International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 5059

One of the aims of the spacegroup tables of Chapter 2.3 is to represent the symmetry operations of each of the 17 plane groups and 230 space groups. The following sections offer a short description of the symbols of the symmetry operations, their listings and their graphical representations as found in the spacegroup tables of Chapter 2.3 . For a detailed discussion of crystallographic symmetry operations and their matrix–column presentation the reader is referred to Chapter 1.2 .
Given the analytical description of the symmetry operations by matrix–column pairs , their geometric meaning can be determined following the procedure discussed in Section 1.2.2 . The notation scheme of the symmetry operations applied in the spacegroup tables was designed by W. Fischer and E. Koch, and the following description of the symbols partly reproduces the explanations by the authors given in Section 11.1.2 of IT A5. Further explanations of the symbolism and examples are presented in Section 2.1.3.9 .
The symbol of a symmetry operation indicates the type of the operation, its screw or glide component (if relevant) and the location of the corresponding geometric element (cf. Section 1.2.3 and Table 1.2.3.1 for a discussion of geometric elements). The symbols of the symmetry operations explained below are based on the Hermann–Mauguin symbols (cf. Section 1.4.1.4), modified and supplemented where necessary.
The notation scheme is extensively applied in the symmetryoperations blocks of the spacegroup descriptions in the tables of Chapter 2.3 . The numbering of the entries of the symmetryoperations block corresponds to that of the coordinate triplets of the general position, and in space groups with primitive cells the two lists contain the same number of entries. As an example consider the symmetryoperations block of the space group shown in Fig. 1.4.2.1. The four entries correspond to the four coordinate triplets of the generalposition block of the group and provide the geometric description of the symmetry operations chosen as coset representatives of with respect to its translation subgroup.
For space groups with conventional centred cells, there are several (2, 3 or 4) blocks of symmetry operations: one block for each of the translations listed below the subheading `Coordinates'. Consider, for example, the four symmetryoperations blocks of the space group Fmm2 (42) reproduced in Fig. 1.4.2.2. They correspond to the four sets of coordinate triplets of the general position obtained by the translations , , and , cf. Fig. 1.4.2.2. The numbering scheme of the entries in the different symmetryoperations blocks follows that of the general position. For example, the geometric description of entry (4) in the symmetryoperations block under the heading `For set' of Fmm2 corresponds to the coordinate triplet , which is obtained by adding to the translation part of the printed coordinate triplet (cf. Fig. 1.4.2.2).
Apart from the notation for the geometric interpretation of the matrix–column representation of symmetry operations discussed in detail in the previous section, there is another notation which has been adopted and is widely used by solidstate physicists and chemists. This is the socalled Seitz notation introduced by Seitz in a series of papers on the matrixalgebraic development of crystallographic groups (Seitz, 1935).
Seitz symbols reflect the fact that spacegroup operations are affine mappings and are essentially shorthand descriptions of the matrix–column representations of the symmetry operations of the space groups. They consist of two parts: a rotation (or linear) part and a translation part . The Seitz symbol is specified between braces and the rotational and the translational parts are separated by a vertical line. The translation parts correspond exactly to the columns of the coordinate triplets of the generalposition blocks of the spacegroup tables. The rotation parts consist of symbols that specify (i) the type and the order of the symmetry operation, and (ii) the orientation of the corresponding symmetry element with respect to the basis. The orientation is denoted by the direction of the axis for rotations or rotoinversions, or the direction of the normal to reflection planes. (Note that in the latter case this is different from the way the orientation of reflection planes is given in the symmetryoperations block.)
The linear parts of Seitz symbols are denoted in many different ways in the literature (Litvin & Kopsky, 2011). According to the conventions approved by the Commission of Crystallographic Nomenclature of the International Union of Crystallography (Glazer et al., 2014) the symbol is 1 and for the identity and the inversion, m for reflections, the symbols 2, 3, 4 and 6 are used for rotations and , and for rotoinversions. For rotations and rotoinversions of order higher than 2, a superscript + or − is used to indicate the sense of the rotation. Subscripts of the symbols denote the characteristic direction of the operation: for example, the subscripts 100, 010 and refer to the directions [100], [010] and , respectively.
Examples
The linear parts R of the Seitz symbols of the spacegroup symmetry operations are shown in Tables 1.4.2.1–1.4.2.3. Each symbol R is specified by the shorthand notation of its (3 × 3) matrix representation (also known as the Jones' faithful representation symbol, cf. Bradley & Cracknell, 1972), the type of symmetry operation and its orientation as described in the corresponding symmetryoperations block of the spacegroup tables of this volume. The sequence of R symbols in Table 1.4.2.1 corresponds to the numbering scheme of the generalposition coordinate triplets of the space groups of the crystal class, while those of Table 1.4.2.2 and Table 1.4.2.3 correspond to the generalposition sequences of the space groups of 6/mmm and (rhombohedral axes) crystal classes, respectively.



The same symbols R can be used for the construction of Seitz symbols for the symmetry operations of subperiodic layer and rod groups (Litvin & Kopsky, 2014), and magnetic groups, or for the designation of the symmetry operations of the point groups of space groups. [One should note that the Seitz symbols applied in the first and second editions of IT E and in the IUCr ebook on magnetic groups (Litvin, 2012) differ from the standard symbols adopted by the Commission of Crystallographic Nomenclature.]
The Seitz symbols for plane groups are constructed following similar rules to those for space groups. The rotation part R is 1 for the identity, m for reflections, and 2, 3, 4 and 6 are used for rotations. The orientation of a reflection line is specified by a subscript indicating the direction of its `normal'. Obviously, the direction indicators are of no relevance for the rotation points. The linear parts R of the Seitz symbols of the planegroup symmetry operations are shown in Tables 1.4.2.4 and 1.4.2.5. Each symbol R is specified by the shorthand notation of its (2 × 2) matrix representation, the type of symmetry operation and, if applicable, its orientation as described in the corresponding symmetryoperations block of the planegroup tables of this volume. The sequence of R symbols in Table 1.4.2.4 corresponds to the numbering scheme of the generalposition coordinate doublets of the plane group p4mm, while those of Table 1.4.2.5 correspond to the generalposition sequence of the plane group p6mm. The same symbols R can be used for the construction of Seitz symbols for the symmetry operations of subperiodic frieze groups (Litvin & Kopsky, 2014).


As illustrated in the examples above, zero translations are normally specified by a single zero in the Seitz symbols, but in cases where it is unclear whether the symbol refers to a space or a planegroup symmetry operation, an explicit indication of the components of the translation vector is recommended.
From the description given above, it is clear that Seitz symbols can be considered as shorthand modifications of the matrix–column presentation of symmetry operations discussed in detail in Chapter 1.2 : the translation parts of and coincide, while the different (3 × 3) matrices W are represented by the symbols R shown in Tables 1.4.2.1–1.4.2.3. As a result, the expressions for the product and the inverse of symmetry operations in Seitz notation are rather similar to those of the matrix–column pairs discussed in detail in Chapter 1.2 :
Similarly, the action of a symmetry operation on the column of point coordinates x is given by [cf. Chapter 1.2, equation (1.2.2.4) ].
The rotation parts of the Seitz symbols partly resemble the geometricdescription symbols of symmetry operations described in Section 1.4.2.1 and listed in the symmetryoperation blocks of the spacegroup tables of this volume: R contains the information on the type and order of the symmetry operation, and its characteristic direction. The Seitz symbols do not directly indicate the location of the symmetry operation, nor its glide or screw component, if any.
