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The transformation matrix

Consider the space group G = {(W, w)} referred to a coordinate system defined by the set of basis vectors (a b c) and an origin O. The transformation of the space group data to other basis (a' b' c') with origin O', can be performed by a matrix pair (P,p). The square (3x3) matrix P describes the change of the basis:
(a' b' c') = (a b c) P,

and the column p describes the origin shift
O'= O + p

In order to obtain the symmetry operations (W',w') of G referred to the new basis, the following relation has to be applied:

(W',w') = (P,p)-1 (W,w) (P,p)

Each group-subgroup chain G > H is characterized by the transformation matrix (P,p) which gives the relation between the group (a b c) and the subgroup (a' b' c') reference (default) coordinate systems:

(a' b' c') = (a b c) P

If the subgroup data HG is given with respect to the reference basis of G, then the transformation to the reference subgroup basis is achieved by applying the relation:

(P,p)-1 HG (P,p) = Href

The same relation transforms the elements of the group G from the reference system of this group to the one of the subgroup:

(P,p)-1 Gref (P,p) = GH

[*] For more information: International Tables for Crystallography. Vol. A, Space Group Symmetry. Ed. Theo Hahn (3rd ed.), Dordrecht, Kluwer Academic Publishers, Section "Transformations in crystallography", 1995.

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