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. 4273
doi: 10.1107/97809553602060000922 Chapter 1.4. Space groups and their descriptions
B. Souvignier,^{d} H. Wondratschek,^{e}^{‡} M. I. Aroyo,^{a}^{*} G. Chapuis^{b} and A. M. Glazer^{c}
^{a}Departamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain,^{b}École Polytechnique Fédérale de Lausanne, BSP/Cubotron, CH1015 Lausanne, Switzerland,^{c}Department of Physics, University of Oxford, Parks Road, Oxford, United Kingdom,^{d}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands, and ^{e}Laboratorium für Applikationen der Synchrotronstrahlung (LAS), Universität Karlsruhe, Germany This chapter provides an introduction to the various crystallographic items used for the presentation of the symmetry data in the spacegroup tables of this volume. It starts with a detailed introduction to the Hermann–Mauguin symbols for space, plane and crystallographic point groups, and to their Schoenflies symbols. A description is given of the symbols of the symmetry operations applied in the volume, and their listings in the generalposition and the symmetryoperations blocks. This is followed by analysis of some specific features of the symmetryelement and generalposition graphical representations of space groups. Seitz symbols for crystallographic symmetry operations are discussed and illustrated, along with the socalled additional symmetry operations of space groups, which result from the periodicity of the space groups. The classification of points in direct space into general and special Wyckoff positions, and the study of their sitesymmetry groups and Wyckoff multiplicities are presented in detail. In addition, more advanced topics like Wyckoff sets, eigensymmetry groups and noncharacteristic orbits are treated. The final sections offer a useful introduction to twodimensional sections and projections of space groups and their symmetry properties. 
Space groups describe the symmetries of crystal patterns; the point group of the space group is the symmetry of the macroscopic crystal. Both kinds of symmetry are characterized by symbols of which there are different kinds. In this section the spacegroup numbers as well as the Schoenflies symbols and the Hermann–Mauguin symbols of the space groups and point groups will be dealt with and compared, because these are used throughout this volume. They are rather different in their aims. For the Fedorov symbols, mainly used in Russian crystallographic literature, cf. Chapter 3.3 . In that chapter the Hermann–Mauguin symbols and their use are also discussed in detail. For computeradapted symbols of space groups implemented in crystallographic software, such as Hall symbols (Hall, 1981a,b) or explicit symbols (Shmueli, 1984), the reader is referred to Chapter 1.4 of International Tables for Crystallography, Volume B (2008).
For the definition of space groups and plane groups, cf. Chapter 1.3 . The plane groups characterize the symmetries of twodimensional periodic arrangements, realized in sections and projections of crystal structures or by periodic wallpapers or tilings of planes. They are described individually and in detail in Chapter 2.2 . Groups of one and twodimensional periodic arrangements embedded in twodimensional and threedimensional space are called subperiodic groups. They are listed in Vol. E of International Tables for Crystallography (2010) (referred to as IT E) with symbols similar to the Hermann–Mauguin symbols of plane groups and space groups, and are related to these groups as their subgroups. The space groups sensu stricto are the symmetries of periodic arrangements in threedimensional space, e.g. of normal crystals, see also Chapter 1.3 . They are described individually and in detail in the spacegroup tables of Chapter 2.3 . In the following, if not specified separately, both space groups and plane groups are covered by the term space group.
The description of each space group in the tables of Chapter 2.3 starts with two headlines in which the different symbols of the space group are listed. All these names are explained in this section with the exception of the data for Patterson symmetry (cf. Chapter 1.6 and Section 2.1.3.5 for explanations of Patterson symmetry).
The spacegroup numbers were introduced in International Tables for Xray Crystallography (1952) [referred to as IT (1952)] for plane groups (Nos. 1–17) and space groups (Nos. 1–230). They provide a short way of specifying the type of a space group uniquely, albeit without reference to its symmetries. They are particularly convenient for use with computers and have been in use since their introduction.
There are no numbers for the point groups.
The Schoenflies symbols were introduced by Schoenflies (1891, 1923). They describe the pointgroup type, also known as the geometric crystal class or (for short) crystal class (cf. Section 1.3.4.2 ), of the space group geometrically. The different spacegroup types within the same crystal class are denoted by a superscript index appended to the pointgroup symbol.
Schoenflies derived the point groups as groups of crystallographic symmetry operations, but described these crystallographic point groups geometrically by their representation through axes of rotation or rotoreflection and reflection planes (also called mirror planes), i.e. by geometric elements; for geometric elements of symmetry elements, cf. Section 1.2.3 , de Wolff et al. (1989, 1992) and Flack et al. (2000). Rotation axes dominate the description and planes of reflection are added when necessary. Rotoreflection axes are also indicated when necessary. The orientation of a reflection plane, whether horizontal, vertical or diagonal, refers to the plane itself, not to its normal.
A coordinate basis may be chosen by the user: the basis vectors start at the origin which is placed in front of the user. The basis vector c points vertically upwards, the basis vectors a and b lie more or less horizontal; the basis vector a pointing at the user, b pointing to the user's righthand side, i.e. the basis vectors a, b and c form a righthanded set. Such a basis will be called a conventional crystallographic basis in this chapter. (In the usual basis of mathematics and physics the basis vector a points to the righthand side and b points away from the user.) The lengths of the basis vectors, the inclination of the ab plane relative to the c axis and the angles between the basis vectors are determined by the symmetry of the point group and the specific values of the lattice parameters of the crystal structure.
The letter C is used for cyclic groups of rotations around a rotation axis which is conventionally c. The order n of the rotation is appended as a subscript index: ; Fig. 1.4.1.1(a) represents . The values of n that are possible in the rotation symmetry of a crystal are 1, 2, 3, 4 and 6 (cf. Section 1.3.3.1 for a discussion of this basic result). The axis of an nfold rotoreflection, i.e. an nfold rotation followed or preceded by a reflection through a plane perpendicular to the rotation axis (such that neither the rotation nor the reflection is in general a symmetry operation) is designated by , see Fig. 1.4.1.1(b) for .
The following types of point groups exist:
The description of crystal classes using Schoenflies symbols is intuitive and much more graphic than that by Hermann–Mauguin symbols. It is useful for morphological studies investigating the symmetry of the ideal shape of crystals. Schoenflies symbols of crystal classes are also still used traditionally by physicists and chemists, in particular in spectroscopy and quantum chemistry.
Different space groups of the same crystal class are distinguished by their superscript index, for example ; or .
Schoenflies symbols display the spacegroup symmetry only partly. Therefore, they are nowadays rarely used for the description of the symmetry of crystal structures. In comparison with the Schoenflies symbols, the Hermann–Mauguin symbols are more indicative of the spacegroup symmetry and that of the crystal structures.
The Hermann–Mauguin symbols, abbreviated as HM symbols in the following sections, were proposed by Hermann (1928, 1931) and Mauguin (1931), and introduced to the Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) according to the decision of the corresponding Programme Committee (Ewald, 1930). There are different kinds of HM symbols of a space group. One distinguishes short HM symbols, full HM symbols and extended HM symbols. The full HM symbols will be the basis of this description. They form the most transparent kind of HM symbols and their use will minimize confusion, especially for those who are new to crystallography.
As the name suggests, the short HM symbols are mostly shortened versions of the full HM symbols: some symmetry information of the full HM symbols is omitted such that these symbols are more convenient in daily use. The full HM symbol can be reconstructed from the short symbol. In the extended HM symbols the symmetry of the space group is listed in a more complete fashion (cf. Section 1.5.4 ). They are rarely used in crystallographic practice.
In the next section general features of the HM symbols will be discussed. Thereafter, the HM symbols for each crystal system will be presented in a separate section, because the appearance of the HM symbols depends strongly on the crystal system to which the space group belongs.
The Hermann–Mauguin symbol for a space group consists of a sequence of letters and numbers, here called the constituents of the HM symbol. The first constituent is always a symbol for the conventional cell of the translation lattice of the space group (cf. Section 1.3.2.1 for the definition of the translation lattice); the following constituents, namely rotations, screw rotations, rotoinversions, reflections and glide reflections, are marked by conventional symbols, cf. Table 2.1.2.1 .^{1} Together with the generating translations of the lattice, the set of these symmetry operations forms a set of generating symmetry operations of the space group. The space group can thus be generated from its HM symbol.
The symmetry operations of the constituents are referred to the lattice basis that is used conventionally for the crystal system of the space group. The kind of symmetry operation can be read from its symbol; the orientation of its geometric element, cf. de Wolff et al. (1989, 1992), i.e. its invariant axis or plane normal, can be concluded from the position of the corresponding constituent in the HM symbol, as the examples in the following sections will show. The origin is not specified. It is chosen by the user, who selects it in such a way that the matrices of the symmetry operations appear in the most convenient form. This is often, but not necessarily, the conventional origin chosen in the spacegroup tables of this volume. The choice of a different origin may make other tasks, e.g. the derivation of the space group from its generators, particularly easy and transparent.
The first constituent (the lattice symbol) characterizes the lattice of the space group referred to the conventional coordinate system. (Each lattice can be referred to a lattice basis, also called a primitive basis: the lattice vectors have only integer coefficients and the lattice is called a primitive lattice.) Lattice vectors with noninteger coefficients can occur if the lattice is referred to a nonprimitive basis. In this way similarities and relations between different spacegroup types are emphasized.
The lattice symbol of a primitive basis consists of an uppercase letter P (primitive). Lattices with conventional nonprimitive bases are called centred lattices, cf. Section 1.3.2.4 and Table 2.1.1.2 . For these other letters are used: if the ab plane of the unit cell is centred with a lattice vector , the letter is C; for ca centring [ as additional centring vector] the letter is B, and A is the letter for centring the bc plane of the unit cell by . The letter is F for centring all side faces of the cell with centring vectors , and . It is I (German: innenzentriert) for body centring by the vector and R for the rhombohedral centring of the hexagonal cell by the vectors and . In 1985, the letter S was introduced as a settingindependent `centring symbol' for monoclinic and orthorhombic Bravais lattices (cf. de Woff et al., 1985).
To describe the structure of the HM symbols the introduction of the term symmetry direction is useful.
Definition
A direction is called a symmetry direction of a crystal structure if it is parallel to an axis of rotation, screw rotation or rotoinversion or if it is parallel to the normal of a reflection or glidereflection plane. A symmetry direction is thus the direction of the geometric element of a symmetry operation when the normal of a symmetry plane is used for the description of its orientation.
The corresponding symmetry operations [the element set of de Wolff et al. (1989 & 1992)] specify the type of the symmetry direction. The symmetry direction is always a lattice direction of the space group; the shortest lattice vector in the symmetry direction will be called q.
If q represents both a rotation or screw rotation and a reflection or glide reflection, then their symbols are connected in the HM symbol by a slash `/', e.g. 2/m or 4_{1}/a etc.
The symmetry directions of a space group form sets of equivalent symmetry directions under the symmetry of the space group. For example, in a cubic space group the a, b and c axes are equivalent and form the set of six directions : [100], , [010] etc. Another set of equivalent directions is formed by the eight space diagonals : [111], . If there are twofold rotations around the twelve face diagonals , as in the space group of the crystal structure of NaCl, forms a third set of 12 symmetry directions.^{2}
Instead of listing the symmetry operations (element set) for each symmetry direction of a set of symmetry directions, it is sufficient to choose one representative direction of the set. In the HM symbol, generators for the element set of each representative direction are listed.
It can be shown that there are zero (triclinic space groups), one (monoclinic), up to two (trigonal and rhombohedral) or up to three (most other space groups) sets of symmetry directions in each space group and thus zero, one, two or three representative symmetry directions.
The nontranslation generators of a symmetry direction may include only one kind of symmetry operation, e.g. for twofold rotations 2 in space group P121, but they may also include several symmetry operations, e.g. 2, 2_{1}, m and a in space group C12/m1. To search for such directions it is helpful simply to look at the spacegroup diagrams to find out whether more than one kind of symmetry operation belongs to the generators of a symmetry direction. In general, only the simplest symbols are listed (simplestoperation rule): if we use `>' to mean `has priority', then pure rotations > screw rotations; pure rotations > rotoinversions; reflection .^{3} The space group mentioned above is conventionally called C12/m1 and not or C12/a1 or .
The position of a plane is fixed by one parameter if its orientation is known. On the other hand, fixing an axis of known direction needs two parameters. Glide components also show twodimensional variability, whereas there is only one parameter of a screw component. Therefore, reflections and glide reflections can better express the geometric relations between the symmetry operations than can rotations and screw rotations; reflections and glide reflections are more important for HM symbols than are rotations and screw rotations. The latter are frequently omitted to form short HM symbols from the full ones.
The second part of the full HM symbol of a space group consists of one position for each of up to three representative symmetry directions. To each position belong the generating symmetry operations of their representative symmetry direction. The position is thus occupied either by a rotation, screw rotation or rotoinversion and/or by a reflection or glide reflection.
The representative symmetry directions are different in the different crystal systems. For example, the directions of the basis vectors a, b and c are symmetry independent in orthorhombic crystals and are thus all representative, whereas a and b are symmetry equivalent and thus dependent in tetragonal crystals. All three directions are symmetry equivalent in cubic crystals; they belong to the same set and are represented by one of the directions. Therefore, the symmetry directions and their sequence in the HM symbols depend on the crystal system to which the crystal and thus its space group belongs.
Table 1.4.1.1 gives the positions of the representative latticesymmetry directions in the HM symbols for the different crystal systems.

