International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 
International Tables for Crystallography (2011). Vol. A1, ch. 1.7, pp. 5866
Section 1.7.3. Group–subgroup and group–supergroup relations between space groups^{a}Departamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E48080 Bilbao, Spain, and ^{b}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany 
If two space groups and form a group–subgroup pair , it is always possible to represent their relation by a chain of intermediate maximal subgroups : > > > . For a specified index of in there are, in general, a number of possible chains relating both groups, and a number of different subgroups isomorphic to . We have developed two basic tools for the analysis of the group–subgroup relations between space groups: SUBGROUPGRAPH (Ivantchev et al., 2000) and HERMANN (Capillas, 2006). Given the spacegroup types and and an index [i], both programs determine all different subgroups of with the given index and their distribution into classes of conjugate subgroups with respect to . Owing to its importance in a number of group–subgroup problems, the program COSETS is included as an independent application. It performs the decomposition of a space group in cosets with respect to one of its subgroups. Apart from these basic tools, there are two complementary programs which are useful in specific crystallographic problems that involve group–subgroup relations between space groups. The program CELLSUB calculates the subgroups of a space group for a given multiple of the unit cell. The common subgroups of two or three space groups are calculated by the program COMMONSUBS.
This program is based on the data for the maximal subgroups of index 2, 3 and 4 of the space groups of IT A1. These data are transformed into a graph with 230 nodes corresponding to the 230 spacegroup types. If two nodes in the graph are connected by an edge, the corresponding space groups form a group–maximal subgroup pair. Each one of these pairs is characterized by a group–subgroup index. The different maximal subgroups of the same spacegroup type are distinguished by corresponding matrix–column pairs (P, p) which give the relations between the standard coordinate systems of the group and the subgroup. The index and the set of transformation matrices are considered as attributes of the edge connecting the group with the subgroup.
The specification of the group–subgroup pair leads to a reduction of the total graph to a subgraph with as the top node and as the bottom node, see Example 1.7.3.1.1 at the end of this section. In addition, the subgraph, referred to as the general graph, contains all possible groups which appear as intermediate maximal subgroups between and . It is important to note that in the general graphs the spacegroup symbols indicate spacegroup types, i.e. all space groups belonging to the same spacegroup type are represented by one node on the graph. Such graphs are called contracted. The contracted graphs have to be distinguished from the complete graphs where all space groups occurring in a group–subgroup graph are indicated by different spacegroup nodes.
The number of the nodes in the general graph may be further reduced if the index of in is specified. The subgraph obtained is again of the contracted type.
The comparison of complete graphs and contracted graphs shows that the use of contracted graphs for the analysis of specific group–subgroup relations can be very misleading (see Example 1.7.3.1.1, Fig. 1.7.3.2 and Fig. 1.7.3.3). The complete graphs produced by SUBGROUPGRAPH are equal for subgroups of a conjugacy class; the different orientations and/or origin shifts of the conjugate subgroups are manifested by the different transformation matrices listed by the program.
Different chains of maximal subgroups for the group–subgroup pair are obtained following the possible paths connecting the top of the graph (the group ) with the bottom (the group ). Each group–maximal subgroup pair determines one step of this chain. The index of in equals the product of the indices for each one of the intermediate edges. The transformation matrices relating the standard bases of and are obtained by multiplying the matrices of each step of the chain. Thus, for each pair of group–subgroup types with a given index there is a set of transformation matrices (P, p)_{j}, where each matrix corresponds to a subgroup isomorphic to . Some of these subgroups could coincide. To find the different of , the program transforms the elements of the subgroup to the basis of the group using the different matrices (P, p)_{j} and compares the elements of the subgroups in the basis of . Two subgroups that are characterized by different transformation matrices are considered identical if their elements, transformed to the basis of the group , coincide.
The different subgroups are distributed into classes of conjugate subgroups with respect to by checking directly their conjugation relations with elements of .
Input to SUBGROUPGRAPH:
Output of SUBGROUPGRAPH:
The output is illustrated by Example 1.7.3.1.1.