The classifications of space groups introduced in Chapter 1.3 allow one to reduce the practically unlimited number of possible space groups to a finite number of spacegroup types. However, each individual spacegroup type still consists of an infinite number of symmetry operations generated by the set of all translations of the space group. A practical way to represent the symmetry operations of space groups is based on the coset decomposition of a space group with respect to its translation subgroup, which was introduced and discussed in Section 1.3.3.2 . For our further considerations, it is important to note that the listings of the general position in the spacegroup tables can be interpreted in two ways:
With reference to a conventional coordinate system, the set of symmetry operations of a space group is described by the set of matrix–column pairs . The set of all translations forms the translation subgroup , which is a normal subgroup of of finite index [i]. If is a fixed symmetry operation, then all the products = = of translations with have the same rotation part W. Conversely, every symmetry operation of with the same matrix part W is represented in the set . The infinite set of symmetry operations is called a coset of the right coset decomposition of with respect to , and its coset representative. In this way, the symmetry operations of can be distributed into a finite set of infinite cosets, the elements of which are obtained by the combination of a coset representative and the infinite set of translations (cf. Section 1.3.3.2 ):where is omitted. Obviously, the coset representatives of the decomposition represent in a clear and compact way the infinite number of symmetry operations of the space group . Each coset in the decomposition is characterized by its linear part and its entries differ only by lattice translations. The translations form the first coset with the identity as a coset representative. The symmetry operations with rotation part form the second coset etc. The number of cosets equals the number of different matrices of the symmetry operations of the space group. This number [i] is always finite and is equal to the order of the point group of the space group (cf. Section 1.3.3.2 ).
For each space group, a set of coset representatives of the decomposition is listed under the generalposition block of the spacegroup tables. In general, any element of a coset may be chosen as a coset representative. For convenience, the representatives listed in the spacegroup tables are always chosen such that the components , of the translation parts fulfil (by subtracting integers). To save space, each matrix–column pair is represented by the corresponding coordinate triplet (cf. Section 1.2.2.3 for the shorthand notation of matrix–column pairs).
Example
The right coset decomposition of , No. 14 (unique axis b, cell choice 1) with respect to its translation subgroup is shown in Table 1.4.2.6. All possible symmetry operations of are distributed into four cosets:

The coordinate triplets of the generalposition block of (unique axis b, cell choice 1) (cf. Fig. 1.4.2.1) correspond to the coset representatives of the decomposition of the group listed in the first line of Table 1.4.2.6.
When the space group is referred to a primitive basis (which is always done for `P' space groups), each coordinate triplet of the generalposition block corresponds to one coset of , i.e. the multiplicity of the general position and the number of cosets is the same. If, however, the space group is referred to a centred cell, then the complete set of generalposition coordinate triplets is obtained by the combinations of the listed coordinate triplets with the centring translations. In this way, the total number of coordinate triplets per conventional unit cell, i.e. the multiplicity of the general position, is given by the product , where [i] is the index of in and [p] is the index of the group of integer translations in the group of all (integer and centring) translations.
Example
The listing of the general position for the space–group type Fmm2 (42) of the spacegroup tables is reproduced in Fig. 1.4.2.2. The four entries, numbered (1) to (4), are to be taken as they are printed [indicated by (0, 0, 0)+]. The additional 12 more entries are obtained by adding the centring translations to the translation parts of the printed entries [indicated by , and , respectively]. Altogether there are 16 entries, which is announced by the multiplicity of the general position, i.e. by the first number in the row. (The additional information specified on the left of the generalposition block, namely the Wyckoff letter and the site symmetry, will be dealt with in Section 1.4.4.)
The symmetry operations of a space group are conveniently partitioned into the cosets with respect to the translation subgroup. All operations which belong to the same coset have the same linear part and, if a single operation from a coset is given, all other operations in this coset are obtained by composition with a translation. However, not all symmetry operations in a coset with respect to the translation subgroup are operations of the same type and, furthermore, they may belong to element sets of different symmetry elements. In general, one can distinguish the following cases:
In order to distinguish the different cases, a closer analysis of the type of a symmetry operation and its symmetry element is required. These types, however, might be obscured by two obstacles:
These issues can be overcome by decomposing the translation part w of a symmetry operation into an intrinsic translation part which is fixed by the linear part W of and thus parallel to the geometric element of , and a location part , which is perpendicular to the intrinsic translation part. Note that the subspace of vectors fixed by W and the subspace perpendicular to this space of fixed vectors are complementary subspaces, i.e. their dimensions add up to 3, therefore this decomposition is always possible.
