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. 1.5, pp. 91106

1.5.4. Synoptic tables of plane and space groups^{2}
It is already clear from Section 1.5.3.1 that the Hermann–Mauguin symbols of a space group depend on the choice of the basis vectors. The purpose of this section is to give an overview of a large selection of possible alternative settings of space groups and their Hermann–Mauguin symbols covering most practical cases. In particular, the synoptic tables include two main types of information:

In order to interpret (or even determine) the extended symbol for a space group, one has to recall that all operations that belong to the same coset with respect to the translation subgroup have the same linear part, but that not all symmetry operations within a coset are operations of the same type. Furthermore, symmetry operations in one coset can belong to element sets of different symmetry elements.
In this section, a procedure for determining the types of symmetry operations and the corresponding symmetry elements is explained. It is a development of the method of geometrical interpretation discussed in Section 1.2.2.4 . The procedure is based on the originshift transformations discussed in Sections 1.5.1 and 1.5.2, and provides an efficient way of analysing the additional symmetry operations and symmetry elements. The key to the procedure is the decomposition of the translation part of a symmetry operation into an intrinsic translation part , which is fixed by the linear part of and thus parallel to the geometric element of , and a location part , which is perpendicular to the intrinsic translation part. Note that the space fixed by and the space perpendicular to this fixed space are complementary, i.e. their dimensions add up to 3, therefore this decomposition is always possible.
As described in Section 1.2.2.4 , the determination of the intrinsic translation part of a symmetry operation with linear part of order k is based on the fact that the kth power of 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 . This terminology is justified by the following observation: As explained in detail in Sections 1.5.1.3 and 1.5.2.3, under an origin shift by , a column of point coordinates is transformed tomaking in particular the new origin, and a matrix–column pair is transformed toApplied to the symmetry operation , known as the reduced symmetry operation in which the full translation part is replaced by the location part (thereby neglecting the intrinsic translation part), an origin shift by results inThis means that if it is possible to find an origin shift p such that , then with respect to the new origin the reduced symmetry operation is transformed to . But since the subspace perpendicular to the fixed space of clearly does not contain any vector fixed by , the restriction of to this subspace is an invertible linear transformation, and therefore for every location part there is indeed a suitable perpendicular to the fixed space of such that .
The fact that an origin shift by transforms the translation part of the reduced symmetry operation to is equivalent to being a fixed point of , which can also be seen directly becauseNote that for one fixed point of the reduced symmetry operation , the full set of fixed points, as defined in Section 1.2.4 , is obtained by adding to the fixed vectors of , because for an arbitrary fixed point of one has and since also one finds , i.e. the difference between two fixed points is a vector that is fixed by . In other words, the geometric element of is the space fixed by , translated such that it runs through .
Finally, in order to determine the symmetry element of the symmetry operation correctly, it may be necessary to reduce the intrinsic translation part by a lattice translation in the fixed space of .
Summarizing, the types of symmetry operations and their symmetry elements can be identified as follows:
This analysis allows one to read off the types of the symmetry operations and of the corresponding symmetry elements that occur for the coset of . The following two sections provide examples illustrating that in some cases the coset does not contain symmetry operations belonging to symmetry elements of different type, while in others it does.
In cases where the linear part of a symmetry operation fixes only the origin, all elements in the coset are of the same type. This is due to the fact that the translation part is decomposed as and . Since fixes only the origin, is invertible and a fixed point of the reduced operation can be found, as . This situation occurs when is an inversion or a three, four or sixfold rotoinversion. The element set of the symmetry element of an inversion consists only of this inversion; the element set of a rotoinversion consists of the rotoinversion and its inverse (the latter belonging to a different coset). Therefore, in these cases each symmetry operation in the coset of belongs to the element set of a different symmetry element (of the same type, namely an inversion centre or a rotoinversion axis).
Note that the above argument does not apply to twofold rotoinversions, since these are in fact reflections which fix a plane perpendicular to the rotoinversion axis and not only a single point. The following two examples illustrate that translations from a primitive lattice do not give rise to symmetry elements of different type in the cases of either a reflection or glide reflection with normal vector along one of the coordinate axes, or of a rotation or screw rotation with rotation axis along one of the coordinate axes.
