International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 2.1, pp. 150172

The presentation of the planegroup and spacegroup data in Chapters 2.2 and 2.3 follows the style of the previous editions of International Tables. The data for most of the space groups are displayed on one page or on two facing pages. A typical distribution of the data is shown below and is illustrated by the example of Cccm (66) provided inside the front and back covers.
It is important to note that the symmetry data are displayed in the same sequence for all the space groups. The actual distribution of the data between pages can vary depending on the amount and nature of the data that are shown.
The symmetry data for the ten space groups of the crystal class [ (221) to (230)] are displayed on four pages. Additional generalposition diagrams in tilted projection are shown on the fourth page, providing a threedimensionalstyle view of these complicated generalposition diagrams.
For several space groups, more than one description is available. Three cases occur:

Coordinate transformations between different spacegroup descriptions are treated in detail in Section 1.5.3 .
The description of each plane group or space group starts with a headline consisting of two (sometimes three) lines which contain the following information, when read from left to right.
First line

Second line
Third line
This line is used, where appropriate, to indicate origin choices, settings, cell choices and coordinate axes (see Section 2.1.3.2). For five orthorhombic space groups, an entry `Former spacegroup symbol' is given; cf. Section 2.1.2.
(For more details, cf. Section 1.4.1 and Chapter 3.3 .)
Current symbols. Both the short and the full Hermann–Mauguin symbols consist of two parts: (i) a letter indicating the centring type of the conventional cell, and (ii) a set of characters indicating symmetry elements of the space group (modified pointgroup symbol).

Short and full Hermann–Mauguin symbols differ only for the plane groups of class m, for the monoclinic space groups, and for the space groups of crystal classes mmm, , , , and . In the full symbols, symmetry axes and symmetry planes for each symmetry direction are listed; in the short symbols, symmetry axes are suppressed as much as possible. Thus, for space group No. 62, the full symbol is and the short symbol is Pnma. For No. 194, the full symbol is and the short symbol is . For No. 230, the full symbol is and the short symbol is .
Many space groups contain more kinds of symmetry elements than are indicated in the full symbol (`additional symmetry operations and elements', cf. Sections 1.4.2.4 and 1.5.4.1 ). A listing of additional symmetry operations is given in Tables 1.5.4.3 and 1.5.4.4 under the heading Extended full symbols. Note that a centre of symmetry is never explicitly indicated (except for space group ); its presence or absence, however, can be readily inferred from the spacegroup symbol.
Changes in Hermann–Mauguin spacegroup symbols as compared with the 1952 and 1935 editions of International Tables. Extensive changes in the spacegroup symbols were applied in IT (1952) as compared with the original Hermann–Mauguin symbols of IT (1935), especially in the tetragonal, trigonal and hexagonal crystal systems. Moreover, new symbols for the caxis setting of monoclinic space groups were introduced. All these changes are recorded on pp. 51 and 543–544 of IT (1952). In the present edition, the symbols of the 1952 edition are retained, except for the following four cases (cf. Section 3.3.4 ).

The entry Patterson symmetry in the headline gives the symmetry of the `vector set' generated by the operation of the space group on an arbitrary set of general positions. More prosaically, it may be described as the symmetry of the set of the interatomic vectors of a crystal structure with the selected space group. The Patterson symmetry is a crystallographic space group denoted by its Hermann–Mauguin symbol. It is in fact one of the 24 centrosymmetric symmorphic space groups (see Section 1.3.3.3 ) in three dimensions and one of 7 in two dimensions. For each of the 230 space groups, the Patterson symmetry has the same Bravaislattice type as the space group itself and its point group is the lowestindex centrosymmetric supergroup of the point group of the space group. The `pointgroup part' of the symbol of the Patterson symmetry represents the Laue class to which the plane group or space group belongs (cf. Table 2.1.2.1). By way of examples: space group No. 100, P4bm, has a Bravais lattice of type tP and point group 4mm. The centrosymmetric supergroup of 4mm (see Fig. 3.2.1.3 ) is 4/mmm, so the Patterson symmetry is P4/mmm; space group No. 66, Cccm, has a Bravais lattice of type oC and point group mmm. This point group is centrosymmetric, so the Patterson symmetry is Cmmm.
Note: For the four space groups Amm2 (38), Aem2 (39), Ama2 (40) and Aea2 (41), the standard symbol for their Patterson symmetry, Cmmm, is added (between parentheses) after the actual symbol Ammm in the spacegroup tables.
The Patterson symmetry is intimately related to the symmetry of the Patterson function (see Flack, 2015). The latter, P_{F2}(uvw), is the inverse Fourier transform of the squared structurefactor amplitudes. Patterson functions possess the crystallographic symmetry of the symmorphic spacegroup representative of the arithmetic crystal class (see Section 1.3.4.4.1 ) to which the space group belongs. Table 2.1.3.3 lists these crystallographic symmetries of the Patterson function and the Patterson symmetries for the space groups and plane groups. However, further symmetry is also present, as desribed below for the three common forms of the Patterson function:

The spacegroup diagrams serve two purposes: (i) to show the relative locations and orientations of the symmetry elements and (ii) to illustrate the arrangement of a set of symmetryequivalent points of the general position.
With the exception of generalposition diagrams in perspective projection for some space groups (cf. Section 2.1.3.6.8), all of the diagrams are orthogonal projections, i.e. the projection direction is perpendicular to the plane of the figure. Apart from the descriptions of the rhombohedral space groups with `rhombohedral axes' (cf. Section 2.1.3.6.6), the projection direction is always a cell axis. If other axes are not parallel to the plane of the figure, they are indicated by the subscript p, as or in the case of one or two axes for monoclinic and triclinic space groups, respectively (cf. Figs. 2.1.3.1 to 2.1.3.3), or by the subscript rh for the three rhombohedral axes in Fig. 2.1.3.9.
The graphical symbols for symmetry elements, as used in the drawings, are displayed in Tables 2.1.2.2 to 2.1.2.7.
In the diagrams, `heights' h above the projection plane are indicated for symmetry planes and symmetry axes parallel to the projection plane, as well as for centres of symmetry. The heights are given as fractions of the shortest lattice translation normal to the projection plane and, if different from 0, are printed next to the graphical symbols. Each symmetry element at height h is accompanied by another symmetry element of the same type at height (this does not apply to the horizontal fourfold axes in the diagrams for the cubic space groups). In the spacegroup diagrams, only the symmetry element at height h is indicated (cf. Section 2.1.2).
Schematic representations of the diagrams, displaying the origin, the labels of the axes, and the projection direction [uvw], are given in Figs. 2.1.3.1 to 2.1.3.10 (except Fig. 2.1.3.6). The generalposition diagrams are indicated by the letter .
Each description of a plane group contains two diagrams, one for the symmetry elements (left) and one for the general position (right). The two axes are labelled a and b, with a pointing downwards and b running from left to right.
For each of the two triclinic space groups, three elevations (along a, b and c) are given, in addition to the generalposition diagram (projected along c) at the lower right of the set, as illustrated in Fig. 2.1.3.1.
The diagrams represent a reduced cell of type II for which the three interaxial angles are nonacute, i.e. . For a cell of type I, all angles are acute, i.e. . For a discussion of the two types of reduced cells, see Section 3.1.3 .
The `complete treatment' of each of the two settings contains four diagrams (Figs. 2.1.3.2 and 2.1.3.3). Three of them are projections of the symmetry elements, taken along the unique axis (upper left) and along the other two axes (lower left and upper right). For the general position, only the projection along the unique axis is given (lower right).
The `synoptic descriptions' of the three cell choices (for each setting) are headed by a pair of diagrams, as illustrated in Fig. 2.1.3.4. The drawings on the left display the symmetry elements and the ones on the right the general position (labelled ). Each diagram is a projection of four neighbouring unit cells along the unique axis. It contains the outlines of the three cell choices drawn as heavy lines. For the labelling of the axes, see Fig. 2.1.3.4. The headline of the description of each cell choice contains a smallscale drawing, indicating the basis vectors and the cell that apply to that description.

Monoclinic space groups, cell choices 1, 2, 3. Upper pair of diagrams: setting with unique axis b. Lower pair of diagrams: setting with unique axis c. The numbers 1, 2, 3 within the cells and the subscripts of the labels of the axes indicate the cell choice (cf. Section 2.1.3.15). The unique axis points upwards from the page. = generalposition diagram. 
The spacegroup tables contain a set of four diagrams for each orthorhombic space group. The set consists of three projections of the symmetry elements [along the c axis (upper left), the a axis (lower left) and the b axis (upper right)] in addition to the generalposition diagram, which is given only in the projection along c (lower right). The projected axes, the origins and the projection directions of these diagrams are illustrated in Fig. 2.1.3.5. They refer to the socalled `standard setting' of the space group, i.e. the setting described in the spacegroup tables and indicated by the `standard Hermann–Mauguin symbol' in the headline.

Orthorhombic space groups. Diagrams for the `standard setting' as described in the spacegroup tables ( = generalposition diagram). 
For each orthorhombic space group, six settings exist, i.e. six different ways of assigning the labels a, b, c to the three orthorhombic symmetry directions; thus the shape and orientation of the cell are the same for each setting. These settings correspond to the six permutations of the labels of the axes (including the identity permutation); cf. Section 1.5.4.3 : The symbol for each setting, here called `setting symbol', is a shorthand notation for the (3 × 3) transformation matrix P of the basis vectors of the standard setting, a, b, c, into those of the setting considered (cf. Chapter 1.5 for a detailed discussion of coordinate transformations). For instance, the setting symbol cab stands for the cyclic permutation or where a′, b′, c′ is the new set of basis vectors. An interchange of two axes reverses the handedness of the coordinate system; in order to keep the system righthanded, each interchange is accompanied by the reversal of the sense of one axis, i.e. by an element in the transformation matrix. Thus, denotes the transformation The six orthorhombic settings correspond to six Hermann–Mauguin symbols which, however, need not all be different; cf. Table 2.1.3.4.^{1}

In the earlier (1935 and 1952) editions of International Tables, only one setting was illustrated, in a projection along c, so that it was usual to consider it as the `standard setting' and to accept its cell edges as crystal axes and its spacegroup symbol as the `standard Hermann–Mauguin symbol'. In the present edition, following IT A (2002), however, all six orthorhombic settings are illustrated, as explained below.
The three projections of the symmetry elements can be interpreted in two ways. First, in the sense indicated above, that is, as different projections of a single (standard) setting of the space group, with the projected basis vectors a, b, c labelled as in Fig. 2.1.3.5. Second, each one of the three diagrams can be considered as the projection along c′ of either one of two different settings: one setting in which b′ is horizontal and one in which b′ is vertical (a′, b′, c′ refer to the setting under consideration). This second interpretation is used to illustrate in the same figure the spacegroup symbols corresponding to these two settings. In order to view these projections in conventional orientation (b′ horizontal, a′ vertical, origin in the upper left corner, projection down the positive c′ axis), the setting with b′ horizontal can be inspected directly with the figure upright; hence, the corresponding spacegroup symbol is printed above the projection. The other setting with b′ vertical and a′ horizontal, however, requires turning the figure by 90°, or looking at it from the side; thus, the spacegroup symbol is printed at the left, and it runs upwards.
The `setting symbols' for the six settings are attached to the three diagrams of Fig. 2.1.3.6, which correspond to those of Fig. 2.1.3.5. In the orientation of the diagram where the setting symbol is read in the usual way, a′ is vertical pointing downwards, b′ is horizontal pointing to the right, and c′ is pointing upwards from the page. Each setting symbol is printed in the position that in the spacegroup tables is actually occupied by the corresponding full Hermann–Mauguin symbol. The changes in the spacegroup symbol that are associated with a particular setting symbol can easily be deduced by comparing Fig. 2.1.3.6 with the diagrams for the space group under consideration.

