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_{1}a+p_{2}b+p_{3}c
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