with every " → " serving as certain distinctive normal covering. This condition can be satisfied unless G is simple (Z p of prime order). Within proceeding development, we will only consider G as finite groups or those space groups of the translation groups which are of finite order, thus for our purposes, all space groups can be regarded as solvable, while a few finite groups may not. For space groups, it is considered first to reduce their lattice group Z k of finite dimension k as X 1 = X/Z k , since translational symmetry is always trivial in crystalline phases. These normal covering series will play a vital role in the following calculation of persistent homology over quotient spaces, but involving inconsistent global symmetry by transforming kernel ker p # (G i ) in Proposition 5.2. It will then be applied to practical scenarios where symmetry rules paticipates in distinctive levels. Proposition 4.3. (p is a normal covering here) The Proposition 4.1 indicates that K l /G = (K/G) l , thus: