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. 1.3, pp. 3741
Section 1.3.4.4. Other classifications of space groups
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Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands 
In this section we summarize a number of other classification schemes which are perhaps of slightly lower significance than those of spacegroup types, geometric crystal classes and Bravais types of lattices, but also play an important role for certain applications.
We have already seen that every space group can be assigned to a symmorphic space group in a natural way by setting the translation parts of coset representatives with respect to the translation subgroup to . The groups assigned to a symmorphic space group in this way all have the same translation lattice and the same point group but the different possibilities for the interplay between these two parts are ignored.
If we want to collect together all space groups that correspond to symmorphic space groups of the same type, we arrive at the classification into arithmetic crystal classes. This can also be seen as a classification of the symmorphic spacegroup types. The distribution of the space groups into arithmetic classes, represented by the corresponding symmorphic spacegroup types, is given in Table 2.1.3.3 .
The crucial observation for characterizing this classification is that space groups that correspond to the same symmorphic space group all have translation lattices of the same Bravais type. This means that the freedom in the choice of a basis transformation of the underlying vector space is restricted, because a primitive basis has to be mapped again to a primitive basis. Assuming that the point groups are written with respect to primitive bases, this means that the basis transformation is an integral matrix with determinant .
Definition
Two space groups and with point groups and , respectively, both written with respect to primitive bases of their translation lattices, are said to lie in the same arithmetic crystal class if can be obtained from by an integral basis transformation of determinant , i.e. if there is an integral 3 × 3 matrix with such that
Also, two integral matrix groups and are said to belong to the same arithmetic crystal class if they are conjugate by an integral 3 × 3 matrix with .
Example
Letbe reflections in the planes x = 0, y = 0 and x = y, respectively, and let , and be the integral matrix groups generated by these reflections. Then and belong to the same arithmetic crystal class because they are transformed into each other by the basis transformationinterchanging the x and y axes. But belongs to a different arithmetic crystal class, because is not conjugate to by an integral matrix of determinant . The two groups and belong, however, to the same geometric crystal class, because and are transformed into each other by the basis transformationwhich has determinant . This basis transformation shows that and can be interpreted as the action of the same reflection on a primitive lattice and on a Ccentred lattice.
As explained above, the number of arithmetic crystal classes is equal to the number of symmorphic spacegroup types: in dimension 2 there are 13 such classes, in dimension 3 there are 73 arithmetic crystal classes. The Hermann–Mauguin symbol of the symmorphic spacegroup type to which a space group belongs is obtained from the symbol for the spacegroup type of by replacing any screwrotation axis symbol N_{m} by the corresponding rotation axis symbol N and every glideplane symbol a, b, c, d, e, n by the symbol m for a mirror plane.
It is clear that the classification into arithmetic crystal classes refines both the classifications into geometric crystal classes and into Bravais classes, since in the first case only the point groups and in the second case only the translation lattices are taken into account, whereas for the arithmetic crystal classes the combination of point groups and translation lattices is considered. Note, however, that for the determination of the arithmetic crystal class of a space group it is not sufficient to look only at the type of the point group and the Bravais type of the translation lattice. It is crucial to consider the action of the point group on the translation lattice.
Example
Let and be space groups of types P3m1 (156) and P31m (157), respectively. Since and are symmorphic space groups of different types, they must belong to different arithmetic classes. The point groups and of and both belong to the same geometric crystal class with symbol 3m and the translation lattices of both space groups are primitive hexagonal lattices, and thus of the same Bravais type. It is the different action on the translation lattice which causes and to lie in different arithmetic classes:
In the conventional setting, the point group of contains the threefold rotationand the reflectionswhereas the point group of contains the same rotation and the reflectionsSince the threefold rotation is represented by the same matrix in both groups, the lattice basis for both groups can be taken as the conventional basis of a hexagonal lattice, with a and b of the same length and enclosing an angle of 120° and c perpendicular to the plane spanned by a and b. One now sees that in the reflection planes of , and contain the vectors , a and b, respectively, whereas in these vectors are just perpendicular to the reflection planes. In the socalled hexagonally centred lattice with primitive basis , , , the vectors and are perpendicular to the vectors a and b. The group can thus be regarded as the action of on the hexagonally centred lattice, showing that and are actions of the same group on different lattices which therefore belong to different arithmetic crystal classes.
