International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 144148

As already introduced in Section 1.2.3 , a `symmetry element' (of a given structure or object) is defined as a concept with two components; it is the combination of a `geometric element' (that allows the fixed points of a reduced symmetry operation to be located and oriented in space) with the set of symmetry operations having this geometric element in common (`element set'). The element set of a symmetry element is represented by the socalled `defining operation', which is the simplest symmetry operation from the element set that suffices to identify the geometric element. The alphanumeric and graphical symbols of symmetry elements and the related symmetry operations used throughout the tables of plane (Chapter 2.2 ) and space groups (Chapter 2.3 ) are listed in Tables 2.1.2.1 to 2.1.2.7. For detailed discussion of the definition and symbols of symmetry elements, cf. Section 1.2.3 , de Wolff et al. (1989, 1992) and Flack et al. (2000).
^{†}In the rhombohedral spacegroup symbols (161) and (167), the symbol c refers to the description with `hexagonal axes'; i.e. the glide vector is , along [001]. In the description with `rhombohedral axes', this glide vector is , along [111], i.e. the symbol of the glide plane would be n: cf. Table 1.5.4.4
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^{‡}Glide planes `e' occur in orthorhombic A, C and Fcentred space groups, tetragonal Icentred and cubic F and Icentred space groups. The geometric element of an eglide plane is a plane shared by glide reflections with perpendicular glide vectors, with at least one glide vector along a crystal axis [cf. Section 1.2.3 and de Wolff et al. (1992)]. ^{§}Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector. ^{¶}Only the symbol m is used in the Hermann–Mauguin symbols, for both point groups and space groups. ^{††}The inversion point is a centre of symmetry if n is odd. 
^{†}The graphical symbols of the `e'glide planes are applied to the diagrams of seven orthorhombic A, C and Fcentred space groups, five tetragonal Icentred space groups, and five cubic F and Icentred space groups.
^{‡}Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector. 
^{†}The symbols are given at the upper left corner of the spacegroup diagrams. A fraction h attached to a symbol indicates two symmetry planes with `heights' h and above the plane of projection; e.g. stands for and . No fraction means and (cf. Section 2.1.3.6).
^{‡}The graphical symbols of the `e'glide planes are applied to the diagrams of seven orthorhombic A, C and Fcentred space groups, five tetragonal Icentred space groups, and five cubic F and Icentred space groups. ^{§}Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector. 
^{†}The symbols represent orthographic projections. In the cubic spacegroup diagrams, complete orthographic projections of the symmetry elements around highsymmetry points, such as ; ; , are given as `inserts'.
^{‡}In the space groups (216), (225) and (227), the shortest lattice translation vectors in the glide directions are or and or , respectively. ^{§}Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors, for instance and . The second power of a glide reflection d is a centring vector. ^{¶}The glide vector is half of a centring vector, i.e. one quarter of the diagonal of the conventional bodycentred cell in space groups (220) and (230). 
^{†}
Notes on the `heights' h of symmetry points , , and : (1) Centres of symmetry and , as well as inversion points and on and axes parallel to [001], occur in pairs at `heights' h and . In the spacegroup diagrams, only one fraction h is given, e.g. stands for and . No fraction means and . In cubic space groups, however, because of their complexity, both fractions are given for vertical axes, including and . (2) Symmetries and contain vertical and axes; their and inversion points coincide with the centres of symmetry. This is not indicated in the spacegroup diagrams. (3) Symmetries and also contain vertical and axes, but their and inversion points alternate with the centres of symmetry; i.e. points at h and interleave with or points at and . In the tetragonal and hexagonal spacegroup diagrams, only one fraction for and one for or is given. In the cubic diagrams, all four fractions are listed for ; e.g. (223): : ; : . 
^{†}The symbols for horizontal symmetry axes are given outside the unit cell of the spacegroup diagrams. Twofold axes always occur in pairs, at `heights' h and above the plane of projection; here, a fraction h attached to such a symbol indicates two axes with heights h and . No fraction stands for and . The rule of pairwise occurrence, however, is not valid for the horizontal fourfold axes in cubic space groups; here, all heights are given, including and . This applies also to the horizontal axes and the inversion points located on these axes.

^{†}The dots mark the intersection points of axes with the plane at . In some cases, the intersection points are obscured by symbols of symmetry elements with height ; examples: (203), origin choice 2; (222), origin choice 2; (223); (229); (230).

