& $\Uparrow$\
$\Bigg($ &

  ----------
   Clifford
   algebra
  ----------

& $\hookrightarrow$ &

  ------------------
       Extended
   Clifford algebra
  ------------------

& $\Bigg)$ & $\mapsto$ &

  ---------------
   All Hilbert-
   space actions
  ---------------

\
\
& ${{\mathrm{Cl}}}_{p,q}$ & & ${{\mathrm{Cl}}}_{p+1,q}$ & & & ${\mathcal{R}}_{p-q+2}$

In this paper we generalize this classification scheme as to include the effects of all point group symmetries of crystalline matter.

One highly successful approach towards solving the classification problem is the use of symmetry indicators of elementary band representations [@Dong2016Classification; @po2017symmetry; @bradlyn2017topological; @Watanabe2018Structure; @Bradlyn2018Band; @song2018quantitative; @Cano2018Building; @Vergniory2017Graph; @Ono2018Unified; @vergniory2018high; @Khalaf2018Symmetry; @Khalaf2018Higher]. In this approach, the “symmetry indicators", which are a generalization of the Fu-Kane [@Fu2007Topological; @Fu2011Topological] formula ${\mathbb{Z}}_2$ invariant, are used to analyse the band structure into elementary band representation of the corresponding crystallographic group. It is shown that bands which form one subpart of a disconnected elementary band representation must always be topological [@Cano2018Building]. This is a very prolific classification scheme which is used to categorize numerous topological crystalline phases. However, this technique requires one to perform Berry-phase analyses of individual electronic Bloch wavefunctions along various Wilson loops in order to detect different topological phases which correspond to the same band representation, e.g., the integer quantum Hall effect.

An alternate 