As we have seen, the assignment of a space group to its arithmetic crystal class is equivalent to the assignment to its corresponding symmorphic space group, which in turn can be seen as an assignment to the combination of a point group and a lattice on which this point group acts. This correspondence between arithmetic crystal classes and point group/lattice combinations is reflected in the symbol for an arithmetic crystal class suggested in de Wolff et al. (1985), which is the symbol of the symmorphic space group with the letter for the lattice moved to the end, e.g. 4mmP for the arithmetic crystal class containing the symmorphic space groups of type P4mm (99) and the nonsymmorphic groups derived from this symmorphic group, i.e. the groups of spacegroup type P4bm, P4_{2}cm, P4_{2}nm, P4cc, P4nc, P4_{2}mc and P4_{2}bc (100–106).
Recall that the members of one arithmetic crystal class are space groups with the same translation lattice and the same point group, possibly written with respect to different primitive bases. If the point group happens to be the Bravais group of the translation lattice, this is independent of the chosen primitive basis and thus being a Bravais group is clearly a property of the full arithmetic crystal class.
Definition
The arithmetic crystal class of a space group is called a Bravais arithmetic crystal class if the point group of is the Bravais group of the translation lattice of .
The arithmetic crystal class of an integral matrix group is a Bravais arithmetic crystal class if is maximal among the integral matrix groups with the same space of metric tensors , i.e. if for any integral matrix group properly containing as a subgroup, the space of metric tensors is strictly smaller than that of . This amounts to saying that must act on a lattice with specialized metric.
Note that in the previous edition of IT A the shorter term Bravais class was used as a synonym for Bravais arithmetic crystal class. However, in this edition the term Bravais class is reserved for the classification of spacegroup types according to their lattices (see Section 1.3.4.3).
Since the lattice types are characterized by their Bravais groups, the Bravais arithmetic crystal classes are in onetoone correspondence with the Bravais types of lattices. The 14 Bravais arithmetic crystal classes (given by the symbol for the arithmetic class, with the number of the associated symmorphic spacegroup type in brackets) and the corresponding lattice types are: P (2), triclinic; 2/mP (10), primitive monoclinic; 2/mC (12), centred monoclinic; mmmP (47), primitive orthorhombic; mmmC (65), singlefacecentred orthorhombic; mmmF (69), allfacecentred orthorhombic; mmmI (71), bodycentred orthorhombic; 4/mmmP (123), primitive tetragonal; 4/mmmI (139), bodycentred tetragonal; (166), rhombohedral; 6/mmmP (191), hexagonal; (221), primitive cubic; (225), facecentred cubic; and (229), bodycentred cubic.
In the classification of space groups according to their translation lattices, the point groups play only a secondary role (as groups acting on the lattices). From the perspective of arithmetic crystal classes, this classification can now be reformulated in terms of integral matrix groups. The crucial point is that every arithmetic crystal class can be assigned to a Bravais arithmetic crystal class in a natural way: If is a point group, there is a unique Bravais arithmetic crystal class containing a Bravais group of minimal order with . Conversely, a Bravais group acting on a lattice is grouped together with its subgroups that do not act on a more general lattice, i.e. on a lattice with more free parameters than . This observation gives rise to the concept of Bravais flocks, which is mainly applied to matrix groups.
Definition
Two integral matrix groups and belong to the same Bravais flock if they are both conjugate by an integral basis transformation to subgroups of a common Bravais group, i.e. if there exists a Bravais group and integral 3 × 3 matrices and such that for all and for all . Moreover, , and must all have spaces of metric tensors of the same dimension.
Each Bravais flock consists of the union of the arithmetic crystal class of a Bravais group and the arithmetic crystal classes of the subgroups of that do not act on a more general lattice than .