The procedure for determining the intrinsic translation part of a symmetry operation is described in Section 1.2.2.4 , and is based on the fact that the kth power of a symmetry operation with linear part W of order k must be a pure translation, i.e. for some lattice translation . The intrinsic translation part of is then defined as .
The difference is perpendicular to and it is called the location part of w. This terminology is justified by the fact that the location part can be reduced to o by an origin shift, i.e. the location part indicates whether the origin of the chosen coordinate system lies on the geometric element of .
The transformation of point coordinates and matrix–column pairs under an origin shift is explained in detail in Sections 1.5.1.3 and 1.5.2.3 , and the complete procedure for determining the additional symmetry operations will be discussed in the context of the synoptic tables in Section 1.5.4 . In this section we will restrict ourselves to a detailed discussion of two examples which illustrate typical phenomena.
Example 1
Consider a space group of type Fmm2 (42). The information on the general position and on the symmetry operations given in the spacegroup tables are reproduced in Fig. 1.4.2.2. From this information one deduces that coset representatives with respect to the translation subgroup are the identity element , a rotation with the c axis as geometric element, a reflection with the plane as geometric element and a reflection with the plane as geometric element (with the indices following the numbering in the table).
Composing these coset representatives with the centring translations , and gives rise to elements in the same cosets, but with different types of symmetry operations and symmetry elements in several cases.
In this example, all additional symmetry operations are listed in the symmetryoperations block of the spacegroup tables of Fmm2 because they are due to compositions of the coset representatives with centring translations.
The additional symmetry operations can easily be recognized in the symmetryelement diagrams (cf. Section 1.4.2.5). Fig. 1.4.2.3 shows the symmetryelement diagram of Fmm2 for the projection along the c axis. One sees that twofold rotation axes alternate with twofold screw axes and that mirror planes alternate with `double' or eglide planes, i.e. glide planes with two glide vectors. For example, the dot–dashed lines at and in Fig. 1.4.2.3 represent the b and c glides with normal vector along the a axis [for a discussion of eglide notation, see Sections 1.2.3 and 2.1.2 , and de Wolff et al., 1992].
Example 2
In a space group of type P4mm (99), representatives of the space group with respect to the translation subgroup are the powers of a fourfold rotation and reflections with normal vectors along the a and the b axis and along the diagonals [110] and (cf. Fig. 1.4.2.4).

Generalposition and symmetryoperations blocks as given in the spacegroup tables for space group P4mm (99). 
In this case, additional symmetry operations occur although there are no centring translations. Consider for example the reflection with the plane as geometric element. Composing this reflection with the translation gives rise to the symmetry operation represented by . This operation maps a point with coordinates to and is thus a glide reflection with the plane as geometric element and as glide vector. In a similar way, composing the other diagonal reflection with translations yields further glide reflections.
These glide reflections are symmetry operations which are not listed in the symmetryoperations block, although they are clearly of a different type to the operations given there. However, in the symmetryelement diagram as shown in Fig. 1.4.2.5, the corresponding symmetry elements are displayed as diagonal dashed lines which alternate with the solid diagonal lines representing the diagonal reflections.
In the spacegroup tables of Chapter 2.3 , for each space group there are at least two diagrams displaying the symmetry (there are more diagrams for space groups of low symmetry). The symmetryelement diagram displays the location and orientation of the symmetry elements of the space group. The generalposition diagrams show the arrangement of a set of symmetryequivalent points of the general position. Because of the periodicity of the arrangements, the presentation of the contents of one unit cell is sufficient. Both types of diagrams are orthogonal projections of the spacegroup unit cell onto the plane of projection along a basis vector of the conventional crystallographic coordinate system. The symmetry elements of triclinic, monoclinic and orthorhombic groups are shown in three different projections along the basis vectors. The thin lines outlining the projection are the traces of the side planes of the unit cell.
Detailed explanations of the diagrams of space groups are found in Section 2.1.3.6 . In this section, after a very brief introduction to the diagrams, we will focus mainly on certain important but very often overlooked features of the diagrams.
The graphical symbols of the symmetry elements used in the diagrams are explained in Section 2.1.2 . The heights along the projection direction above the plane of the diagram are indicated for rotation or screw axes and mirror or glide planes parallel to the projection plane, for rotoinversion axes and inversion centres. The heights (if different from zero) are given as fractions of the shortest translation vector along the projection direction. In Fig. 1.4.2.6 (left) the symmetry elements of (unique axis b, cell choice 1) are represented graphically in a projection of the unit cell along the monoclinic axis b. The directions of the basis vectors c and a can be read directly from the figure. The origin (upper left corner of the unit cell) lies on a centre of inversion indicated by a small open circle. The black lenticular symbols with tails represent the twofold screw axes parallel to b. The cglide plane at height along b is shown as a bent arrow with the arrowhead pointing along c.

Symmetryelement diagram (left) and generalposition diagram (right) for the space group , No. 14 (unique axis b, cell choice 1). 
The crystallographic symmetry operations are visualized geometrically by the related symmetry elements. Whereas the symmetry element of a symmetry operation is uniquely defined, more than one symmetry operation may belong to the same symmetry element (cf. Section 1.2.3 ). The following examples illustrate some important features of the diagrams related to the fact that the symmetryelement symbols that are displayed visualize all symmetry operations that belong to the element sets of the symmetry elements.
Examples
The graphical presentations of the spacegroup symmetries provided by the generalposition diagrams consist of a set of generalposition points which are symmetry equivalent under the symmetry operations of the space group. Starting with a point in the upper left corner of the unit cell, indicated by an open circle with a sign `+', all the displayed points inside and near the unit cell are images of the starting point under some symmetry operation of the space group. Because of the onetoone correspondence between the image points and the symmetry operations, the number of generalposition points in the unit cell (excluding the points that are equivalent by integer translations) equals the multiplicity of the general position. The coordinates of the points in the projection plane can be read directly from the diagram. For all systems except cubic, only one parameter is necessary to describe the height along the projection direction. For example, if the height of the starting point above the projection plane is indicated by a `+' sign, then signs `+', `−' or their combinations with fractions (e.g. , etc.) are used to specify the heights of the image points. A circle divided by a vertical line represents two points with different coordinates along the projection direction but identical coordinates in the projection plane. A comma `,' in the circle indicates an image point obtained by a symmetry operation of the second kind [i.e. with , cf. Section 1.2.2 ].
Example
The generalposition diagram of (unique axis b, cell choice 1) is shown in Fig. 1.4.2.6 (right). The open circles indicate the location of the four symmetryequivalent points of the space group within the unit cell along with additional eight translationequivalent points to complete the presentation. The circles with a comma inside indicate the image points generated by operations of the second kind – inversions and glide planes in the present case. The fractions and signs close to the circles indicate their heights in units of b of the symmetryequivalent points along the monoclinic axis. For example, is a shorthand notation for .
Notes:

References
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Litvin, D. B. (2012). Magnetic Group Tables. IUCr ebook. http://www.iucr.org/publ/978–09553602–20 .
Litvin, D. B. & Kopský, V. (2011). Seitz notation for symmetry operations of space groups. Acta Cryst. A67, 415–418.
Litvin, D. B. & Kopsky, V. (2014). Seitz symbols for symmetry operations of subperiodic groups. Acta Cryst. A70, 677–678.
Müller, U. (2012). Personal communication.
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Suescun, L. & Nespolo, M. (2012). From patterns to space groups and the eigensymmetry of crystallographic orbits: a reinterpretation of some symmetry diagrams in IUCr Teaching Pamphlet No. 14. J. Appl. Cryst. 45, 834–837.
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Hahn, Th., Senechal, M., Shoemaker, D. P., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1992). Symbols for symmetry elements and symmetry operations. Final report of the International Union of Crystallography AdHoc Committee on the Nomenclature of Symmetry. Acta Cryst. A48, 727–732.