Example 1.7.3.1.1
Consider the group–subgroup relations between the groups , No. 92, and , No. 4. If no index is specified then the graph of maximal subgroups that relates and is represented as a table indicating the spacegroup types of the possible intermediate space groups and the corresponding indices. The contracted general graph is shown in Fig. 1.7.3.1. Two edges with opposite arrows between a group–subgroup pair correspond to group–subgroup relations in both directions, e.g. the pair and .
When the index [i] of the subgroup in the group is specified, the resultant graph is reduced to the chains of maximal subgroups that correspond to the value of [i]. For example, in Fig. 1.7.3.2 the contracted graph of index 4 is shown. The data in Table 1.7.3.1 and the complete graph shown in Fig. 1.7.3.3 indicate that there are three different subgroups of of index 4, distributed over two classes of conjugate subgroups. The three different subgroups of spacegroup type of index 4 correspond to three sets of twofold screw axes in : those pointing along [100] and [010] of the tetragonal cell give rise to the two conjugate subgroups, and the third one (forming a class of conjugate subgroups by itself) is along the tetragonal axis. Their full Hermann–Mauguin symbols are , and . The corresponding transformations are listed in Table 1.7.3.1. The complete graph , index 4 (Fig. 1.7.3.3) also shows that three different maximal subgroup chains end at the same subgroup, each of them specified by a different transformation matrix (Table 1.7.3.1). The three different transformation matrices are related by elements of the normalizer of the subgroup.
The method of calculation of the subgroups and their distribution into classes of conjugate subgroups used in SUBGROUPGRAPH is not adequate for group–subgroup pairs of indices greater than 50. The program HERMANN has been developed to treat the cases of such considerable reduction of symmetry. It is a modification of SUBGROUPGRAPH and is based on Hermann's theorem (cf. Lemma 1.2.8.1.1 ). Consider a group–subgroup pair , with a general subgroup of index [i]. The existence and the uniqueness of Hermann's group , , implies the possibility of factorizing the group–subgroup chain and its index [i] into two subchains of smaller indices with . The first one, the socalled translationengleiche or tchain , is related to the reduction of the pointgroup symmetry in the subgroup. The second one is known as the klassengleiche or kchain , and it takes account of the loss of translations. (The index is equal to the cellmultiplication factor of the volumes of the primitive cells of the lattices.)
The program HERMANN calculates all subgroups of of index [i] and distributes them into conjugacy classes with respect to . In addition, the program indicates the corresponding Hermann groups. It is important to note that for a given pair of spacegroup types and index [i], Hermann groups of different spacegroup types can exist that belong to the same crystal class, cf. Example 1.7.3.1.2. However, there is a unique Hermann group for any group–subgroup pair of specific space groups.
The spacegroup types of the possible Hermann groups for the pair are determined by the following conditions: (i) the groups are subgroups of of index where is equal to the ratio of the pointgroup orders of and ; (ii) the point groups of coincide with the point group of , ; (iii) the groups have subgroups of index . The complete set of the subgroups of is calculated by the consecutive application of SUBGROUPGRAPH to the two subchains and . At this stage, the subgroups are distributed into conjugacy classes with respect to the corresponding Hermann groups only. Finally, the program determines the conjugacy classes of with respect to : these either coincide with conjugacy classes relative to , or some of the classes relative to merge together to form conjugacy classes with respect to .
Input to HERMANN:
The data needed are the spacegroup types of and and their index.
Output of HERMANN:
Essentially, the output of the program is a list of all subgroups of of index , distributed into conjugacy classes with respect to . The classes of conjugate subgroups are listed in different blocks depending on the spacegroup type of the Hermann groups . The subgroups in each class are listed explicitly and are distinguished by the transformation matrices . Options for transforming the elements of to the basis of and for decomposing in right cosets with respect to are available. In addition, it is possible to calculate the splittings of the Wyckoff positions of relative to (cf. Section 1.7.4, program WYCKSPLIT). There is an optional link to the program SYMMODES, which carries out a symmetry analysis of the possible distortions compatible with for the symmetry break .