The classification of space groups into Bravais flocks is the same as that according to the Bravais types of lattices and as that into Bravais classes. If the point groups and of two space groups and belong to the same Bravais flock, then the space groups are also said to belong to the same Bravais flock, but this is the case if and only if and belong to the same Bravais class.
Example
For the bodycentred tetragonal lattice the Bravais arithmetic crystal class is the arithmetic crystal class 4/mmmI and the corresponding symmorphic spacegroup type is I4/mmm (139). The other arithmetic crystal classes in this Bravais flock are (with the number of the corresponding symmorphic space group in brackets): 4I (79), (82), 4/mI (87), 422I (97), 4mmI (107), (119) and (121).
It is sometimes convenient to group together those Bravais types of lattices for which the Bravais groups belong to the same holohedry.
Definition
Two lattices belong to the same lattice system if their Bravais groups belong to the same geometric crystal class (which is thus a holohedry).
Remark : The lattice systems were called Bravais systems in earlier editions of this volume.
Example
The primitive cubic, facecentred cubic and bodycentred cubic lattices all belong to the same lattice system, because their Bravais groups all belong to the holohedry with symbol .
On the other hand, the hexagonal and the rhombohedral lattices belong to different lattice systems, because their Bravais groups are not even of the same order and lie in different holohedries (with symbols 6/mmm and , respectively).
From the definition it is obvious that lattice systems classify lattices because they consist of full Bravais types of lattices. On the other hand, the example of the geometric crystal class shows that lattice systems do not classify point groups, because depending on the chosen basis a point group in this geometric crystal class belongs to either the hexagonal or the rhombohedral lattice system.
However, since the translation lattices of space groups in the same Bravais class belong to the same Bravais type of lattices, the lattice systems can also be regarded as a classification of space groups in which full Bravais classes are grouped together.
Definition
Two Bravais classes belong to the same lattice system if the corresponding Bravais arithmetic crystal classes belong to the same holohedry.
More precisely, two space groups and belong to the same lattice system if the point groups and are contained in Bravais groups and , respectively, such that and belong to the same holohedry and such that , , and all have spaces of metric tensors of the same dimension.
Every lattice system contains the lattices of precisely one holohedry and a holohedry determines a unique lattice system, containing the lattices of the Bravais arithmetic crystal classes in the holohedry. Therefore, there is a onetoone correspondence between holohedries and lattice systems. There are four lattice systems in dimension 2 and seven lattice systems in dimension 3. The lattice systems in threedimensional space are displayed in Table 1.3.4.1. Along with the name of each lattice system, the Bravais types of lattices contained in it and the corresponding holohedry are given.

The point groups contained in a geometric crystal class can act on different Bravais types of lattices, which is the reason why lattice systems do not classify point groups. But the action on different types of lattices can be exploited for a classification of point groups by joining those geometric crystal classes that act on the same Bravais types of lattices. For example, the holohedry acts on primitive, facecentred and bodycentred cubic lattices. The other geometric crystal classes that act on these three types of lattices are 23, , 432 and .
Definition
Two space groups and with point groups and , respectively, belong to the same crystal system if the sets of Bravais types of lattices on which and act coincide. Since point groups in the same geometric crystal class act on the same types of lattices, crystal systems consist of full geometric crystal classes and the point groups and are also said to belong to the same crystal system.
Remark : In the literature there are many different notions of crystal systems. In International Tables, only the one defined above is used.
In many cases, crystal systems collect together geometric crystal classes for point groups that are in a group–subgroup relation and act on lattices with the same number of free parameters. However, this condition is not sufficient. If a point group is a subgroup of another point group , it is clear that acts on each lattice on which acts. But may in addition act on different types of lattices on which does not act.
Note that it is sufficient to consider the action on lattices with the maximal number of free parameters, since the action on these lattices implies the action on lattices with a smaller number of free parameters (corresponding to metric specializations).
Example
The holohedry of type 4/mmm acts on tetragonal and bodycentred tetragonal lattices. The crystal system containing this holohedry thus consists of all the geometric crystal classes in which the point groups act on tetragonal and bodycentred tetragonal lattices, but not on lattices with more than two free parameters. This is the case for all geometric crystal classes with point groups containing a fourfold rotation or rotoinversion and that are subgroups of a point group of type 4/mmm. This means that the crystal system containing the holohedry 4/mmm consists of the geometric classes of types 4, , 4/m, 422, 4mm, and 4/mmm.
This example is typical for the situation in threedimensional space, since in threedimensional space usually all the arithmetic crystal classes contained in a holohedry are Bravais arithmetic crystal classes. In this case, the geometric crystal classes in the crystal system of the holohedry are simply the classes of those subgroups of a point group in the holohedry that do not act on lattices with a larger number of free parameters.
The only exceptions from this situation are the Bravais arithmetic crystal classes for the hexagonal and rhombohedral lattices.
Example
A point group containing a threefold rotation but no sixfold rotation or rotoinversion acts both on a hexagonal lattice and on a rhombohedral lattice. On the other hand, point groups containing a sixfold rotation only act on a hexagonal but not on a rhombohedral lattice. The geometric crystal classes of point groups containing a threefold rotation or rotoinversion but not a sixfold rotation or rotoinversion form a crystal system which is called the trigonal crystal system. The geometric crystal classes of point groups containing a sixfold rotation or rotoinversion form a different crystal system, which is called the hexagonal crystal system.
The classification of the pointgroup types into crystal systems is summarized in Table 1.3.4.2.

Remark : Crystal systems can contain at most one holohedry and in dimensions 2 and 3 it is true that every crystal system does contain a holohedry. However, this is not true in higher dimensions. The smallest counterexamples exist in dimension 5, where two (out of 59) crystal systems do not contain any holohedry.
The classification into crystal systems has many important applications, but it has the disadvantage that it is not compatible with the classification into lattice systems. Space groups that belong to the hexagonal lattice system are distributed over the trigonal and the hexagonal crystal system. Conversely, space groups in the trigonal crystal system belong to either the rhombohedral or the hexagonal lattice system. It is therefore desirable to define a further classification level in which the classes consist of full crystal systems and of full lattice systems, or, equivalently, of full geometric crystal classes and full Bravais classes. Since crystal systems already contain only geometric crystal classes with spaces of metric tensors of the same dimension, this can be achieved by the following definition.
Definition
For a space group with point group the crystal family of is the union of all geometric crystal classes that contain a space group that has the same Bravais type of lattices as .
The crystal family of thus consists of those geometric crystal classes that contain a point group such that and are contained in a common supergroup (which is a Bravais group) and such that , and all act on lattices with the same number of free parameters.
In twodimensional space, the crystal families coincide with the crystal systems and in threedimensional space only the trigonal and hexagonal crystal system are merged into a single crystal family, whereas all other crystal systems again form a crystal family on their own.
Example
The trigonal and hexagonal crystal systems belong to a single crystal family, called the hexagonal crystal family, because for both crystal systems the number of free parameters of the corresponding lattices is 2 and a point group of type in the trigonal crystal system is a subgroup of a point group of type 6/mmm in the hexagonal crystal system.
A space group in the hexagonal crystal family belongs to either the trigonal or the hexagonal crystal system and to either the rhombohedral or the hexagonal lattice system. A group in the hexagonal crystal system cannot belong to the rhombohedral lattice system, but all other combinations of crystal system and lattice system are possible. The distribution of the space groups in the hexagonal crystal family over these different combinations is displayed in Table 1.3.4.3.

Remark : Up to dimension 3 it seems exceptional that a crystal family contains more than one crystal system, since the only instance of this phenomenon is the hexagonal crystal family consisting of the trigonal and the hexagonal crystal systems. However, in higher dimensions it actually becomes rare that a crystal family consists only of a single crystal system.
For the space groups within one crystal family the same coordinate system is usually used, which is called the conventional coordinate system (for this crystal family). However, depending on the application it may be useful to work with a different coordinate system. To avoid confusion, it is recommended to state explicitly when a coordinate system differing from the conventional coordinate system is used.
References
Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravaislattice types and arithmetic classes. Report of the International Union of Crystallography AdHoc Committee on the Nomenclature of Symmetry. Acta Cryst. A41, 278–280.Google Scholar