Bilbao Crystallographic Server Transformation matrix 
The relation between an arbitrary setting of a space group (given by a set of basis vectors (a, b, c) and an origin O) and a reference (default) coordinate system, defined by the set (a', b', c') and the origin O^{'}, is determined by a (3x4) matrix  column pair (P,p). The (3x3) linear matrix P
P = 

describes the transformation of the row of basis vectors (a, b, c) to the reference basis vectors (a', b', c').
a' = P_{11}a + P_{21}b + P_{31}c 
b' = P_{12}a + P_{22}b + P_{32}c 
c' = P_{13}a + P_{23}b + P_{33}c 
which is often written as
(a', b', c') = (a, b, c)P
The (3x1) column p = (p_{1}, p_{2}, p_{3}) determines the origin shift of O^{'} with respect the origin O:
O^{'} = O + p
The transformation matrix pair (P,p) of a groupsubgroup chain G > H used by the programs on Bilbao Crystallographic Server always describes the transformation from the reference (default) coordinate system of the group G to that of the subgroup H. Let (a,b,c)_{G} be the row of the basis vectors of G and (a',b',c')_{H} the basis row of H. The basis (a',b',c') of H is expressed in the basis (a,b,c)_{G} by the system of equations:
(a', b', c')_{H} = (a, b, c)_{G} P
The column p describes the origin shift between the default origin O_{G} of G to that of H, O^{'}_{H};
O^{'}_{H} = O_{G} + p
In some of the applications on the Bilbao Crystallographic Server the data on the matrixcolumn pair (P,p) is listed in the following concise form:
P_{11}a + P_{21}b + P_{31}c, P_{12}a + P_{22}b + P_{32}c, P_{13}a + P_{23}b + P_{33}c ; p_{1}, p_{2}, p_{3}
Note: As the bases (a', b', c') and (a, b, c) are written as rows, in each of the sums a column of the matrix P is listed. The matrix part of concise form of (P,p) is left empty if there is no change of basis, i.e. if P is (3x3) unit matrix. The 'origin shift' part is empty is there is no origin shift, i.e. p is a column consisting of zeros.
Example: b,c,a ; 0,1/4,1/4
The first column of the linear part of the transformation matrix is (0 1 0). The second column of the linear part of the transformation matrix is (0 0 1). The third column of the linear part of the transformation matrix is (1 0 0) or
the linear part of the transformation matrix is: 
 
and the origin shift is: 

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