Example 1.7.3.1.2
Consider the pair of group–subgroup types P422 (No. 89) > P2_{1} (No. 4), index [i] = 8 (Fig. 1.7.3.4). The factorization of the index into and follows from the ratio of the orders of the point groups of and . The two spacegroup types P2, No. 3, and C2, No. 5, satisfy the conditions for Hermann subgroups for the pair . There are three P2 Hermann subgroups of distributed in two conjugacy classes: the subgroups with twofold axes along [100] and [010] form a conjugacy class, and is a normal subgroup in . The two C2 Hermann subgroups with twofold axes along [110] and form a single conjugacy class. Each Hermann subgroup has just one normal subgroup of type of index . Altogether there are five different subgroups of of index [i] = 8, distributed in three conjugacy classes, following the conjugacy relationships of the corresponding Hermann groups.
The coset decomposition of a group with respect to a subgroup is a basic step in many problems that involve group–subgroup relations between space groups, e.g. the distribution of subgroups into conjugacy classes (cf. Section 1.2.6.3 ) or the determination of supergroups of space groups (cf. Section 2.1.7.4 ). The procedure for the coset decomposition is well defined: one transforms to the basis of by the transformation matrix that relates the two bases, and then one distributes the transformed elements of into left or right cosets with respect to (cf. Section 1.2.4.2 ). The elements and are called the coset representatives. Consider the spacegroup pair with the corresponding transformation matrix . The coset decomposition can be easily achieved if both and are decomposed into cosets with respect to the translation subgroup of , consisting of integer translations only (i.e. the coset decomposition is performed with respect to a primitive lattice of ). The determination of the coset representatives of for the rightcoset decomposition is simplified by the fact that two elements of that differ by an integer translation of belong to the same coset . This is, however, not in general true for the leftcoset decomposition: in that case, two elements of belong to the same coset if they differ by the translation where . In other words, two elements of (with the same rotational part) belong to the same coset if the difference between their translational parts satisfies .
Given the spacegroup pair with the corresponding transformation matrix , the program COSETS decomposes into left or right cosets with respect to .
Input to COSETS:
The data that are needed are the group , the subgroup and the transformation matrix that relates the default settings of and . The user can choose between right or leftcoset decomposition of with respect to .
Output of COSETS:
The output data consist of the list of the cosets of the decomposition of with respect to . The first coset corresponds to the subgroup represented by the set of its generalposition triplets, , where is the order of the point group and f is the socalled centring factor that equals the number of lattice points per cell. The triplets of the rest of the cosets are of the form for the leftcoset decomposition or for the rightcoset decomposition, with s, r = 1,, .
Example 1.7.3.1.3
Consider the group–subgroup pair (No. 166) > P2_{1}/c (No. 14) of index 6, with the transformation matrixrelating the bases defined for the default settings of the group and the subgroup (hexagonal axes setting for , and the unique axis b setting for ). The decomposition of with respect to consists of four cosets. (For P space groups, coincides with .) There are 36 coset representatives of , with consisting of integer translations only. From the determinant of the transformation matrix it follows that there are cosets in the decomposition of , i.e. some of the coset representatives of belong to the same cosets with respect to . The distribution of the coset representatives of into cosets of with respect to is different for the right and leftcoset decomposition. Consider the three coset representatives of corresponding to the threefold rotation, namely , and . In the basis of the subgroup, the rotational partis transformed by toand the translational parts by to , and , correspondingly. The elements with translational parts , belong to the same coset in the case of rightcoset decomposition, and can be taken as a representative of a different coset [the program chooses the element ( as a coset representative]. In the case of leftcoset decomposition, the coset representatives are and : the elements and belong to the same coset as the difference in their translational parts satisfies the condition , with and .
The output of COSETS gives the distribution of the 24 coset representatives of into the six cosets of the decomposition of with respect to . In the case of rightcoset decomposition, the elements , , , , , can be selected as coset representatives. The elements , , , are valid coset representatives also for the leftcoset decomposition of with respect to , while have to be substituted by , .
