International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 11.2, pp. 812816
https://doi.org/10.1107/97809553602060000523 Chapter 11.2. Derivation of symbols and coordinate triplets^{1}^{a}Institut für Mineralogie, Petrologie und Kristallographie, PhilippsUniversität, D35032 Marburg, Germany Chapter 11.2 describes a method for deriving the symbol of a symmetry operation (i.e. its type, the location of its symmetry element, and possibly its glide or screw vector) from its coordinate triplet as given in the `general position' of a space group. The reverse procedure is also explained. A table shows the coordinate triplets and the 3 × 3 matrices for all symmetry operations of crystallographic point groups and for all orientations of the corresponding symmetry elements. 
In the spacegroup tables, all symmetry operations with are listed explicitly. As a consequence, the number of entries under the heading Symmetry operations equals the multiplicity of the general position. For space groups with centred unit cells, may result if the centring translations are applied to the explicitly listed coordinate triplets. In those cases, all w values have been reduced modulo 1 for the derivation of the corresponding symmetry operations (see Section 2.2.9 ). In addition to the tabulated symmetry operations, each space group contains an infinite number of further operations obtained by application of integral lattice translations. In many cases, it is not trivial to obtain the additional symmetry operations (cf. Part 4 ) from the ones listed. Therefore, a general procedure is described below by which symbols for symmetry operations as described in Section 11.1.2 may be derived from coordinate triplets or, more specifically, from the corresponding matrix pairs (W, w). [For a similar treatment of this topic, see Wondratschek & Neubüser (1967).] This procedure may also be applied to cases where space groups are given in descriptions not contained in International Tables. In practice, two cases may be distinguished:

The sense of a pure or screw rotation or of a rotoinversion may be calculated as follows: One takes two arbitrary points and on the rotation axis, having the lower value for the free parameter of the axis. One takes a point not lying on the axis and generates from by the symmetry operation under consideration. One calculates the determinant d of the matrix composed of the components of vectors and . For rotations or screw rotations, the sense is positive for and negative for . For rotoinversions, the sense is positive for and negative for .
Example
According to the example in (b) above, the triplet represents a threefold screw rotation with screw part and screw axis at . To obtain the sense of the rotation, the points and are used as and on the axis and the points 000 and as and outside the axis. The resulting vectors are
A particular symmetry operation is uniquely described by its symbol, as introduced in Section 11.1.2 , and the coordinate system to which it refers. In the examples of the previous section, the symbols have been derived from the coordinate triplets representing the respective symmetry operations. Inversely, the pair (W, w) of the symmetry operation and the coordinate triplet of the image point can be deduced from the symbol.
