindler *et al.* \[\]. A recent survey by Vergniory *et al.* \[\] had found that a staggering 24% of materials in nature have some nontrivial band structure topology.

In this paper we engage the vital challenge of achieving a complete classification of all possible topological phases in presence of all possible material symmetries.

A major milestone in our understanding of topological phases came with the discovery of the periodic table of topological insulators and superconductors [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @Hasan2010Colloquium; @moore2010birth; @stone2010symmetries; @Teo2010Topological; @franz2013topological; @witten2015three; @Chiu2016Classification]; see Table \[tab:per\]. The table provides a complete classification of all topological phases of non-interacting fermions in presence of non-spatial symmetries. The presence and absence of these symmetries are categorized 