Orthorhombic space groups. The three projections of the symmetry elements with the six setting symbols (see text). For setting symbols printed vertically, the page has to be turned clockwise by 90° or viewed from the side. Note that in the actual spacegroup tables instead of the setting symbols the corresponding full Hermann–Mauguin spacegroup symbols are printed. 
Not all of the 59 orthorhombic space groups have all six projections distinct, i.e. have different Hermann–Mauguin symbols for the six settings. This aspect is treated in Table 2.1.3.4. Only 22 space groups have six, 25 have three, 2 have two different symbols, while 10 have all symbols the same. This information can be of help in the early stages of a crystalstructure analysis.
The six setting symbols, i.e. the six permutations of the labels of the axes, form the column headings of the orthorhombic entries in Table 1.5.4.4 , which contains the extended Hermann–Mauguin symbols for the six settings of each orthorhombic space group. Note that some of these setting symbols exhibit different sign changes compared with those in Fig. 2.1.3.6.
The pairs of diagrams for these space groups are similar to those in the previous editions of IT. Each pair consists of a generalposition diagram (right) and a diagram of the symmetry elements (left), both projected along c, as illustrated in Figs. 2.1.3.7 and 2.1.3.8.
The seven rhombohedral space groups are treated in two versions, the first based on `hexagonal axes' (obverse setting), the second on `rhombohedral axes' (cf. Sections 2.1.1.2 and 2.1.3.2). The pairs of diagrams are similar to those in IT (1952) and IT A (2002); the left or top one displays the symmetry elements, the right or bottom one the general position. This is illustrated in Fig. 2.1.3.9, which gives the axes a and b of the triple hexagonal cell and the projections of the axes of the primitive rhombohedral cell, labelled a_{rh}, b_{rh} and c_{rh}. For convenience, all `heights' in the spacegroup diagrams are fractions of the hexagonal c axis. For `hexagonal axes', the projection direction is [001], for `rhombohedral axes' it is [111]. In the generalposition diagrams, the circles drawn in heavier lines represent atoms that lie within the primitive rhombohedral cell (provided the symbol `−' is read as rather than as ).

Rhombohedral space groups. Obverse triple hexagonal cell with `hexagonal axes' a, b and primitive rhombohedral cell with projections of `rhombohedral axes' a_{rh}, b_{rh}, c_{rh}. Note: In the actual spacegroup diagrams the edges of the primitive rhombohedral cell (dashed lines) are only indicated in the generalposition diagram of the rhombohedralaxes description ( = generalposition diagram). 
The symmetryelement diagrams for the hexagonal and the rhombohedral descriptions of a space group are the same. The edges of the primitive rhombohedral cell (cf. Fig. 2.1.3.9) are only indicated in the generalposition diagram of the rhombohedral description.
For each cubic space group, one projection of the symmetry elements along [001] is given, Fig. 2.1.3.10; for details of the diagrams, see Section 2.1.2 and Buerger (1956). For facecentred lattices F, only a quarter of the unit cell is shown; this is sufficient since the projected arrangement of the symmetry elements is translationequivalent in the four quarters of an F cell. It is important to note that symmetry axes inclined to the projection plane are indicated where they intersect the plane of projection. Symmetry planes inclined to the projection plane that occur in classes and are shown as `inserts' around the highsymmetry points, such as ; ; etc.

Cubic space groups. = generalposition diagram, in which the equivalent positions are shown as the vertices of polyhedra. 
The cubic diagrams given in IT (1935) are different from the ones used here. No drawings for cubic space groups were provided in IT (1952).
Noncubic space groups. In these diagrams, the `heights' of the points are z coordinates, except for monoclinic space groups with unique axis b where they are y coordinates. For rhombohedral space groups, the heights are always fractions of the hexagonal c axis. The symbols and − stand for and (or and ) in which z or y can assume any value. For points with symbols or − preceded by a fraction, e.g. or , the relative z or y coordinate is etc. higher than that of the point with symbol or −.
Where a mirror plane exists parallel to the plane of projection, the two positions superimposed in projection are indicated by the use of a ring divided through the centre. The information given on each side refers to one of the two positions related by the mirror plane, as in .
Diagrams for cubic space groups (Fig. 2.1.3.10). Following the approach of IT (1935), for each cubic space group a diagram showing the points of the general position as the vertices of polyhedra is given. In these diagrams, the polyhedra are transparent, but the spheres at the vertices are opaque. For most of the space groups, `starting points' with the same coordinate values, x = 0.048, y = 0.12, z = 0.089, have been used. The origins of the polyhedra are chosen at special points of highest site symmetry, which for most space groups coincide with the origin (and its equivalent points in the unit cell). Polyhedra with origins at sites have been chosen for the space groups (212) and (214), and for (213). The two diagrams shown for the space groups (220) and (230) correspond to polyhedra with origins chosen at two different special sites with sitesymmetry groups of equal (32 versus in ) or nearly equal order (3 versus in ). The height h of the centre of each polyhedron is given on the diagram, if different from zero. For spacegroup Nos. 198, 199 and 220, h refers to the special point to which the polyhedron (triangle) is connected. Polyhedra with height 1 are omitted in all the diagrams. A grid of four squares is drawn to represent the four quarters of the basal plane of the cell. For space groups (219), (226) and (228), where the number of points is too large for one diagram, two diagrams are provided, one for the upper half and one for the lower half of the cell.
Notes:

An additional generalposition diagram is shown on the fourth page for each of the ten space groups of the crystal class. To provide a clearer threedimensionalstyle overview of the arrangements of the polyhedra, these generalposition diagrams are shown in tilted projection (in contrast to the orthogonalprojection diagrams described above).
The generalposition diagrams of the cubic groups in both orthogonal and tilted projections were generated using the program VESTA (Momma & Izumi, 2011).
Readers who wish to compare other approaches to spacegroup diagrams and their history are referred to IT (1935), IT (1952), the fifth edition of IT A (2002) (where generalposition stereodiagrams of the cubic space groups are shown) and the following publications: Astbury & Yardley (1924), Belov et al. (1980), Buerger (1956), Fedorov (1895; English translation, 1971), Friedel (1926), Hilton (1903), Niggli (1919) and Schiebold (1929).
The determination and description of crystal structures and particularly the application of direct methods are greatly facilitated by the choice of a suitable origin and its proper identification. This is even more important if related structures are to be compared or if `chains' of group–subgroup relations are to be constructed. In this volume, as well as in IT (1952) and IT A (2002), the origin of the unit cell has been chosen according to the following conventions (cf. Sections 2.1.1 and 2.1.3.2):

There are several ways of determining the location and site symmetry of the origin. First, the origin can be inspected directly in the spacegroup diagrams (cf. Section 2.1.3.6). This method permits visualization of all symmetry elements that intersect the chosen origin.
Another procedure for finding the site symmetry at the origin is to look for a special position that contains the coordinate triplet or that includes it for special values of the parameters, e.g. position 1a: 0, 0, z in space group P4 (75), or position ; ; in space group (152). If such a special position occurs, the symmetry at the origin is given by the oriented sitesymmetry symbol (see Section 2.1.3.12) of that special position; if it does not occur, the site symmetry at the origin is 1. For most practical purposes, these two methods are sufficient for the identification of the site symmetry at the origin.
Origin statement. In the line Origin immediately below the diagrams, the site symmetry of the origin is stated, if different from the identity. A further symbol indicates all symmetry elements (including glide planes and screw axes) that pass through the origin, if any. For space groups with two origin choices, for each of the two origins the location relative to the other origin is also given. An example is space group Ccce (68).
In order to keep the notation as simple as possible, no rigid rules have been applied in formulating the origin statements. Their meaning is demonstrated by the examples in Table 2.1.3.5, which should be studied together with the appropriate spacegroup diagrams.

These examples illustrate the following points:

An asymmetric unit of a space group is a (simply connected) smallest closed part of space from which, by application of all symmetry operations of the space group, the whole of space is filled. This implies that mirror planes and rotation axes must form boundary planes and boundary edges of the asymmetric unit. A twofold rotation axis may bisect a boundary plane. Centres of inversion must either form vertices of the asymmetric unit or be located at the midpoints of boundary planes or boundary edges. For glide planes and screw axes, these simple restrictions do not hold. An asymmetric unit contains all the information necessary for the complete description of the crystal structure. In mathematics, an asymmetric unit is called `fundamental region' or `fundamental domain'.
Example
The boundary planes of the asymmetric unit in space group Pmmm (47) are fixed by the six mirror planes x, y, 0; ; x, 0, z; ; 0, y, z; and . For space group (19), on the other hand, a large number of connected regions, each with a volume of (cell), may be chosen as asymmetric unit.
In cases where the asymmetric unit is not uniquely determined by symmetry, its choice may depend on the purpose of its application. For the description of the structures of molecular crystals, for instance, it is advantageous to select asymmetric units that contain one or more complete molecules. In the spacegroup tables of this volume, following IT A (2002), the asymmetric units are chosen in such a way that Fourier summations can be performed conveniently.
For all triclinic, monoclinic and orthorhombic space groups, the asymmetric unit is chosen as a parallelepiped with one vertex at the origin of the cell and with boundary planes parallel to the faces of the cell. It is given by the notation where stands for x, y or z.
For space groups with higher symmetry, cases occur where the origin does not coincide with a vertex of the asymmetric unit or where not all boundary planes of the asymmetric unit are parallel to those of the cell. In all these cases, parallelepipeds are given that are equal to or larger than the asymmetric unit. Where necessary, the boundary planes lying within these parallelepipeds are given by additional inequalities, such as , etc.
In the trigonal, hexagonal and especially the cubic crystal systems, the asymmetric units have complicated shapes. For this reason, they are also specified by the coordinates of their vertices. Drawings of asymmetric units for cubic space groups have been published by Koch & Fischer (1974). Fig. 2.1.3.11 shows the boundary planes occurring in the tetragonal, trigonal and hexagonal systems, together with their algebraic equations.

Boundary planes of asymmetric units occurring in the spacegroup tables. (a) Tetragonal system. (b) Trigonal and hexagonal systems. The point coordinates refer to the vertices in the plane . 
Examples

Fourier syntheses. For complicated space groups, the easiest way to calculate Fourier syntheses is to consider the parallelepiped listed, without taking into account the additional boundary planes of the asymmetric unit. These planes should be drawn afterwards in the Fourier synthesis. For the computation of integrated properties from Fourier syntheses, such as the number of electrons for parts of the structure, the values at the boundaries of the asymmetric unit must be applied with a reduced weight if the property is to be obtained as the product of the content of the asymmetric unit and the multiplicity.
Example
In the parallelepiped of space group Pmmm (47), the weights for boundary planes, edges and vertices are , and , respectively.
Asymmetric units of the plane groups have been discussed by Buerger (1949, 1960) in connection with Fourier summations.
As explained in Sections 1.3.3.2 and 1.4.2.3 , the coordinate triplets of the General position of a space group may be interpreted as a shorthand description of the symmetry operations in matrix notation. The geometric description of the symmetry operations is found in the spacegroup tables under the heading Symmetry operations.
Numbering scheme. The numbering of the entries in the blocks Symmetry operations and General position (first block below Positions) is the same. Each listed coordinate triplet of the general position is preceded by a number between parentheses (p). The same number (p) precedes the corresponding symmetry operation. For space groups with primitive cells, the two lists contain the same number of entries.
For space groups with centred cells, several (2, 3 or 4) blocks of Symmetry operations correspond to the one General position block. The numbering scheme of the general position is applied to each one of these blocks. The number of blocks equals the multiplicity of the centred cell, i.e. the number of centring translations below the subheading Coordinates, such as .
Whereas for the Positions the reader is expected to add these centring translations to each printed coordinate triplet themselves (in order to obtain the complete general position), for the Symmetry operations the corresponding data are listed explicitly. The different blocks have the subheadings `For (0, 0, 0)+ set', `For set', etc. Thus, an obvious onetoone correspondence exists between the analytical description of a symmetry operation in the form of its generalposition coordinate triplet and the geometrical description under Symmetry operations. Note that the coordinates are reduced modulo 1, where applicable, as shown in the example below.
Example: Ibca (73)
The centring translation is . Accordingly, above the general position one finds and . In the block Symmetry operations, under the subheading `For set', entry (2) refers to the coordinate triplet . Under the subheading `For set', however, entry (2) refers to . The triplet is selected rather than , because the coordinates are reduced modulo 1.
The coordinate triplets of the general position represent the symmetry operations chosen as coset representatives of the decomposition of the space group with respect to its translation subgroup (cf. Section 1.4.2 for a detailed discussion). In space groups with two origins the origin shift may lead to the choice of symmetry operations of different types as coset representatives of the same coset (e.g. mirror versus glide plane, rotation versus screw axis, see Tables 1.4.2.2 and 1.4.2.3 ) and designated by the same number (p) in the generalposition blocks of the two descriptions. Thus, in (129), (p) = (7) represents a 2 and a 2_{1} axis, both in , whereas (p) = (16) represents a g and an m plane, both in .
Designation of symmetry operations. An entry in the block Symmetry operations is characterized as follows.

