## 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 **H**_{G} 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} **H**_{G} `(P,p)` = **H**_{ref}
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} **G**_{ref} `(P,p)` = **G**_{H}

[*] 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.