Examples of full HM symbols are (from triclinic to cubic) , P12/c1, A112/m, F2/d2/d2/d, , , , P3m1, , , , and .
There are crystal systems, for example tetragonal, for which the highsymmetry space groups display symmetry in all symmetry directions whereas lowersymmetry space groups display symmetry in only some of them. In such cases, the symmetry of the `empty' symmetry direction is denoted by the constituent 1 or it is simply omitted. For example, instead of three symmetry directions in P4mm, there is only one in , for which the HM symbol is usually written . However, in some trigonal space groups the designation of a symmetry direction by `1' () is necessary to maintain the uniqueness of the HM symbols.^{4}
The HM symbols can not only describe the space groups in their conventional settings but they can also indicate the setting of the space group relative to the conventional coordinate system mentioned in Section 1.4.1.3.1. For example, the orthorhombic space group may appear as or or or or depending on its orientation relative to the conventional coordinate basis. On the one hand this is an advantage, because the HM symbols include some indication of the orientation of the space group and form a more powerful tool than being just a spacegroup nomenclature. On the other hand, it is sometimes not easy to recognize the spacegroup type that is described by an unconventional HM symbol. In Section 1.4.1.4.5 an example is provided which deals with this problem.
The full HM symbols describe the symmetry of a space group in a transparent way, but they are redundant. They can be shortened to the short HM symbols such that the set of generators is reduced to a necessary set. Examples will be displayed for the different crystal systems. The conventional short HM symbols still provide a unique description and enable the generation of the space group. For the monoclinic space groups with their many conventional settings they are not variable and are taken as standard for their spacegroup types. Monoclinic short HM symbols may look quite different from the full HM symbol, e.g. Cc instead of A1n1 or I1a1 or B11n or I11b.
The extended HM symbols display the additional symmetry that is often generated by lattice centrings. The full HM symbol denotes only the simplest symmetry operations for each symmetry direction, by the `simplest symmetry operation' rule; the other operations can be found in the extended symbols, which are treated in detail in Section 1.5.4 and are listed in Tables 1.5.4.3 (plane groups) and 1.5.4.4 (space groups).
From the HM symbol of the space group, the full or short HM symbol for a crystal class of a space group is obtained easily: one omits the lattice symbol, cancels all screw components such that only the symbol for the rotation is left and replaces any letter for a glide reflection by the letter m for a reflection. Examples are and .
If one is not yet familiar with the HM symbols, it is recommended to start with the orthorhombic space groups in Section 1.4.1.4.5. In the orthorhombic crystal system all crystal classes have the same number of symmetry directions and the HM symbols are particularly transparent. Therefore, the orthorhombic HM symbols are explained in more detail than those of the other crystal systems.
The following discussion treats mainly the HM symbols of space groups in conventional settings; for nonconventional descriptions of space groups the reader is referred to Chapter 1.5 .
There is no symmetry direction in a triclinic space group. Therefore, the basis vectors of a triclinic space group can always be chosen to span a primitive cell and the HM symbols are P1 (without inversions) and (with inversions). The HM symbol is the only one which displays the inversion explicitly. Sometimes nonconventional centred lattice descriptions may be used, especially when comparing crystal structures.
Monoclinic space groups have exactly one symmetry direction, often called the monoclinic axis. The b axis is the symmetry direction of the (most frequently used) conventional setting, called the baxis setting. Another conventional setting has c as its symmetry direction (caxis setting). In earlier literature, the uniqueaxis c setting was called the first setting and the uniqueaxis b setting the second setting (cf. Section 2.1.3.15 ). In addition to the primitive lattice P there is a centred lattice which is taken as C in the baxis setting, A in the caxis setting. The (possible) glide reflections are c (or a). In this volume, more settings are described, cf. Sections 1.5.4 and 2.1.3.15 and the spacegroup tables of Chapter 2.3 .
The full HM symbol consists of the lattice symbol and three possible positions for the symmetry directions. The symmetry in the a direction is described first, followed by the symmetry in the b direction and last in the c direction. The two positions of the HM symbol that are not occupied by the monoclinic symmetry direction are marked by 1. The symbol is thus similar to the orthorhombic HM symbol and the monoclinic axis is clearly visible. P1m1 or P11m may designate the same space group but in different settings. Pm11 is a possible but not conventional setting.
The short HM symbols of the monoclinic space groups are independent of the setting of the space group. They form the monoclinic standard symbols and are not variable: P2, , C2, Pm, Pc, Cm, Cc, P2/m, , C2/m, P2/c, and C2/c. Altogether there are 13 monoclinic spacegroup types.
There are several reasons for the many conventional settings.
To the orthorhombic crystal system belong the crystal classes 222, mm2 and 2/m2/m2/m with the Bravais types of lattices P, C, A, F and I. Four space groups with a P lattice belong to the crystal class 222, ten to mm2 and 16 to 2/m2/m2/m. Each of the basis vectors marks a symmetry direction; the lattice symbol is followed by characters representing the symmetry operations with respect to the symmetry directions along a, b and c.
We start with the full HM symbols. For a space group of crystal class 222 with a P lattice the HM symbol is thus `', where , , = 2 or . Conventionally one chooses a setting with the symbols P222, , and .
For the generation of the space groups of this crystal class only two nontranslational generators are necessary, say and . However, it is not possible to indicate in the HM symbol whether the axes and intersect or not. This is decided by the third (screw) rotation : if , the axes and intersect, if , they do not. For this reason, is sometimes called an indicator. However, any two of the three rotations or screw rotations can be taken as the generators and the third one is then the indicator. Mathematically each element of a generating set is a generator independent of its possible redundancy.
In the space groups of crystal class mm2 the two reflections or glide reflections are the generators, the twofold rotation or screw rotation is generated by composition of the (glide) reflections. The position of the rotation axis relative to the intersection line of the two planes as well as its screw component are determined uniquely by the glide components of the reflections or glide reflections.
The rotation or screw rotation in the HM symbols of space groups of the crystal class mm2 could be omitted, and were omitted in older HM symbols. Nowadays they are included to make the orthorhombic HM symbols more homogeneous. Conventional symbols are, among others, Pmm2, , Pba2 and .
The 16 space groups with a P lattice in crystal class 2/m2/m2/m are similarly obtained by starting with the letter P and continuing with the pointgroup symbol, modified by the possible replacements for 2 and a, b, c or n for m. The conventional symbols are, among others, P2/m2/m2/m, , , or . The symbols and designate different spacegroup types, as is easily seen by looking at the screw rotations: has screw axes in the direction of c only, has screw axes in all three symmetry directions.
If the lattice is centred, the constituents in the same symmetry direction are not unique. In this case, according to the `simplest symmetry operation' rule, in general the simplest operation is chosen, cf. Section 1.5.4 .
Examples
In the HM symbol there are in addition screw rotations in the first two symmetry directions; additional glide reflections b occur in the first, and n in the second and third symmetry directions.
In I2/b2/a2/m, all rotations 2 are accompanied by screw rotations 2_{1}; b and a are accompanied by c and m is accompanied by n. The symmetry operations that are not listed in the full HM symbol can be derived by composition of the listed operations with a centring translation, cf. Section 1.4.2.4.
There are two exceptions to the `simplest symmetry operation' rule. If the I centring is added to the P space groups of the crystal class 222, one obtains two different space groups with an I lattice, each has 2 and 2_{1} operations in each of the symmetry directions. One space group is derived by adding the I centring to the space group P222, the other is obtained by adding the I centring to a space group . In the first case the twofold axes intersect, in the second they do not. According to the rules both should get the HM symbol I222, but only the space group generated from P222 is named I222, whereas the space group generated from is called . The second exception occurs among the cubic space groups and is due to similar reasons, cf. Section 1.4.1.4.8.
The short HM symbols for the space groups of the crystal classes 222 and mm2 are the same as the full HM symbols. In the short HM symbols for the space groups of the crystal class 2/m2/m2/m the symbols for the (screw) rotations are omitted, resulting in the short symbols Pmmm, Pmma, Pmna, Pbam, Pnma, Cmcm and Ibam for the space groups mentioned above.
These are HM symbols of space groups in conventional settings. It is less easy to find the conventional HM symbol and the spacegroup type from an unconventional short HM symbol. This may be seen from the following example:
Question: Given the short HM symbols Pman, Pmbn and Pmcn, what are the conventional descriptions of their spacegroup types, and are they identical or different?
Answer: A glance at the HM symbols shows that the second symbol does not describe any spacegroup type at all. The second symmetry direction is b; the glide plane is perpendicular to it and the glide component may be , or , but not .
In this case it is convenient to define the intersection of the three (glide) reflection planes as the site of the origin. Then all translation components of the generators are zero except the glide components.
There are seven tetragonal crystal classes. The lattice may be P or I. The space groups of the three crystal classes 4, and 4/m have only one symmetry direction, [001]. The other four classes, 422, 4mm, and 4/m2/m2/m display three symmetry directions which are listed in the sequence [001], [100] and .^{5}
In the space groups of the crystal class 4, rotation or screw rotation axes run in direction [001]; in the space groups of crystal class these are rotoinversion axes ; and in crystal class 4/m both occur. The rotation 4 of the point group may be replaced by screw rotations 4_{1}, 4_{2} or 4_{3} in the space groups with a P lattice. If the lattice is Icentred, 4 and 4_{2} or 4_{1} and 4_{3} occur simultaneously, together with rotoinversions.
In the space groups of crystal class 4/m with a P lattice, the rotations 4 can be replaced by the screw rotations 4_{2} and the reflection m by the glide reflection n such that four spacegroup types with a P lattice exist: P4/m, , P4/n and . Two more are based on an I lattice: I4/m and . In all these six space groups the short HM symbols and full HM symbols are the same.
There are four crystal classes with three symmetry directions each. In the corresponding spacegroup symbols the constituents 2, 4 and m may be replaced by , with k = 1, 2 or 3, and a, b, c, n or d, respectively. The constituent persists. Full HM symbols of space groups are, among others, , , and .
The full and short HM symbols agree for the space groups that belong to the crystal classes 422, 4mm and . Only for the space groups of 4/m2/m2/m have the short HM symbols lost their twofold rotations or screw rotations leading, e.g., to the symbol instead of .
Example
In P4mm, to the primary symmetry direction [001] belong the rotation 4 and its powers, to the secondary symmetry direction [100] belongs the reflection . However, in the tertiary symmetry direction , there occur reflections m and glide reflections g with a glide vector . Such glide reflections are not listed in the `symmetry operations' blocks of the spacegroup tables if they are composed of a representing general position and an integer translation, as happens here (cf. Section 1.4.2.4 and Section 1.5.4 for a detailed discussion of the additional symmetry operations generated by combinations with integer translations). Glide reflections may have complicated glide vectors. If these do not fit the labels a, b, c, n or d, they are frequently called g.
Hexagonal and trigonal space groups are referred to a hexagonal coordinate system P with basis vector . The basis vectors a and b span a hexagonal net and form an angle of 120°. The sequence of the representatives of the (up to three) symmetry directions is [001], [100] and . Usually, the seven trigonal space groups of the rhombohedral lattice system (or rhombohedral space groups for short) are described either with respect to a hexagonal coordinate system (triple hexagonal cell) or to a rhombohedral coordinate system (primitive rhombohedral cell).
Trigonal space groups are characterized by threefold rotation or screw rotation or rotoinversion axes in [001]. There may be in addition 2 and 2_{1} axes in [100] or , but only in one of these two directions. The same holds for reflections m or glide reflections c. The different possibilities are:
Hexagonal space groups have either one or three representative symmetry directions. The space groups of crystal classes 6, and 6/m have [001] as their single symmetry direction for the axis 6 or 6_{k} for or , and for the plane m with its normal along [001]. The short and full HM symbols are the same. Examples are P6, , and .
Space groups of crystal classes 622, 6mm, and 6/m2/m2/m have the representative symmetry directions [001], [100] and . As opposed to the trigonal HM symbols, in the hexagonal HM symbols no symmetry direction is `empty' and occupied by `1'.
In space groups of the crystal classes 622, 6mm and the short and full HM symbols are the same; in 6/m2/m2/m the short symbols are deprived of the parts `2/' of the full symbols. The full HM symbol is shortened to the short HM symbol , the full HM symbol is shortened to . The two denote different spacegroup types.
The rhombohedral lattice may be understood as an Rcentred hexagonal lattice and then referred to the hexagonal basis. It has two kinds of symmetry directions, which coincide with the primary and secondary symmetry directions of the hexagonal lattice (owing to the R centrings, no symmetry operation along the tertiary symmetry direction of the hexagonal lattice is compatible with the rhombohedral lattice). On the other hand, the rhombohedral lattice may be referred to a (primitive) rhombohedral coordinate system with the lattice parameters a = b = c and α = β = γ. The HM symbol of a rhombohedral space group starts with R, its representative symmetry directions are or and or . In this section the rhombohedral primitive cell is used. The rotations 3 and the rotoinversions are accompanied by screw rotations 3_{1} and 3_{2}. Rotations 2 about horizontal axes always alternate with 2_{1} screw rotations and reflections m are accompanied by different glide reflections g with unconventional glide components. The additional operations mentioned are not listed in the full HM symbols.
The seven rhombohedral space groups belong to the five crystal classes 3, , 32, 3m and . In R3 and only the first of the symmetry directions is occupied and listed in the full and short HM symbols. In the space groups of the other crystal classes the second symmetry direction is occupied by `2' or `m' or `c' or `2/m' or `2/c', leading to the full HM symbols R32, R3m, R3c, and . In the short HM symbols the `2/' parts of the last two symbols are skipped: and .
There are five cubic crystal classes combined with the three types of lattices P, F and I in which the cubic space groups are classified. The two symmetry directions [100] and [111] are the representative directions in the space groups of the crystal classes 23 and . A third representative symmetry direction, , is added for space groups of the crystal classes 432, and .^{6}
In the full HM symbol the symmetry is described as usual. Examples are , , , , and finally No. 230, . The short HM symbols of the noncentrosymmetric space groups (those of crystal classes 23, 432 and ) are the same as the full HM symbols. In the short HM symbols of centrosymmetric space groups of the crystal classes and the rotations or screw rotations are omitted with the exception of the rotations 3 and rotoinversions which represent the symmetry in direction [111]. Thus, in the examples listed above, , and are the short HM symbols differing from the full HM symbols.
As in the orthorhombic space groups I222 and , there is the pair I23 and in which the `simplest symmetry operation' rule is violated. In both space groups twofold rotations and screw rotations around a, b and c occur simultaneously. In I23 the rotation axes intersect, in they do not. The first space group can be generated by adding the Icentring to the space group P23, the second is obtained by adding the Icentring to the space group .
The principles of the HM symbols for space groups are retained in the HM symbols for plane groups (also known as wallpaper groups). The rotation axes along c of three dimensions are replaced by rotation points in the ab plane; the possible orders of rotations are the same as in threedimensional space: 2, 3, 4 and 6. The lattice (sometimes called net) of a plane group is spanned by the two basis vectors a and b, and is designated by a lowercase letter. The choice of a lattice basis, i.e. of a minimal cell, leads to a primitive lattice p, in addition a ccentred lattice is conventionally used. The nets are listed in Table 3.1.2.1 . The reflections and glide reflections through planes of the space groups are replaced by reflections and glide reflections through lines. Glide reflections are called g independent of the direction of the glide line. The arrangement of the constituents in the HM symbol is displayed in Table 1.4.1.2.

Short HM symbols are used only if there is at most one symmetry direction, e.g. p411 is replaced by p4 (no symmetry direction), p1m1 is replaced by pm (one symmetry direction) etc.
There are four crystal systems of plane groups, cf. Table 3.2.3.1 . The analogue of the triclinic crystal system is called oblique, the analogues of the monoclinic and orthorhombic crystal systems are rectangular. Both have rotations of order 2 at most. The presence of reflection or glide reflection lines in the rectangular crystal system allows one to choose a rectangular basis with one basis vector perpendicular to a symmetry line and one basis vector parallel to it. The square crystal system is analogous to the tetragonal crystal system for space groups by the occurrence of fourfold rotation points and a square net. Plane groups with threefold and sixfold rotation points are united in the hexagonal crystal system with a hexagonal net.
Plane groups occur as sections and projections of the space groups, cf. Section 1.4.5. In order to maintain the relations to the space groups, the symmetry directions of the symmetry lines are determined by their normals, not by the directions of the lines themselves. This is important because the normal of the line, not the direction of the line itself, determines the position in the HM symbol.
The sequence of spacegroup entries in the spacegroup tables follows that introduced by Schoenflies (1891) and is thus established historically. Within each geometric crystal class, Schoenflies numbered the spacegroup types in an obscure way. As early as 1919, Niggli (1919) considered this Schoenflies sequence to be unsatisfactory and suggested that another sequence might be more appropriate. Fedorov (1891) used a different sequence in order to distinguish between symmorphic, hemisymmorphic and asymmorphic space groups (cf. Section 1.3.3.3 for a detailed discussion of symmorphic space groups).
The basis of the Schoenflies symbols and thus of the Schoenflies listing is the geometric crystal class. For the present spacegroup tables, a sequence might have been preferred in which, in addition, spacegroup types belonging to the same arithmetic crystal class were grouped together. It was decided, however, that the longestablished sequence in the earlier editions of International Tables should not be changed.
In Table 1.4.1.3, those geometric crystal classes are listed in which the Schoenflies sequence separates space groups belonging to the same arithmetic crystal class (cf. Section 1.3.4.4 for the definition and discussion of arithmetic crystal classes). The space groups are rearranged in such a way that space groups of the same arithmetic crystal class are grouped together. The arithmetic crystal classes are separated by rules spanning the last three columns of the table and the geometric crystal classes are separated by rules spanning the full width of the table. In all cases not listed in Table 1.4.1.3, the Schoenflies sequence, as used in the spacegroup tables, does not break up arithmetic crystal classes. Nevertheless, some rearrangement would be desirable in other arithmetic crystal classes too. For example, the symmorphic space group should always be the first entry of each arithmetic crystal class.

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:

In group theory, a set of generators of a group is a set of group elements such that each group element may be obtained as a finite ordered product of the generators. For space groups of one, two and three dimensions, generators may always be chosen and ordered in such a way that each symmetry operation can be written as the product of powers of h generators (j = ). Thus,where the powers are positive or negative integers (including zero). The description of a group by means of generators has the advantage of compactness. For instance, the 48 symmetry operations in point group can be described by two generators. Different choices of generators are possible. For the spacegroup tables, generators and generating procedures have been chosen such as to make the entries in the blocks `General position' (cf. Section 2.1.3.11 ) and `Symmetry operations' (cf. Section 2.1.3.9 ) as transparent as possible. Space groups of the same crystal class are generated in the same way (see Table 1.4.3.1 for the sequences that have been chosen), and the aim has been to accentuate important subgroups of space groups as much as possible. Accordingly, a process of generation in the form of a composition series has been adopted, see Ledermann (1976). The generator is defined as the identity operation, represented by (1) . The generators , , and are the translations with translation vectors a, b and c, respectively. Thus, the coefficients , and may have any integral value. If centring translations exist, they are generated by translations (and in the case of an F lattice) with translation vectors d (and e). For a C lattice, for example, d is given by . The exponents (and ) are restricted to the following values:

As a consequence, any translation of with translation vectorcan be obtained as a productwhere are integers determined by . The generators and are enclosed between parentheses because they are effective only in centred lattices.
The remaining generators generate those symmetry operations that are not translations. They are chosen in such a way that only terms or occur. For further specific rules, see below.
The process of generating the entries of the spacegroup tables may be demonstrated by the example in Table 1.4.3.2, where denotes the group generated by . For , the next generator is introduced when , because in this case no new symmetry operation would be generated by . The generating process is terminated when there is no further generator. In the present example, completes the generation: (178).

For the nontranslational generators, the following sequence has been adopted:

For the space groups with lattice symbol P, the generation procedure has given the same triplets (except for their sequence) as in IT (1952). In nonP space groups, the triplets listed sometimes differ from those of IT (1952) by a centring translation.
One of the first tasks in the analysis of crystal patterns is to determine the actual positions of the atoms. Since the full crystal pattern can be reconstructed from a single unit cell or even an asymmetric unit, it is clearly sufficient to focus on the atoms inside such a restricted volume. What one observes is that the atoms typically do not occupy arbitrary positions in the unit cell, but that they often lie on geometric elements, e.g. reflection planes or lines along rotation axes. It is therefore very useful to analyse the symmetry properties of the points in a unit cell in order to predict likely positions of atoms.
We note that in this chapter all statements and definitions refer to the usual threedimensional space , but also can be formulated, mutatis mutandis, for plane groups acting on and for higherdimensional groups acting on ndimensional space .
Since the operations of a space group provide symmetries of a crystal pattern, two points X and Y that are mapped onto each other by a spacegroup operation are regarded as being geometrically equivalent. Starting from a point , infinitely many points Y equivalent to X are obtained by applying all spacegroup operations to X: = .
Definition
For a space group acting on the threedimensional space , the (infinite) setis called the orbit of X under .
The orbit of X is the smallest subset of that contains X and is closed under the action of . It is also called a crystallographic orbit.
Every point in direct space belongs to precisely one orbit under and thus the orbits of partition the direct space into disjoint subsets. It is clear that an orbit is completely determined by its points in the unit cell, since translating the unit cell by the translation subgroup of entirely covers .
It may happen that two different symmetry operations and in map X to the same point. Since implies that , the point X is fixed by the nontrivial operation in .
Definition
The subgroup of symmetry operations from that fix X is called the sitesymmetry group of X in .
Since translations, glide reflections and screw rotations fix no point in , a sitesymmetry group never contains operations of these types and thus consists only of reflections, rotations, inversions and rotoinversions. Because of the absence of translations, contains at most one operation from a coset relative to the translation subgroup of , since otherwise the quotient of two such operations and would be the nontrivial translation (see Chapter 1.3 for a discussion of coset decompositions). In particular, the operations in all have different linear parts and because these linear parts form a subgroup of the point group of , the order of the sitesymmetry group is a divisor of the order of the point group of .
The sitesymmetry group of a point X is thus a finite subgroup of the space group , a subgroup which is isomorphic to a subgroup of the point group of .
Example
For a space group of type , the sitesymmetry group of the origin is clearly generated by the inversion in the origin: . On the other hand, the point is fixed by the inversion in Y, i.e.The symmetry operation also belongs to and generates the sitesymmetry group of Y. The sitesymmetry groups of X and of Y are thus different subgroups of order 2 of which are isomorphic to the point group of (which is generated by ).
The order of the sitesymmetry group is closely related to the number of points in the orbit of X that lie in the unit cell. An application of the orbit–stabilizer theorem (see Section 1.1.7 ) yields the crucial observation that each point in the orbit of X under is obtained precisely times as an orbit point: for each one has = and conversely implies that and thus for an operation in .
Assuming first that we are dealing with a space group described by a primitive lattice, each coset of relative to the translation subgroup contains precisely one operation such that lies in the primitive unit cell. Since the number of cosets equals the order of the point group of and since each orbit point is obtained times, it follows that the number of orbit points in the unit cell is .
If we deal with a space group with a centred unit cell, the above result has to be modified slightly. If there are k − 1 centring vectors, the lattice spanned by the conventional basis is a sublattice of index k in the full translation lattice. The conventional cell therefore is built up from k primitive unit cells (spanned by a primitive lattice basis) and thus in particular contains k times as many points as the primitive cell (see Chapter 1.3 for a detailed discussion of conventional and primitive bases and cells).
Proposition
Let be a space group with point group and let be the sitesymmetry group of a point X in . Then the number of orbit points of the orbit of X which lie in a conventional cell for is equal to the product , where k is the volume of the conventional cell divided by the volume of a primitive unit cell.
As already mentioned, one of the first issues in the analysis of crystal structures is the determination of the actual atom positions. Energetically favourable configurations in inorganic compounds are often achieved when the atoms occupy positions that have a nontrivial sitesymmetry group. This suggests that one should classify the points in into equivalence classes according to their sitesymmetry groups.
Definition
A point is called a point in a general position for the space group if its sitesymmetry group contains only the identity element of . Otherwise, X is called a point in a special position.
The distinctive feature of a point in a general position is that the points in its orbit are in onetoone correspondence with the symmetry operations of the group by associating the orbit point with the group operation . For different group elements and , the orbit points and must be different, since otherwise would be a nontrivial operation in the sitesymmetry group of X. Therefore, the entries listed in the spacegroup tables for the general positions can not only be interpreted as a shorthand notation for the symmetry operations in (as seen in Section 1.4.2.3), but also as coordinates of the points in the orbit of a point X in a general position with coordinates x, y, z (up to translations).
Whereas points in general positions exist for every space group, not every space group has points in a special position. Such groups are called fixedpointfree space groups or Bieberbach groups and are precisely those groups that may contain glide reflections or screw rotations, but no proper reflections, rotations, inversions and rotoinversions.
Example
The group of type (33) has a point group of order 4 and representatives for the nontrivial cosets relative to the translation subgroup are the twofold screw rotation , the a glide and the n glide . No operation in the coset of the twofold screw rotation can have a fixed point, since such an operation maps the z component to for an integer , and this is never equal to z. The same argument applies to the x component of the a glide and to the y component of the n glide, hence this group contains no operation with a fixed point (apart from the identity element) and is thus a fixedpointfree space group.
The distinction into general and special positions is of course very coarse. In a finer classification, it is certainly desirable that two points in the same orbit under the space group belong to the same class, since they are symmetry equivalent. Such points have conjugate sitesymmetry groups (cf. the orbit–stabilizer theorem in Section 1.1.7 ).
Lemma
Let X and Y be points in the same orbit of a space group and let such that . Then the sitesymmetry groups of X and Y are conjugate by the operation mapping X to Y, i.e. one has .
The classification motivated by the conjugacy relation between the sitesymmetry groups of points in the same orbit is the classification into Wyckoff positions.
Definition
Two points X and Y in belong to the same Wyckoff position with respect to if their sitesymmetry groups and are conjugate subgroups of .
In particular, the Wyckoff position containing a point X also contains the full orbit of X under .
Remark: It is built into the definition of Wyckoff positions that points that are related by a symmetry operation of belong to the same Wyckoff position. However, a single sitesymmetry group may have more than one fixed point, e.g. points on the same rotation axis or in the same reflection plane. These points are in general not symmetry related but, having identical sitesymmetry groups, clearly belong to the same Wyckoff position. This situation can be analyzed more explicitly:
Let be the sitesymmetry group of the point X and assume that Y is another point with the same sitesymmetry group . Choosing a coordinate system with origin X, the operations in all have translational part equal to zero and are thus matrix–column pairs of the form . In particular, these operations are linear operations, and since both points X and Y are fixed by all operations in , the vector is also fixed by the linear operations in . But with the vector v each scaling of v is fixed as well, and therefore all the points on the line through X and Y are fixed by the operations in . This shows that the Wyckoff position of X is a union of infinitely many orbits if has more than one fixed point.
Lemma
Let be the sitesymmetry group of X in :

The spacegroup tables of Chapter 2.3 contain the following information about the Wyckoff positions of a space group :

The entries in the last column, the reflection conditions, are discussed in detail in Chapter 1.6 . This column lists the conditions for the reflection indices hkl for which the corresponding structure factor is not systematically zero.
Examples

Points belonging to the same Wyckoff position have conjugate sitesymmetry groups and thus in particular all those points are collected together that lie in one orbit under the space group . However, in addition, points that are not symmetryrelated by a symmetry operation in may still play geometrically equivalent roles, e.g. as intersections of rotation axes with certain reflection planes.
Example
In the conventional setting, the fourfold axes of a space group of type P4 (75) intersect the ab plane in the points and for integers , as can be seen from the spacegroup diagram in Fig. 1.4.4.3.
The points lie in one orbit under the translation subgroup of , and thus belong to the same Wyckoff position, labelled 1a. For the same reason, the points belong to a single Wyckoff position, namely to position 1b. The points and do not belong to the same Wyckoff position, because the sitesymmetry group is generated by the fourfold rotation 4_{001} and conjugating this by an operation results in a fourfold rotation with axis parallel to the c axis and running through w. But since the translation parts of all operations in are integral, such an axis can not contain and thus and are not conjugate in .
However, the translation by conjugates to , while fixing the group as a whole. This shows that there is an ambiguity in choosing the origin either at or , since these points are geometrically indistinguishable (both being intersections of a fourfold axis with the ab plane).
The ambiguity in the origin choice in the above example can be explained by the affine normalizer of the space group (see Section 1.1.8 for a general introduction to normalizers). The full group of affine mappings acts via conjugation on the set of space groups and the space groups of the same affine type are obtained as the orbit of a single group of that type under .
Definition
The group of affine mappings that fix a space group under conjugation is called the affine normalizer of , i.e.The affine normalizer is the largest subgroup of such that is a normal subgroup of .
Conjugation by operations of the affine normalizer results in a permutation of the operations of , i.e. in a relabelling without changing their geometric properties. The additional translations contained in the affine normalizer can in fact be derived from the spacegroup diagrams, because shifting the origin by such a translation results in precisely the same diagram. More generally, an element of the affine normalizer can be interpreted as a change of the coordinate system that does not alter the spacegroup diagrams.
A more thorough description of the affine normalizers of space groups is given in Chapter 3.5 , where tables with the affine normalizers are also provided.
Since the affine normalizer of a space group is in general a group containing as a proper subgroup, it is possible that subgroups of that are not conjugate by any operation of may be conjugate by an operation in the affine normalizer. As a consequence, the sitesymmetry groups and of two points X and Y belonging to different Wyckoff positions of may be conjugate under the affine normalizer of . This reveals that the points X and Y are in fact geometrically equivalent, since they fall into the same orbit under the affine normalizer of . Joining the equivalence classes of these points into a single class results in a coarser classification with larger classes, which are called Wyckoff sets.
Definition
Two points X and Y belong to the same Wyckoff set if their sitesymmetry groups and are conjugate subgroups of the affine normalizer of .
In particular, the Wyckoff set containing a point X also contains the full orbit of X under the affine normalizer of .
Example
Let be the space group of type (17) generated by the translations of an orthorhombic lattice, the twofold rotation and the twofold screw rotation . Note that the composition of these two elements is the twofold rotation with the line as its geometric element. The group has four different Wyckoff positions with a sitesymmetry group generated by a twofold rotation; representatives of these Wyckoff positions are the points (Wyckoff position 2a, sitesymmetry symbol 2..), (position 2b, symbol 2..), (position 2c, symbol .2.) and (position 2d, symbol .2.).
From the tables of affine normalizers in Chapter 3.5 , but also by a careful analysis of the spacegroup diagrams in Fig. 1.4.4.4, one deduces that the affine normalizer of contains the additional translations , and , since all the diagrams are invariant by a shift of along any of the coordinate axes. Moreover, the symmetry operation which interchanges the a and b axes and shifts the origin by along the c axis belongs to the affine normalizer, because it precisely interchanges the twofold rotations around axes parallel to the a and to the b axes. The translation maps to , and hence and have sitesymmetry groups which are conjugate under the affine normalizer of and thus belong to the same Wyckoff set. Analogously, and belong to the same Wyckoff set, because maps to . Finally, the operation found in the affine normalizer maps to . This shows that the points of all four Wyckoff positions actually belong to the same Wyckoff set.

Symmetryelement diagrams for the space group (17) for orthogonal projections along [001], [010], [100] (left to right). 
Geometrically, the positions in this Wyckoff set can be described as those points that lie on a twofold rotation axis.
The assignments of Wyckoff positions of plane and space groups to Wyckoff sets are discussed and tabulated in Chapter 3.4 .
Remark: The previous example deserves some further discussion. The group of type belongs to the orthorhombic crystal family, and the conventional unit cell is spanned by three basis vectors a, b, c with lengths a, b, c and right angles between each pair of basis vectors. Unless the parameters a and b are equal because of some metric specialization, the operation of the affine normalizer is not an isometry but changes lengths. If it is desired that the metric properties are preserved, the full affine normalizer cannot be taken into account, but only the subgroup that consists of isometries. This subgroup is called the Euclidean normalizer of . (A detailed discussion of Euclidean normalizers of space groups and their tabulation are given in Chapter 3.5 .)
Taking conjugacy of the sitesymmetry groups under the Euclidean normalizer as a condition results in a notion of equivalence which lies between that of Wyckoff positions and Wyckoff sets. In the above example, the four Wyckoff positions would be merged into two classes represented by and , but and would not be regarded as equivalent, since they are not related by an operation of the Euclidean normalizer.
It turns out, however, that in many cases this intermediate classification coincides with the Wyckoff sets, because points belonging to different Wyckoff positions are often related to each other by a translation contained in the affine normalizer. Since translations are always isometries, the translations contained in the affine normalizer always belong to the Euclidean normalizer as well.
A crystallographic orbit has been defined as the set of points obtained by applying all operations of some space group to a point . From that it is clear that the set is invariant as a whole under the action of operations in , since for some point in the orbit and one has , which is again contained in because belongs to . However, it is possible that the orbit is also invariant under some isometries of that are not contained in . Since the composition of two such isometries still keeps the orbit invariant, the set of all isometries leaving invariant forms a group which contains as a subgroup.
Definition
Let be the orbit of a point under a space group . Then the group of isometries of which leave invariant as a whole is called the eigensymmetry group of .
Since the orbit is a discrete set, the eigensymmetry group has to be a space group itself. One distinguishes the following cases:

Noncharacteristic orbits are closely related to the concept of lattice complexes, which are discussed in Chapter 3.4 . An extensive listing of noncharacteristic orbits of space groups can be found in Engel et al. (1984).
The fact that an orbit of a space group has a larger eigensymmetry group is an important example of a pair of groups that are in a group–subgroup relation. Knowledge of subgroups and supergroups of a given space group play a crucial role in the analysis of phase transitions, for example, and are discussed in detail in Chapter 1.7 .
The occurrence of noncharacteristic orbits does not require the point X to be chosen at a special position. Even the general position of a space group may give rise to a noncharacteristic orbit. Moreover, special values of the coordinates of the general position may give rise to additional eigensymmetries without the position becoming a special position. Conversely, the orbit of a point at a special position need not be noncharacteristic.
Example
We compare space groups of types (76) and (77). For a space group of type , the general position with generic coordinates gives rise to a characteristic orbit, whereas the generalposition orbit for a space group of type consists of the points , , and . An inversion in 0, 0, z interchanges and , and maps to , which is clearly equivalent to under a translation. This shows that the generalposition orbit for a space group of type is a noncharacteristic orbit, and the eigensymmetry group of this orbit is of type (84), where the origin has to be shifted to the inversion point 0, 0, z to obtain the conventional setting. Since the unit cell and the orbit are unchanged, but the point group of is a subgroup of index 2 in the point group of , the orbit points must belong to a special position for , namely the position labelled 4j. In the conventional setting of , a point belonging to this Wyckoff position is given by x, y, 0 and one finds that the orbit of this point in special position is characteristic, i.e. its eigensymmetry group is just .
If we assume that the metric of the space group is not special, the eigensymmetry group is restricted to the same crystal family (for the definition of `specialized' metrics, cf. Section 1.3.4.3 and Chapter 3.5 ). Therefore, a space group for which the point group is a holohedry can only have noncharacteristic orbits by additional translations, i.e. extraordinary orbits. However, if we allow specialized metrics, the eigensymmetry group may belong to a higher crystal family. For example, if a space group belongs to the orthorhombic family, but the unit cell has equal parameters a = b, then the eigensymmetry group of an orbit can belong to the tetragonal family.
Note: A space group is equal to the intersection of the eigensymmetry groups of the orbits of all its positions. If none of the positions of a space group gives rise to a characteristic orbit, this means that each single orbit under does not have as its symmetry group, but a larger group that contains as a proper subgroup. It may thus be necessary to have the union of at least two orbits under to obtain a structure that has precisely as its group of symmetry operations.
Examples

Knowledge of the eigensymmetry groups of the different positions for a group is of utmost importance for the analysis of diffraction patterns. Atoms in positions that give rise to noncharacteristic orbits, in particular extraordinary orbits, may cause systematic absences that are not explained by the spacegroup operations. These absences are specified as special reflection conditions in the spacegroup tables of this volume, but only as long as no specialization of the coordinates is involved. For the latter case, the possible existence of systematic absences has to be deduced from the tables of noncharacteristic orbits. Reflection conditions are discussed in detail in Chapter 1.6 .
Example
For the group of type Pccm (49) the special position (Wyckoff position 4p) gives rise to an extraordinary orbit, since it allows the additional translation . The special reflection condition corresponding to this additional translation is the integral reflection condition hkl: l = 2n. However, if the z coordinate in position 4p is set to , the eigensymmetry group also contains the translation . In this case, the special reflection condition becomes hkl: l = 4n.
In crystallography, twodimensional sections and projections of crystal structures play an important role, e.g. in structure determination by Fourier and Patterson methods or in the treatment of twin boundaries and domain walls. Planar sections of threedimensional scattering density functions are used for finding approximate locations of atoms in a crystal structure. They are indispensable for the location of Patterson peaks corresponding to vectors between equivalent atoms in different asymmetric units (the Harker vectors).
A twodimensional section of a crystal pattern takes out a slice of a crystal pattern. In the mathematical idealization, this slice is regarded as a twodimensional plane, allowing one, however, to distinguish its upper and lower side. Depending on how the slice is oriented with respect to the crystal lattice, the slice will be invariant by translations of the crystal pattern along zero, one or two linearly independent directions. A section resulting in a slice with twodimensional translational symmetry is called a rational section and is by far the most important case for crystallography.
Because the slice is regarded as a twosided plane, the symmetries of the full crystal pattern that leave the slice invariant fall into two types:
Therefore, the symmetries of twodimensional rational sections are described by layer groups, i.e. subgroups of space groups with a twodimensional translation lattice. Layer groups are subperiodic groups and for their elaborate discussion we refer to Chapter 1.7 and IT E (2010).
Analogous to twodimensional sections of a crystal pattern, one can also consider the penetration of crystal patterns by a straight line, which is the idealization of a onedimensional section taking out a rod of the crystal pattern. If the penetration line is along the direction of a translational symmetry of the crystal pattern, the rod has onedimensional translational symmetry and its group of symmetries is a rod group, i.e. a subgroup of a space group with a onedimensional translation lattice. Rod groups are also subperiodic groups, cf. IT E for their detailed treatment and listing.
A projection along a direction d into a plane maps a point of a crystal pattern to the intersection of the plane with the line along d through the point. If the projection direction is not along a rational lattice direction, the projection of the crystal pattern will contain points with arbitrarily small distances and additional restrictions are required to obtain a discrete pattern (e.g. the cutandproject method used in the context of quasicrystals). We avoid any such complication by assuming that d is along a rational lattice direction. Furthermore, one is usually only interested in orthogonal projections in which the projection direction is perpendicular to the projection plane. This has the effect that spheres in threedimensional space are mapped to circles in the projection plane.
Although it is also possible to regard the projection plane as a twosided plane by taking into account from which side of the plane a point is projected into it, this is usually not done. Therefore, the symmetries of projections are described by ordinary plane groups.
Sections and projections are related by the projection–slice theorem (Bracewell, 2003) of Fourier theory: A section in reciprocal space containing the origin (the socalled zero layer) corresponds to a projection in direct space and vice versa. The projection direction in the one space is normal to the slice in the other space. This correspondence is illustrated schematically in Fig. 1.4.5.1. The top part shows a rectangular lattice with b/a = 2 and a slice along the line defined by 2x + y = 0. Normalizing a = 1, the distance between two neighbouring lattice points in the slice is . If the pattern is restricted to this slice, the points of the corresponding diffraction pattern in reciprocal space must have distance and this is precisely obtained by projecting the lattice points of the reciprocal lattice onto the slice.
The different, but related, viewpoints of sections and projections can be stated in a simple way as follows: For a section perpendicular to the c axis, only those points of a crystal pattern are considered which have z coordinate equal to a fixed value or in a small interval around . For a projection along the c axis, all points of the crystal pattern are considered, but their z coordinate is simply ignored. This means that all points of the crystal pattern that differ only by their z coordinate are regarded as the same point.
For a space group and a point X in the threedimensional point space , the sitesymmetry group of X is the subgroup of operations of that fix X. Analogously, one can also look at the subgroup of operations fixing a onedimensional line or a twodimensional plane. If the line is along a rational direction, it will be fixed at least by the translations of along that direction. However, it may also be fixed by a symmetry operation that reverses the direction of the line. The resulting subgroup of that fixes the line is a rod group.
Similarly, a plane having a normal vector along a rational direction is fixed by translations of corresponding to a twodimensional lattice. Again, the plane may also be fixed by additional symmetry operations, e.g. by a twofold rotation around an axis lying in the plane, by a rotation around an axis normal to the plane or by a reflection in the plane.
Definition
A rational planar section of a crystal pattern is the intersection of the crystal pattern with a plane containing two linearly independent translation vectors of the crystal pattern. The intersecting plane is called the section plane.
A rational linear section of a crystal pattern is the intersection of the crystal pattern with a line containing a translation vector of the crystal pattern. The intersecting line is called the penetration line.
A planar section is determined by a vector d which is perpendicular to the section plane and a continuous parameter s, called the height, which gives the position of the plane on the line along d.
A linear section is specified by a vector d parallel to the penetration line and a point in a plane perpendicular to d giving the intersection of the line with that plane.
Definition

From now on we will only consider rational sections and omit this attribute. Moreover, we will concentrate on the case of planar sections, since this is by far the most relevant case for crystallographic applications. The treatment of onedimensional sections is analogous, but in general much easier.
Let d be a vector perpendicular to the section plane. In most cases, d is chosen as the shortest lattice vector perpendicular to the section plane. However, in the triclinic and monoclinic crystal family this may not be possible, since the translations of the crystal pattern may not contain a vector perpendicular to the section plane. In that case, we assume that d captures the periodicity of the crystal pattern perpendicular to the section plane. This is achieved by choosing d as the shortest nonzero projection of a lattice vector to the line through the origin which is perpendicular to the section plane. Because of the periodicity of the crystal pattern along d, it is enough to consider heights s with , since for an integer m the sectional layer groups at heights s and s + m are conjugate subgroups of . This is a consequence of the orbit–stabilizer theorem in Section 1.1.7 , applied to the group acting on the planes in . The layer at height s is mapped to the layer at height s + m by the translation through md. Thus, the two layers lie in the same orbit under . According to the orbit–stabilizer theorem, the corresponding stabilizers, being just the layer groups at heights s and s + m, are then conjugate by the translation through md.
Since we assume a rational section, the sectional layer group will always contain translations along two independent directions , which, we assume, form a crystallographic basis for the lattice of translations fixing the section plane. The points in the section plane at height s are then given by . In order to determine whether the sectional layer group contains additional symmetry operations which are not translations, the following simple remark is crucial:
Let be an operation of a sectional layer group. Then the rotational part of maps d either to +d or to −d. In the former case, is sidepreserving, in the latter case it is sidereversing. Moreover, since the section plane remains fixed under , the vectors and are mapped to linear combinations of and by the rotational part of . Therefore, with respect to the (usually nonconventional) basis , , d of threedimensional space and some choice of origin, the operation has an augmented matrix of the formHere, . Moreover, if , i.e. is sidepreserving, then is necessarily zero, since otherwise the plane is shifted along d. On the other hand, if , i.e. is sidereversing, then a plane situated at height s along d is only fixed if .
From these considerations it is straightforward to determine the conditions under which a spacegroup operation belongs to a certain sectional layer group (excluding translations):
The sidepreserving operations will belong to the sectional layer groups for all planes perpendicular to d, independent of the height s:
Sidereversing operations will only occur in the sectional layer groups for planes at special heights along d:

Note that, because of the periodicity along d, a sidereversing operation that occurs at height s gives rise to a sidereversing operation of the same type occurring at height : if is a sidereversing symmetry operation fixing a layer at height s, then maps a point in the layer at height with coordinates (with respect to the layeradapted basis ) to a point with coordinates and hence the composition of with the translation by d maps to , i.e. it fixes the layer at height . This shows that the composition with the translation by d provides a onetoone correspondence between the sidereversing symmetry operations in the layer group at height s with those at height .
If a section allows any sidereversing symmetry at all, then the sidepreserving symmetries of the section form a subgroup of index 2 in the sectional layer group. Since the sidepreserving symmetries exist independently of the height parameter s, the full sectional layer group is always generated by the sidepreserving subgroup and either none or a single sidereversing symmetry.
Summarizing, one can conclude that for a given space group the interesting sections are those for which the perpendicular vector d is parallel or perpendicular to a symmetry direction of the group, e.g. an axis of a rotation or rotoinversion or the normal vector of a reflection or glide reflection.
Example
Consider the space group of type (31). In its standard setting, the cosets of relative to the translation subgroup are represented by the operations given in Table 1.4.5.1.

Since this is an orthorhombic group, it is natural to consider sections along the coordinate axes. The spacegroup diagrams displayed in Fig. 1.4.5.2, which show the orthogonal projections of the symmetry elements along these directions, are very helpful.

As we have seen, a section of a crystal pattern is determined by a vector d and a height s along this vector. Choosing two vectors and perpendicular to d, the points of the section plane at height s are precisely given by the vectors . In contrast to that, a projection of a crystal pattern along d is obtained by mapping an arbitrary point to the point of the plane spanned by and , thereby ignoring the coordinate along the d direction.
Definition
In a projection of a crystal pattern along the projection direction d, a point X of the crystal pattern is mapped to the intersection of the line through X along d with a fixed plane perpendicular to d.
One may think of the projection plane as the plane perpendicular to d and containing the origin, but every plane perpendicular to d will give the same result.
Let be the line along d. If a symmetry operation of a space group maps to a line parallel to , then maps every plane perpendicular to d again to a plane perpendicular to d. This means that points that are projected to a single point (i.e. points on a line parallel to ) are mapped by to points that are again projected to a single point and thus the operation gives rise to a symmetry of the projection of the crystal pattern. Conversely, an operation that maps to a line that is inclined to does not result in a symmetry of the projection, since the points on are projected to a single point, whereas the image points under are projected to a line. In summary, the operations of that map to a line parallel to give rise to symmetries of the projection forming a plane group, sometimes called a wallpaper group.
Let be the subgroup of consisting of those mapping the line to a line parallel to , then is called the scanning group along d. The scanning group can be read off a coset decomposition relative to the translation subgroup of . Since translations map lines to parallel lines, one only has to check whether a coset representative maps to a line parallel to . This is precisely the case if the linear part of maps d to d or to −d. Therefore, is the union of those cosets relative to for which the linear part of maps d to d or to −d.
If the operations of a space group are written as augmented matrices with respect to a (usually nonconventional) basis , , d such that and are perpendicular to d, then an operation of the scanning group is of the formwith (just as for planar sections). Then the action of on the projection along d is obtained by ignoring the z coordinate, i.e. by cutting out the upper 2 × 2 block of the linear part and the first two components of the translation part. This gives rise to the planegroup operation
The mapping that assigns to each operation of the scanning group its action on the projection is in fact a homomorphism from to a plane group and the kernel of this homomorphism are the operations of the formi.e. translations along d and reflections with normal vector parallel to d.
Definition
The symmetry group of the projection along the projection direction d is the plane group of actions on the projection of those operations of that map the line along d to a line parallel to .
This group is isomorphic to the quotient group of the scanning group along d by the group of translations along d and reflections with normal vector parallel to d.
Example
We consider again the space group of type (31) for which the augmented matrices of the coset representatives with respect to the translation subgroup (in the standard setting) are given by
Since the linear parts of all four matrices are diagonal matrices, the scanning group for projections along the coordinate axes is always the full group .
For the projection along the direction [100], one has to cut out the lower 2 × 2 part of the linear parts and the second and third component of the translation part, thus choosing as a basis for the projection plane. This gives as matrices for the projected operationsin which the third and fourth operations are clearly redundant and which is thus a plane group of type p1g1 (plane group No. 4 with short symbol pg).
The projection along the direction [010] gives for the basis , of the projection plane (thus picking out the first and third rows and columns) the matriceswhere the second matrix is the product of the third and fourth. The third operation is a centring translation, the fourth a reflection, thus the resulting plane group is of type c1m1 (plane group No. 5 with short symbol cm).
Finally, the projection along the direction [001] results for the basis of the projection plane in the matriceswhere again the second matrix is the product of two others. The third operation is a glide reflection and the fourth is a reflection, thus the corresponding plane group is of type p2mg (plane group No. 7). Note that in order to obtain the plane group p2mg in its standard setting, the origin has to be shifted to (with respect to the plane basis ).
As for the sectional layer groups, the typical projection directions considered are symmetry directions of the space group , i.e. directions along rotation or screw axes or normal to reflection or glide planes. In order to relate the coordinate system of the plane group to that of the space group, not only the basis vectors perpendicular to the projection direction d have to be given, but also the origin for the plane group. This is done by specifying a line parallel to the projection direction which is projected to the origin of the plane group in its conventional setting. The spacegroup tables list the plane groups for the projections along symmetry directions of the group in the block `Symmetry of special projections'.
It is not hard to determine the corresponding types of planegroup operations for the different types of spacegroup operations, as is shown by the following list of simple rules:

The relationship between the symmetry operations in threedimensional space and the corresponding symmetry operations of a projection as listed above can be seen directly in the diagrams of the corresponding groups. In Fig. 1.4.5.4, the top diagram shows the orthogonal projection of the symmetryelement diagram of along the [001] direction and the bottom diagram shows the diagram for the plane group p2mg, which is precisely the symmetry group of the projection of along [001]. Firstly, one sees immediately that in order to match the two diagrams, the origin in the projection plane has to be shifted to (as already noted in the example above). Secondly, keeping in mind that the projection direction d is perpendicular to the drawing plane, one sees the correspondence between the twofold screw rotations in with the twofold rotations in p2mg [rule (iii)], the correspondence between the reflections with normal vector perpendicular to d in and the reflections in p2mg [rule (vii)] and the correspondence between the diagonal glide reflections in (indicated by the dotdash lines) and the glide reflections in p2mg {rule (vii); note that the diagonal glide vector has a component perpendicular to the projection direction [001]}.
Example
Let be a space group of type (117), then the interesting projection directions (i.e. symmetry directions) are [100], [010], [001], [110] and . However, the directions [100] and [010] are symmetryrelated by the fourfold rotoinversion and thus result in the same projection. The same holds for the directions [110] and . The three remaining directions are genuinely different and the projections along these directions will be discussed in detail below. The corresponding information given in the spacegroup tables under the heading `Symmetry of special projections' is reproduced in Fig. 1.4.5.5 for .

Orthogonal projection along [001] of the symmetryelement diagram for (31) (top) and the diagram for plane group p2mg (7) (bottom). 
Coset representatives of relative to its translation subgroup can be extracted from the generalpositions block in the spacegroup tables of and are given in Table 1.4.5.2.

Note that for directions different from those considered above, additional nontrivial plane groups may be obtained. For example, for the projection direction , the scanning group consists of the cosets of and . The operation acts as a glide reflection and the resulting plane group is of type c1m1 (plane group No. 5).
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