Example 1
Let be an n glide with normal vector along the c axis. For the composition of with an integral translation one obtains a symmetry operation with translation part . The decomposition of into the intrinsic translation part and the location part gives and . This shows that the intrinsic translation part is only changed by the lattice vector and hence is a coplanar equivalent of the symmetry operation , which is an n glide with glide plane normal to the c axis and located at . One concludes that and belong to symmetry elements of the same type. The same conclusion would in fact remain true in the case of a Ccentred lattice, since the composition of with the centring translation would simply result in the intrinsic translation part being changed by the centring translation.
Example 2
As an example of a rotation, let be a fourfold rotation around the c axis. Composing with the translation results in the symmetry operation with intrinsic translation part and location part . Since we assume a primitive lattice, is an integer, hence is a coaxial equivalent of the symmetry operation , which has intrinsic translation part . To locate the geometric element of , one notes that for one hasThe symmetry operation therefore belongs to the symmetry element of a fourfold rotation with the line as geometric element. This analysis shows that all symmetry operations in the coset belong to the same type of symmetry element, since for each of these symmetry operations a coaxial equivalent can be found that has zero screw component.
The examples given in the previous section illustrate that in the case of a translation vector perpendicular to the symmetry axis or symmetry plane of a symmetry operation, the intrinsic translation vector remains unchanged and only the location of the geometric element is altered. In particular, composition with such a translation vector results in symmetry operations and symmetry elements of the same type. On the other hand, composition with translations parallel to the symmetry axis or symmetry plane give rise to coaxial or coplanar equivalents, which also belong to the same symmetry element. Combining these two observations shows that for integral translations, only translations along a direction inclined to the symmetry axis or symmetry plane can give rise to additional symmetry elements. For these cases, the additional symmetry operations and their locations are summarized in Table 1.5.4.1.
In space groups with a centred lattice, the translation subgroup contains also translations with nonintegral components, and these often give rise to symmetry operations and symmetry elements of different types in the same coset. An overview of additional symmetry operations and their locations that occur due to centring vectors is given in Table 1.5.4.2. In rhombohedral space groups all additional types of symmetry elements occur already as a result of combinations with integral lattice translations (cf. Table 1.5.4.1). For this reason, the rhombohedral centring R case is not included in Table 1.5.4.2.
In Section 1.4.2.4 the occurrence of glide reflections in a space group of type P4mm (due to integral translations inclined to a symmetry plane) and of type Fmm2 (due to centring translations) is discussed. We now provide some further examples illustrating the contents of Tables 1.5.4.1 and 1.5.4.2.
Example 3
Let be a threefold rotation along the [111] direction in a cubic (or rhombohedral) space group. Then the coset also contains the symmetry operation . With one sees that and hence the intrinsic translation part isIt follows that the location part is and one finds that for . Thus, the symmetry operation is of a different type to : it is a threefold screw rotation with the line as geometric element.
On the other hand, for an integer , the symmetry operation itself is a screw rotation, but it belongs to a symmetry element of rotation type, since it is a coaxial equivalent of the threefold rotation . The crucial difference between the symmetry operations and is that in the latter case the intrinsic translation part is a lattice vector, whereas for it is not.
This example illustrates in particular the occurrence of symmetry elements of screw or glide type even in the case of symmorphic space groups where all coset representatives with respect to the translation subgroup can be chosen with .
Note that, mainly for historical reasons, the screw rotations resulting from the threefold rotation along the [111] direction are not included in the extended Hermann–Mauguin symbol of cubic space groups, cf. Table 1.5.4.4. However, these screw rotations are represented in the cubic symmetryelement diagrams by the symbols (cf. Table 2.1.2.7 ), as can be observed in the symmetryelement diagram for a group of type P23 (195) in Fig. 1.5.4.1.
Example 4
A twofold rotation with the line as geometric element has linear part The composition of with the translation has intrinsic translation part and location part . Since for , the symmetry operation is a screw rotation with the line as geometric element and is thus of a different type to (cf. Table 1.5.4.1).