The alphanumeric symbols shown in Table 2.1.2.1 correspond to those symmetry elements and symmetry operations which occur in the conventional Hermann–Mauguin symbols of point groups and space groups. Further socalled `additional symmetry elements' are described in Sections 1.4.2.3 and 1.5.4.1 , and Tables 1.5.4.3 and 1.5.4.4 show additional symmetry operations that appear in the socalled `extended Hermann–Mauguin symbols' (cf. Section 1.5.4 ). The symbols of symmetry elements (symmetry operations), except for glide planes (glide reflections), are independent of the choice and the labelling of the basis vectors and of the origin. The symbols of glide planes (glide reflections), however, may change with a change of the basis vectors. For this reason, the possible orientations of glide planes and the glide vectors of the corresponding operations are listed explicitly in columns 2 and 3 of Table 2.1.2.1.
In 1992, following a proposal of the Commission on Crystallographic Nomenclature (de Wolff et al., 1992), the International Union of Crystallography introduced the symbol `e' and graphical symbols for the designation of the socalled `double' glide planes. The double or eglide plane occurs only in centred cells and its geometric element is a plane shared by glide reflections with perpendicular glide vectors related by a centring translation (for details on eglide planes, cf. Section 1.2.3 ). The introduction of the symbol e for the designation of doubleglide planes (cf. de Wolff et al., 1992) results in the modification of the Hermann–Mauguin symbols of five orthorhombic groups:

Since the introduction of its use in IT A (2002) the new symbol is the standard one; it is indicated in the headline of these space groups, while the former symbol is given underneath.
The graphical symbols of symmetry planes are shown in Tables 2.1.2.2 to 2.1.2.4. Like the alphanumeric symbols, the graphical symbols and their explanations (columns 2 and 3) are independent of the projection direction and the labelling of the basis vectors. They are, therefore, applicable to any projection diagram of a space group. The alphanumeric symbols of glide planes (column 4), however, may change with a change of the basis vectors. For example, the dashdotted n glide in the hexagonal description becomes an a, b or c glide in the rhombohedral description. In monoclinic space groups, the `parallel' vector of a glide plane may be along a lattice translation vector that is inclined to the projection plane.
The `e'glide graphical symbols are applied to the diagrams of seven orthorhombic A, C and F centred space groups, five tetragonal Icentred space groups, and five cubic F and Icentred space groups. The `doubledotteddash' symbol for e glides `normal' and `inclined' to the plane of projection was introduced in 1992 (de Wolff et al., 1992), while the `doublearrowed' graphical symbol for eglide planes oriented `parallel' to the projection plane had already been used in IT (1935) and IT (1952).
The graphical symbols of symmetry axes and their descriptions are shown in Tables 2.1.2.5–2.1.2.7. The screw vectors of the defining operations of screw axes are given in units of the shortest lattice translation vectors parallel to the axes. The symbols in the last column of the tables indicate the symmetry elements that are represented by the graphical symbols in the symmetryelement diagrams of the space groups. Two main cases may be distinguished:
The last six entries of Table 2.1.2.5 are combinations of symbols of symmetry axes with that of a centre of inversion. When displayed on the spacegroup diagrams, the combined graphical symbols represent more than one symmetry element. For example, the symbol for a fourfold rotation axis with a centre of inversion (4/m), represents the symmetry elements , 4 and .
The meaning of a graphical symbol on the spacegroup diagrams is often confused with the set of symmetry elements that constitute the sitesymmetry group associated with the symmetry element displayed. As an example, consider the rotoinversion axis (described as `Inversion axis: 6 bar' in Table 2.1.2.5). The sitesymmetry group can be decomposed into three symmetry elements: , 3 and m (cf. de Wolff et al., 1989). However, the graphical symbol of in the diagrams represents the two symmetry elements and 3, as the symmetry element `m' (that `belongs' to ) is represented by a separate graphical symbol.
References
International Tables for Crystallography (2002). Vol. A, 5th ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT A (2002).]International Tables for Xray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]
Flack, H. D., Wondratschek, H., Hahn, Th. & Abrahams, S. C. (2000). Symmetry elements in space groups and point groups. Addenda to two IUCr Reports on the Nomenclature of Symmetry. Acta Cryst. A56, 96–98.
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.
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Senechal, M., Shoemaker, D. P., Wondratschek, H., Hahn, Th., Wilson, A. J. C. & Abrahams, S. C. (1989). Definition of symmetry elements in space groups and point groups. Report of the International Union of Crystallography Adhoc Committee on the Nomenclature of Symmetry. Acta Cryst. A45, 494–499.