Examples

Details on the symbolism and further illustrative examples are presented in Section 1.4.2.1 .
The line Generators selected states the symmetry operations and their sequence, selected to generate all symmetryequivalent points of the General position from a point with coordinates x, y, z. Generating translations are listed as t(1, 0, 0), t(0, 1, 0), t(0, 0, 1); likewise for additional centring translations. The other symmetry operations are given as numbers (p) that refer to the corresponding coordinate triplets of the general position and the corresponding entries under Symmetry operations, as explained in Section 2.1.3.9 [for centred space groups the first block `For (0, 0, 0)+ set' must be used].
For all space groups, the identity operation given by (1) is selected as the first generator. It is followed by the generators t(1, 0, 0), t(0, 1, 0), t(0, 0, 1) of the integral lattice translations and, if necessary, by those of the centring translations, e.g. for a Ccentred lattice. In this way, point x, y, z and all its translationally equivalent points are generated. (The remark `and its translationally equivalent points' will hereafter be omitted.) The sequence chosen for the generators following the translations depends on the crystal class of the space group and is set out in Table 1.4.3.1 .
Example: (14, unique axis b, cell choice 1)
After the generation of (1) x, y, z, the operation (2) which stands for a twofold screw rotation around the axis 0, y, generates point (2) of the general position with coordinate triplet . Finally, the inversion (3) generates point (3) from point (1), and point (4′) from point (2). Instead of (4′), however, the coordinate triplet (4) is listed, because the coordinates are reduced modulo 1.
The example shows that for the space group two operations, apart from the identity and the generating translations, are sufficient to generate all symmetryequivalent points. Alternatively, the inversion (3) plus the glide reflection (4), or the glide reflection (4) plus the twofold screw rotation (2), might have been chosen as generators. The process of generation and the selection of the generators for the spacegroup tables, as well as the resulting sequence of the symmetry operations, are discussed in Section 1.4.3 .
The generating operations for different descriptions of the same space group (settings, cell choices, origin choices) are chosen in such a way that the transformation relating the two coordinate systems also transforms the generators of one description into those of the other (cf. Section 1.5.3 ).
The entries under Positions^{2} (more explicitly called Wyckoff positions) consist of the one General position (upper block) and the Special positions (blocks below). The columns in each block, from left to right, contain the following information for each Wyckoff position.

Detailed treatment of general and special Wyckoff positions, including definitions, theoretical background and examples, is given in Section 1.4.4 .
The two types of positions, general and special, are characterized as follows:

The set of all symmetry operations that map a point onto itself forms a group, known as the `sitesymmetry group' of that point. It is given in the third column by the `oriented sitesymmetry symbol' which is explained in Section 2.1.3.12. General positions always have site symmetry 1, whereas special positions have higher site symmetries, which can differ from one special position to another.
If in a crystal structure the centres of finite objects, such as molecules, are placed at the points of a special position, each such object must display a point symmetry that is at least as high as the site symmetry of the special position. Geometrically, this means that the centres of these objects are located on symmetry elements without translations (centre of symmetry, mirror plane, rotation axis, rotoinversion axis) or at the intersection of several symmetry elements of this kind (cf. the spacegroup diagrams).
Note that the location of an object on a screw axis or on a glide plane does not lead to an increase in the site symmetry and to a corresponding reduction of the multiplicity for that object. Accordingly, a space group that contains only symmetry elements with translation components does not have any special position. Such a space group is called `fixedpointfree' (for further discussion, see Section 1.4.4.2 ).
Example: Space group C12/c1 (15, unique axis b, cell choice 1)
The general position 8f of this space group contains eight equivalent points per cell, each with site symmetry 1. The coordinate triplets of four points, (1) to (4), are given explicitly, the coordinates of the other four points are obtained by adding the components of the Ccentring translation to the coordinate triplets (1) to (4).
The space group has five special positions with Wyckoff letters a to e. The positions 4a to 4d require inversion symmetry, , whereas Wyckoff position 4e requires twofold rotation symmetry, 2, for any object in such a position. For position 4e, for instance, the four equivalent points have the coordinates . The values of x and z are specified, whereas y may take any value. Since each point of position 4e is mapped onto itself by a twofold rotation, the multiplicity of the position is reduced from 8 to 4, whereas the order of the sitesymmetry group is increased from 1 to 2.
From the symmetryelement diagram of C2/c, the locations of the four twofold axes can be deduced as ; ; ; .
From this example, the general rule is apparent that the product of the position multiplicity and the order of the corresponding sitesymmetry group is constant for all Wyckoff positions of a given space group; it is the multiplicity of the general position.
Attention is drawn to ambiguities in the description of crystal structures in a few space groups, depending on whether the coordinate triplets of IT (1952) or of this edition are taken. This problem is analysed by Parthé et al. (1988).
The third column of each Wyckoff position gives the Site symmetry^{3} of that position. The sitesymmetry group is isomorphic to a (proper or improper) subgroup of the point group to which the space group under consideration belongs. The sitesymmetry groups of the different points of the same special position are conjugate (symmetryequivalent) subgroups of the space group. For this reason, all points of one special position are described by the same sitesymmetry symbol. (See Section 1.4.4 for a detailed discussion of sitesymmetry groups.)
Oriented sitesymmetry symbols (cf. Fischer et al., 1973) are employed to show how the symmetry elements at a site are related to the symmetry elements of the crystal lattice. The sitesymmetry symbols display the same sequence of symmetry directions as the spacegroup symbol (cf. Table 2.1.3.1). Sets of equivalent symmetry directions that do not contribute any element to the sitesymmetry group are represented by a dot. In this way, the orientation of the symmetry elements at the site is emphasized, as illustrated by the following examples.
Examples

The above examples show:

To show, for the same type of site symmetry, how the oriented sitesymmetry symbol depends on the space group under discussion, the sitesymmetry group mm2 will be considered in orthorhombic and tetragonal space groups. Relevant crystal classes are mm2, mmm, 4mm, and . The site symmetry mm2 contains two mutually perpendicular mirror planes intersecting in a twofold axis.
For space groups of crystal class mm2, the twofold axis at the site must be parallel to the one direction of the rotation axes of the space group. The sitesymmetry group mm2, therefore, occurs only in the orientation mm2. For space groups of class mmm (full symbol ), the twofold axis at the site may be parallel to a, b or c and the possible orientations of the site symmetry are 2mm, m2m and mm2. For space groups of the tetragonal crystal class 4mm, the twofold axis of the sitesymmetry group mm2 must be parallel to the fourfold axis of the crystal. The two mirror planes must belong either to the two secondary or to the two tertiary tetragonal directions so that 2mm. and 2.mm are possible sitesymmetry symbols. Similar considerations apply to class , which can occur in two settings, and . Finally, for class (full symbol ), the twofold axis of 2mm may belong to any of the three kinds of symmetry directions and possible oriented site symmetries are 2mm., 2.mm, m2m. and m.2m. In the first two symbols, the twofold axis extends along the single primary direction and the mirror planes occupy either both secondary or both tertiary directions; in the last two cases, one mirror plane belongs to the primary direction and the second to either one secondary or one tertiary direction (the other equivalent direction in each case being occupied by the twofold axis).
The Reflection conditions^{4} are listed in the righthand column of each Wyckoff position.
These conditions are formulated here, in accordance with general practice, as `conditions of occurrence' (structure factor not systematically zero) and not as `extinctions' or `systematic absences' (structure factor zero). Reflection conditions are listed for all those three, two and onedimensional sets of reflections for which extinctions exist; hence, for those nets or rows that are not listed, no reflection conditions apply. The theoretical background of reflection conditions and their derivation are discussed in detail in Section 1.6.3 .
There are two types of systematic reflection conditions for diffraction of radiation by crystals:

General reflection conditions. These are due to one of three effects:

Reflection conditions of types (ii) and (iii) are listed in Table 2.1.3.7. They can be understood as follows: Zonal and serial reflections form two or onedimensional sections through the origin of reciprocal space. In direct space, they correspond to projections of a crystal structure onto a plane or onto a line. Glide planes or screw axes may reduce the translation periods in these projections (cf. Section 2.1.3.14) and thus decrease the size of the projected cell. As a consequence, the cells in the corresponding reciprocallattice sections are increased, which means that systematic absences of reflections occur.
^{†}Glide planes d with orientations (100), (010) and (001) occur only in orthorhombic and cubic F space groups. Combination of the integral reflection condition (hkl: all odd or all even) with the zonal conditions for the d glide planes leads to the further conditions given between parentheses.
^{‡}For rhombohedral space groups described with `rhombohedral axes', the three reflection conditions imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin spacegroup symbols, c is always used, as in R3c (161) and , because c glides also occur in the hexagonal description of these space groups. ^{§}For tetragonal P space groups, the two reflection conditions (hhl and with ) imply interleaving of c and n glides. In the Hermann–Mauguin spacegroup symbols, c is always used, irrespective of which glide planes contain the origin: cf. P4cc (103), and . ^{¶}For cubic space groups, the three reflection conditions imply interleaving of c and n glides, a and n glides, and b and n glides, respectively. In the Hermann–Mauguin spacegroup symbols, either c or n is used, depending upon which glide plane contains the origin, cf. , , versus , , . 
For the twodimensional groups, the reasoning is analogous. The reflection conditions for the plane groups are assembled in Table 2.1.3.8.

For the interpretation of observed reflections, the general reflection conditions must be studied in the order (i) to (iii), as conditions of type (ii) may be included in those of type (i), while conditions of type (iii) may be included in those of types (i) or (ii). This is shown in the example below.
In the spacegroup tables, the reflection conditions are given according to the following rules:

Note that the integral reflection conditions for a rhombohedral lattice, described with `hexagonal axes', permit the presence of only one member of the pair hkil and for (cf. Table 2.1.3.6). This applies also to the zonal reflections and , which for the rhombohedral space groups must be considered separately.
Example
For a monoclinic crystal (b unique), the following reflection conditions have been observed:
Line (1) states that the cell used for the description of the space group is C centred. In line (2), the conditions 0kl with , h0l with and hk0 with are a consequence of the integral condition (1), leaving only h0l with as a new condition. This indicates a glide plane c. Line (3) presents no new condition, since h00 with and 0k0 with follow from the integral condition (1), whereas 00l with is a consequence of a zonal condition (2). Accordingly, there need not be a twofold screw axis along [010]. Space groups obeying the conditions are Cc (9, b unique, cell choice 1) and (15, b unique, cell choice 1). Under certain conditions, using methods based on resonant scattering, it is possible to determine whether the structure space group is centrosymmetric or not (cf. Section 1.6.5.1 ).
For a different choice of the basis vectors, the reflection conditions would appear in a different form owing to the transformation of the reflection indices (cf. cell choices 2 and 3 for space groups Cc and in Chapter 2.3 ). The transformations of reflection conditions under coordinate transformations are discussed and illustrated in Sections 1.5.2 and 1.5.3 .
Special or `extra' reflection conditions. These apply either to the integral reflections hkl or to particular sets of zonal or serial reflections. In the spacegroup tables, the minimal special conditions are listed that, on combination with the general conditions, are sufficient to generate the complete set of conditions. This will be apparent from the examples below.
Examples

For the cases where the special reflection conditions are described by means of combinations of `OR' and `AND' instructions, the `AND' condition always has to be evaluated with priority, as shown by the following example.
Example: (218)
Special position or , and .
This expression contains the following two conditions:
(a) ;
(b) and and .
A reflection is `present' (occurring) if either condition (a) is satisfied or if a permutation of the three conditions in (b) are simultaneously fulfilled.
Structural or nonspacegroup absences. Note that in addition nonspacegroup absences may occur that are not due to the symmetry of the space group (i.e. centred cells, glide planes or screw axes). Atoms in general or special positions may cause additional systematic absences if their coordinates assume special values [e.g. `noncharacteristic orbits'; cf. Section 1.4.4.4 and Engel et al. (1984)]. Nonspacegroup absences may also occur for special arrangements of atoms (`false symmetry') in a crystal structure (cf. Templeton, 1956; Sadanaga et al., 1978). Nonspacegroup absences may occur also for polytypic structures; this is briefly discussed by Durovič in Section 9.2.2.2.5 of International Tables for Crystallography (2004), Vol. C. Even though all these `structural absences' are fortuitous and due to the special arrangements of atoms in a particular crystal structure, they have the appearance of spacegroup absences. Occurrence of structural absences thus may lead to an incorrect assignment of the space group. Accordingly, the reflection conditions in the spacegroup tables must be considered as a minimal set of conditions.
The use of reflection conditions and of the symmetry of reflection intensities for spacegroup determination is described in Chapter 1.6 .
Projections of crystal structures are used by crystallographers in special cases. Use of socalled `twodimensional data' (zerolayer intensities) results in the projection of a crystal structure along the normal to the reciprocallattice net. A detailed treatment of projections of space groups, including basic definitions and illustrative examples, is given in Section 1.4.5.3 .
Even though the projection of a finite object along any direction may be useful, the projection of a periodic object such as a crystal structure is only sensible along a rational lattice direction (lattice row). Projection along a nonrational direction results in a constant density in at least one direction.
Data listed in the spacegroup tables. Under the heading Symmetry of special projections, the following data are listed for three projections of each space group; no projection data are given for the plane groups.

Projections of centred cells (lattices). For centred lattices, two different cases may occur:

Projections of symmetry elements. A symmetry element of a space group does not project as a symmetry element unless its orientation bears a special relation to the projection direction; all translation components of a symmetry operation along the projection direction vanish, whereas those perpendicular to the projection direction (i.e. parallel to the plane of projection) may be retained. This is summarized in Table 2.1.3.10 for the various crystallographic symmetry elements. From this table the following conclusions can be drawn:

A detailed discussion of the correspondence between the symmetry elements and their projections is given in Section 1.4.5.3 .
Example: (15, b unique, cell choice 1)
The Ccentred cell has lattice points at 0, 0, 0 and . In all projections, the centre projects as a twofold rotation point.
Projection along [001]: The plane cell is centred; projects as m; the glide component of glide plane c vanishes and thus c projects as m.
Result: Plane group c2mm (9), .
Projection along [100]: The periodicity along b is halved because of the C centring; projects as m; the glide component of glide plane c is retained and thus c projects as g.
Result: Plane group p2gm (7), .
Projection along [010]: The periodicity along a is halved because of the C centring; that along c is halved owing to the glide component of glide plane c; projects as 2.
Result: Plane group p2 (2), .
Further details about the geometry of projections can be found in publications by Buerger (1965) and Biedl (1966).
In this volume, space groups are described by one (or at most two) conventional coordinate systems (cf. Sections 2.1.1.2 and 2.1.3.2). Eight monoclinic space groups, however, are treated more extensively. In order to provide descriptions for frequently encountered cases, they are given in six versions.
The description of a monoclinic crystal structure in this volume, including its Hermann–Mauguin spacegroup symbol, depends upon two choices:
Cell choices. One edge of the cell, i.e. one crystal axis, is always chosen along the monoclinic symmetry direction. The other two edges are located in the plane perpendicular to this direction and coincide with translation vectors in this `monoclinic plane'. It is sensible and common practice (see below) to choose these two basis vectors from the shortest three translation vectors in that plane. They are shown in Fig. 2.1.3.12 and labelled e, f and g, in order of increasing length.^{5} The two shorter vectors span the `reduced mesh' (where mesh means a twodimensional unit cell), here e and f; for this mesh, the monoclinic angle is , whereas for the other two primitive meshes larger angles are possible.

The three primitive twodimensional cells which are spanned by the shortest three translation vectors e, f, g in the monoclinic plane. For the present discussion, the glide vector is considered to be along e and the projection of the centring vector along f. 
Other choices of the basis vectors in the monoclinic plane are possible, provided they span a primitive mesh. It turns out, however, that the spacegroup symbol for any of these (nonreduced) meshes already occurs among the symbols for the three meshes formed by e, f, g in Fig. 2.1.3.12; hence only these cases need be considered. They are designated in this volume as `cell choice 1, 2 or 3' and are depicted in Fig. 2.1.3.4. The transformation matrices for the three cell choices are listed in Table 1.5.1.1 .
Settings. The term setting of a cell or of a space group refers to the assignment of labels (a, b, c) and directions to the edges of a given unit cell, resulting in a set of basis vectors a, b, c. (For orthorhombic space groups, the six settings are described and illustrated in Section 2.1.3.6.4.)
The symbol for each setting is a shorthand notation for the transformation of a given starting set abc into the setting considered. It is called here `setting symbol'. For instance, the setting symbol bca stands for or where a′, b′, c′ is the new set of basis vectors. [Note that the setting symbol bca means that the `old' vector a changes its label to c′ (and not to b′), that the `old' vector b changes its label to a′ (and not to c′) and that the `old' vector c changes its label to b′ (and not to a′).] Transformation of one setting into another preserves the shape of the cell and its orientation relative to the lattice. The matrices of these transformations have one entry +1 or −1 in each row and column; all other entries are 0.
In monoclinic space groups, one axis, the monoclinic symmetry direction, is unique. Its label must be chosen first and, depending upon this choice, one speaks of `unique axis b', `unique axis c' or `unique axis a'.^{6} Conventionally, the positive directions of the two further (`oblique') axes are oriented so as to make the monoclinic angle nonacute, i.e. , and the coordinate system righthanded. For the three cell choices, settings obeying this condition and having the same label and direction of the unique axis are considered as one setting; this is illustrated in Fig. 2.1.3.4.
Note: These three cases of labelling the monoclinic axis are often called somewhat loosely baxis, caxis and aaxis `settings'. It must be realized, however, that the choice of the `unique axis' alone does not define a single setting but only a pair, as for each cell the labels of the two oblique axes can be interchanged.
Table 2.1.3.11 lists the setting symbols for the six monoclinic settings in three equivalent forms, starting with the symbols abc (first line), abc (second line) and abc (third line); the unique axis is underlined. These symbols are also found in the headline of the synoptic Table 1.5.4.4 , which lists the spacegroup symbols for all monoclinic settings and cell choices. Again, the corresponding transformation matrices are listed in Table 1.5.1.1 .