In an Icentred lattice, the composition of with the centring translation has intrinsic translation part and location part . One has for , hence the symmetry operation is a screw rotation with the line as geometric element and is thus of a different type to .
On the other hand, the translation subgroup also contains the translation . In this case, the intrinsic translation part of is , hence is of the same type as , i.e. a twofold rotation. The location part is and since for , the geometric element of is the line .
The analysis illustrates that the combination of the twofold rotation with Icentring translations gives rise to symmetry elements of rotation and of screw rotation type (cf. Table 1.5.4.2).
Example 5
Let be a reflection with the c axis normal to the reflection plane. An Fcentred lattice contains a centring translation and the composition of with this translation is an n glide, since the intrinsic translation part of is and consequently the location part is . The symmetry operation is thus an n glide with the plane as geometric element. However, since the intrinsic translation part is a lattice vector, and are coplanar equivalents and belong to the element set of the same symmetry element, which is a reflection plane.
The composition of with is a b glide, because has intrinsic translation part . The location part is and since for , the geometric element of this glide reflection is the plane . Likewise, the composition of with is an a glide with the same plane as geometric element. The two symmetry operations and , differing only by the lattice vector in their translation parts, are coplanar equivalents and belong to the element set of an eglide plane (cf. Section 1.2.3 for an introduction to eglide notation).
The possible planegroup symbols are listed in Table 1.5.4.3. Two cases of multiple cells are included in addition to the standard cells, namely the c centring in the square system and the h centring in the hexagonal system. The c centring is defined bywith centring points at 0, 0 and . The triple h cell is defined bywith centring points at 0, 0; and . The glide lines g directly listed under the mirror lines m in the extended and multiple cell symbols indicate that the two symmetry elements are parallel and alternate in the perpendicular direction.
Table 1.5.4.4 gives a comprehensive listing of the possible spacegroup symbols for various settings and choices of the unit cell. The data are ordered according to the crystal systems. The extended Hermann–Mauguin symbols provide information on the additional symmetry operations generated by the compositions of the symmetry operations with lattice translations. An extended Hermann–Mauguin symbol is a complex multiline symbol: (i) the first line contains those symmetry operations for which the coordinate triplets are explicitly printed under `Positions' in the spacegroup tables in this volume; (ii) the entries of the lines below indicate the additional symmetry operations generated by the compositions of the symmetry operations of the first line with lattice translations. For example, for A, B, C and Icentred space groups, the entries of the second line of the twoline extended symbol denote the symmetry operations generated by combinations with the corresponding centring translations.^{3}
In the triclinic system the corresponding symbols do not depend on any space direction. Therefore, only the two standard symbols P1 (1) and (2) are listed. One should, however, bear in mind that in some circumstances it might be more appropriate to use a centred cell for comparison purposes, e.g. following a phase transition resulting from a temperature, pressure or composition change.
The monoclinic and orthorhombic systems present the largest number of alternatives owing to various settings and cell choices. In the monoclinic system, three choices of unique axis can occur, namely b, c and a. In each case, two permutations of the other axes are possible, thus yielding six possible settings given in terms of three pairs, namely and , and , and . The unique axes are underlined and the negative sign, placed over the letter, maintains the correct handedness of the reference system. The three possible cell choices indicated in Fig. 1.5.3.1 increase the number of possible symbols by a factor of three, thus yielding 18 different cases for each monoclinic space group, except for five cases, namely P2 (3), (4), Pm (6), P2/m (10) and P2_{1}/m (11) with only six variants.
In monoclinic P lattices, the symmetry operations along the symmetry direction are always unique. Here again, as in the plane groups, the cell centrings give rise to additional entries in the extended Hermann–Mauguin symbols. Consider, for example, the data for monoclinic P12/m1 (10), C12/m1 (12) and C12/c1 (15) in Table 1.5.4.4. For P12/m1 and its various settings there is only one line, which corresponds to the full Hermann–Mauguin symbols; these contain only rotations 2 and reflections m. The first line for C12/m1 is followed by a second line, the first entry of which is the symbol 2_{1}/a, because 2_{1} screw rotations and a glide reflections also belong to this space group. Similarly, in C12/c1 rotations 2 and screw rotations 2_{1} and c and n glide reflections alternate, and thus under the full symbol one finds the entry .
In Table 1.5.4.4 the Hermann–Mauguin symbols of the orthorhombic space groups are listed in six different settings: the standard setting , and the settings , , , and . These six settings result from the possible permutations of the three axes. Let us compare for a few space groups the standard setting with the setting. For Pmm2 (25) the permutation yields the new setting P2mm, reflecting the fact that the twofold axes parallel to the c direction change to the a direction. The mirrors normal to a and b become normal to b and c, respectively.
The case of Cmm2 (35) is slightly more complex due to the centring. As a result of the permutation the C centring becomes an A centring. The changes in the twofold axes and mirrors are similar to those of the previous example and result in the A2mm setting of Cmm2.
The extended Hermann–Mauguin symbol of the centred space group Aem2 (39) reveals the nature of the eglide plane (also called the `double' glide plane): among the set of glide reflections through the same (100) plane, there exist two glide reflections with glide components and (for details of the eglide notation the reader is referred to Section 1.2.3 , see also de Wolff et al., 1992). In the setting, the A centring changes to a B centring and the double glide plane is now normal to b and the glide reflections have glide components and . The corresponding symbol is thus B2em. Note that in the cases of the five orthorhombic space groups whose Hermann–Mauguin symbols contain the eglide symbol, namely Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68), the characters in the first lines of the extended symbols differ from the short symbols because the characters in the extended symbol represent symmetry operations, whereas those in the short and full symbol represent symmetry elements. In all these cases, the extended symbols listed in Table 1.5.4.4 are complemented by the short symbols, given in brackets.
The general discussion in Section 1.5.4.1 about the additional symmetry operations that occur as a result of combinations with lattice translations provides some rules for the construction of the extended Hermann–Mauguin symbols in the orthorhombic crystal system. In orthorhombic space groups with primitive lattices, the symmetry operations of any symmetry direction are always unique: either 2 or 2_{1}, either m or a or b or c or n. In Ccentred lattices, owing to the possible combination of the original symmetry operations with the centring translations, the axes 2 along [100] and [010] alternate with axes 2_{1}. However, parallel to c there are either 2 or 2_{1} axes because the combination of a rotation or screw rotation with a centring translation results in another operation of the same kind. Similarly, alternates with , with , with etc. The reflection plane is simultaneously an glide plane and an glide plane is simultaneously a glide plane. This latter plane with its double role is the glide plane, as found for example in the full symbol of C2/m2/m2/e (67) and the corresponding short symbol Cmme. As another example, consider the space group C2/m2/c2_{1}/m (63). In Table 1.5.4.4, in the line of various settings for this space group the short Hermann–Mauguin symbols are listed, and the rotations or screw rotations do not appear. The , and reflections and glide reflections occur alternating with , and glide reflections, respectively. The entry under Cmcm is thus bnn.
F and I centring cause alternating symmetry operations for all three coordinate axes a, b and c. For these centrings, the permutation of the axes does not affect the symbol F or I of the centring type. However, the number of symmetry operations increases by a factor of four for F centrings and by a factor of two for I centrings when compared to those of a space group with a primitive lattice. In Fmm2 (42) for example, three additional lines appear in the extended symbol, namely ba2, and . These operations are obtained by combining successively the centring translations , and with the symmetry operations of Pmm2. However, in space groups Fdd2 (43) and Fddd (70) the nature of the d planes is not altered by the translations of the Fcentred lattice; for this reason, in Table 1.5.4.4 a twoline symbol for Fdd2 and a oneline symbol for Fddd are sufficient.
In tetragonal space groups with primitive lattices there are no alternating symmetry operations belonging to the symmetry directions [001] and [100]. However, for the symmetry direction the symmetry operations 2 and 2_{1} alternate, as do the reflection m and the glide reflection g [g is the name for a glide reflection with a glide vector ], and the glide reflections c and n. For example, the second line of the extended symbol of (133) contains the expression under the expression .