In the spacegroup tables, only the settings with b and c unique are treated and for these only the lefthand members of the double entries in Table 2.1.3.11. This implies, for instance, that the caxis setting is obtained from the baxis setting by cyclic permutation of the labels, i.e. by the transformation The setting with a unique is also included in the present discussion, as this setting occurs in Table 1.5.4.4 . The aaxis setting (i.e. , , ) is obtained from the caxis setting also by cyclic permutation of the labels and from the baxis setting by the reverse cyclic permutation: .
By the conventions described above, the setting of each of the cell choices 1, 2 and 3 is determined once the label and the direction of the uniqueaxis vector have been selected. Six of the nine resulting possibilities are illustrated in Fig. 2.1.3.4.
Cell choices and settings in the present tables. There are five monoclinic space groups for which the Hermann–Mauguin symbols are independent of the cell choice, viz those space groups that do not contain centred lattices or glide planes: In these cases, description of the space group by one cell choice is sufficient.
For the eight monoclinic space groups with centred lattices or glide planes, the Hermann–Mauguin symbol depends on the choice of the oblique axes with respect to the glide vector and/or the centring vector. These eight space groups are: Here, the glide vector or the projection of the centring vector onto the monoclinic plane is always directed along one of the vectors e, f or g in Fig. 2.1.3.12, i.e. is parallel to the shortest, the secondshortest or the thirdshortest translation vector in the monoclinic plane (note that a glide vector and the projection of a centring vector cannot be parallel). This results in three possible orientations of the glide vector or the centring vector with respect to these crystal axes, and thus in three different full Hermann–Mauguin symbols (cf. Section 2.1.3.4) for each setting of a space group.
Table 2.1.3.12 lists the symbols for centring types and glide planes for the cell choices 1, 2, 3. The order of the three cell choices is defined as follows: The symbols occurring in the familiar `standard short monoclinic spacegroup symbols' (see Section 2.1.3.3) define cell choice 1; for `unique axis b', this applies to the centring type C and the glide plane c, as in Cm (8) and . Cell choices 2 and 3 follow from the anticlockwise order 1–2–3 in Fig. 2.1.3.4 and their spacegroup symbols can be obtained from Table 2.1.3.12. The caxis and the aaxis settings then are derived from the baxis setting by cyclic permutations of the axial labels, as described in this section.

In the two space groups Cc (9) and , glide planes occur in pairs, i.e. each vector e, f, g is associated either with a glide vector or with the centring vector of the cell. For Pc (7), and , which contain only one type of glide plane, the lefthand member of each pair of glide planes in Table 2.1.3.12 applies.
In the spacegroup tables of this volume, the following treatments of monoclinic space groups are given:

All settings and cell choices are identified by the appropriate full Hermann–Mauguin symbols (cf. Section 2.1.3.4), e.g. or . For the two space groups Cc (9) and with pairs of different glide planes, the `simplest operation rule' for reflections (m > a, b, c > n) is not followed (cf. Section 1.4.1 ). Instead, in order to bring out the relations between the various settings and cell choices, the glideplane symbol always refers to that glide plane which intersects the conventional origin.
Example: No. 15, standard short symbol
The full symbols for the three cell choices (rows) and the three unique axes (columns) read Application of the priority rule would have resulted in the following symbols: Here, the transformation properties are obscured.
Comparison with earlier editions of International Tables. In IT (1935), each monoclinic space group was presented in one description only, with b as the unique axis. Hence, only one short Hermann–Mauguin symbol was needed.
In IT (1952), the caxis setting (first setting) was newly introduced, in addition to the baxis setting (second setting). This extension was based on a decision of the Stockholm General Assembly of the International Union of Crystallography in 1951 [cf. Acta Cryst. (1951), 4, 569 and Preface to IT (1952)]. According to this decision, the baxis setting should continue to be accepted as standard for morphological and structural studies. The two settings led to the introduction of full Hermann–Mauguin symbols for all 13 monoclinic space groups (e.g. and ) and of two different standard short symbols (e.g. and ) for the eight space groups with centred lattices or glide planes [cf. p. 545 of IT (1952)]. In the present volume (as in the previous editions of this series), only one of these standard short symbols is retained (see above and Section 2.1.3.3).
The caxis setting (primed labels) was obtained from the baxis setting (unprimed labels) by the following transformation: This corresponds to an interchange of two labels and not to the more logical cyclic permutation, as used in all editions of this series. The reason for this particular transformation was to obtain short spacegroup symbols that indicate the setting unambiguously; thus the lattice letters were chosen as C (baxis setting) and B (caxis setting). The use of A in either case would not have distinguished between the two settings [cf. pp. 7, 55 and 543 of IT (1952); see also Table 2.1.3.12].
As a consequence of the different transformations between b and caxis settings in IT (1952) and in this volume (and all editions of this series), some spacegroup symbols have changed. This is apparent from a comparison of pairs such as & and & in IT (1952) with the corresponding pairs in this volume, & and & . The symbols with Bcentred cells appear now for cell choice 2, as can be seen from Table 2.1.3.12.
Selection of monoclinic cell. In practice, the selection of the (righthanded) unit cell of a monoclinic crystal can be approached in three ways, whereby the axes refer to the bunique setting; for c unique similar considerations apply:

In one dimension, only one crystal family, one crystal system and one Bravais lattice exist. No name or common symbol is required for any of them. All onedimensional lattices are primitive, which is symbolized by the script letter ; cf. Table 2.1.1.1.
There occur two types of onedimensional point groups, 1 and . The latter contains reflections through a point (reflection point or mirror point). This operation can also be described as inversion through a point, thus for one dimension; cf. Section 2.1.2.
Two types of line groups (onedimensional space groups) exist, with Hermann–Mauguin symbols and , which are illustrated in Fig. 2.1.3.13. Line group , which consists of onedimensional translations only, has merely one (general) position with coordinate x. Line group consists of onedimensional translations and reflections through points. It has one general and two special positions. The coordinates of the general position are x and ; the coordinate of one special position is 0, that of the other . The site symmetries of both special positions are . For , the origin is arbitrary, for it is at a reflection point.

The two line groups (onedimensional space groups). Small circles are reflection points; large circles represent the general position; in line group , the vertical bars are the origins of the unit cells. 
The onedimensional point groups are of interest as `edge symmetries' of twodimensional `edge forms'; they are listed in Table 3.2.3.1 . The onedimensional space groups occur as projection and section symmetries of crystal structures.
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