For the space groups in the tetragonal system, the unique axis is always the c axis, thus reducing the number of settings and choices of the unit cell. Two additional multiple cells are considered in this system, namely the C and F cells obtained from the P and I cell by the following relations:The secondary [100] and tertiary [110] symmetry directions are interchanged in this cell transformation. As an example, consider P4/n (85) and its description with respect to a Ccentred basis. Under the transformation , , , the n glide is transformed to an a glide while its coplanar equivalent glide is transformed to a b glide . Thus, the extended symbol of the multiplecell description of P4/n (85) shown in Table 1.5.4.4 is C4/a(b), while in accordance with the eglide convention, the short Hermann–Mauguin symbol becomes C4/e.
In the case of I4/m (87), as a result of the I centring, screw rotations 4_{2} and glide reflections n normal to 4_{2} appear as additional symmetry operations and are shown in the second line of the extended symbol (cf. Table 1.5.4.4). In the multiplecell setting, the space group F4/m exhibits the additional fourfold screw axis and owing to the new orientation of the and axes, which are rotated by 45° relative to the original axes a and b, the n glide of I4/m becomes an a glide in the extended Hermann–Mauguin symbol. The additional b glide obtained from a coplanar n glide is not given explicitly in the extended symbol.
The rhombohedral space groups are listed together with the trigonal space groups under the heading `Trigonal system'. For both representative symmetry directions [001]_{hex} and [100]_{hex}, rotations with screw rotations and reflections with glide reflections or different kinds of glide reflections alternate, so that additional symmetry operations always occur: rotations 3 or rotoinversions are accompanied by 3_{1} and 3_{2} screw rotations; 2 rotations alternate with 2_{1} screw rotations and m reflections or c glide reflections alternate with additional glide reflections. As examples, under the full Hermann–Mauguin symbol R3 (146) one finds and in the line under (167) one finds .
The extended Hermann–Mauguin symbols for space groups of the hexagonal crystal system retain the symbol for the primary symmetry direction [001]. Along the secondary and tertiary symmetry directions every horizontal axis 2 is accompanied by a screw rotation 2_{1}, while the reflections and glide reflections, or different types of glide reflections, alternate.
The list of hexagonal and trigonal spacegroup symbols is completed by a multiple H cell, which is three times the volume of the corresponding P cell. The unitcell transformation is obtained from the relationwith centring points at 0, 0, 0; and . The new vectors and are rotated by −30° in the ab plane with respect to the old vectors and . There are altogether six possible such multiple cells rotated by ±30°, ±90° and ±150° (cf. Table 1.5.1.1 and Fig. 1.5.1.8).
The hexagonal lattice is frequently referred to the orthohexagonal Ccentred cell (cf. Table 1.5.1.1 and Fig. 1.5.1.7). The volume of this centred cell is twice the volume of the primitive hexagonal cell and its basis vectors are mutually perpendicular.
In general, the space groups of the cubic system do not yield any additional orientations and only the short, full and extended symbols are given. The only exception to this general rule is the group (205) with its alternative setting , whose basis vectors are related by a rotation of 90° in the ab plane to the basis vectors of : . The different general reflection conditions of in comparison to those of indicate its importance for diffraction studies (cf. Table 1.6.4.25 ). In some extended symbols of the cubic groups, we note the use of the g or type of glide reflections as in, for example, (219). The g glide is a generic form of a glide plane which is different from the usual glide planes denoted by a, b, c, n, d or e. The symbols g, and indicate specific glide components and orientations that are specified in the Note to Table 1.5.4.4.
References
International Tables for Crystallography (1983). Vol. A, SpaceGroup Symmetry. Edited by Th. Hahn. Dordrecht: D. Reidel Publishing Company.International Tables for Crystallography (1995). Vol. A, SpaceGroup Symmetry. Edited by Th. Hahn, 4th revised ed. Dordrecht: Kluwer Academic Publishers.
International Tables for Xray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
Wolff, P. M. de, Billiet, Y., Donnay, J. D. H., Fischer, W., Galiulin, R. B., Glazer, A. M., Hahn, Th., Senechal, M., Shoemaker, D. P., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1992). Symbols for symmetry elements and symmetry operations. Final Report of the International Union of Crystallography Adhoc Committee on the Nomenclature of Symmetry. Acta Cryst. A48, 727–732.