For several applications, it is of interest to determine the subgroups of a space group for a specific multiple of the cell, i.e. for a given index. This happens, for example, in the search for possible lowsymmetry phases after a phase transition from a known highsymmetry phase with experimental data indicating the reduction of translational symmetry. The program CELLSUB calculates the different subgroups of a space group for a given maximal index in two steps:
The method for obtaining the different subgroup types and indices of a given space group is similar to that used in the SUBGROUPGRAPH module. It is also based on the data for maximal subgroups of the space groups in IT A1. Given the group , the program constructs a graph of maximal subgroups, imposing the additional condition .
Input to CELLSUB:
Output of CELLSUB:
Two space groups and which are not in a group–subgroup relation may be related by common subgroups with . Given and , the program COMMONSUBS determines these groups . Such group–subgroup relations have a subjective component: in general the lattices of both space groups do not fit ideally. There will be some misfit between the lattice parameters of , a subgroup of and those of , a subgroup of . The decision as to how much misfit could be tolerated depends on the specific structural criteria applicable to the problem studied.
The higher the proportion of common symmetry between and , i.e. the smaller the indices and are, the more promising is the search for a relation between the two crystal structures. Owing to the theorem of Hermann both the indices and may be split into the pointgroup index and the lattice index , such that , m = 1, 2. The pointgroup indices are finite, ; the lattice index may have any value in principle. Large indices, however, mean little common symmetry and thus low probability of structural relevance. Therefore, it is reasonable to limit the value of and thus of and by introducing a maximal value .
If two structures with space groups and can be compared within a common subgroup , then the number of formula units of a primitive unit cell of should be the same for both structures. This means that , where and are the numbers of the formula units of the primitive unit cells of the crystal structures 1 (with space group ) and 2 (with space group ). It follows that or where is the order of the point group .
Given the space groups and , the formula units per primitive cells and , and , the program COMMONSUBS calculates the common subgroups in several steps:
The generalization of the procedure for the case of common subgroups of three groups is straightforward and has been also implemented in the program COMMONSUBS.
Input to COMMONSUBS:
The necessary data include the specification of the spacegroup types , the index and the number of formula units per conventional unit cell^{1} (or the ratio of the two indices for the special case of common subgroups of two space groups). The search for common subgroups can be further restricted by specifying the point group, the crystal class or the type of centring desired for the common subgroup.
Output of COMMONSUBS:
The existence of a common subgroup does not automatically mean the existence of a structural relation between the corresponding crystal structures. In general, the existence of a common supergroup of two space groups is mostly taken as more indicative of a structural relation than that of a common subgroup, provided the indices of the common supergroup are comparable with those of the common subgroup. However, the following example shows the utility of the commonsubgroup approach in the search for the possible symmetry of an intermediate phase between two phases with no group–subgroup relation between their space groups.
Example 1.7.3.1.4
The perovskitelike ferroelectric compounds PbZr_{1−x}Ti_{x}O_{3} exhibit a morphotropic^{2} phase boundary around x = 0.45–0.50. At compositions with x < 0.47 they are rhombohedral, at x > 0.47 tetragonal. For x = 0.48 a tetragonaltomonoclinic phase transition has been observed at ~300 K, the space group changing from P4mm, No. 99, to Cm, No. 8 (Noheda et al., 1999, 2000). The monoclinic structure results from the tetragonal one by shifts of the Pb and Zr/Ti atoms along the tetragonal [110] direction. The monoclinic structure can also be envisaged as a distorted variant of the rhombohedral phase, space group R3m, No. 160. There is no group–subgroup relation between P4mm and R3m, but Cm is a common subgroup of both. In this way, the monoclinic structure can be considered as providing a `bridge' between the rhombohedral and tetragonal regions of the morphotropic phase boundary. The application of COMMONSUBS for with , with , and (i.e. no cell multiplication) yields exactly as the common monoclinic subgroup.
Structural relations established through a common subgroup are also being used to model firstorder transformations between phases with no group–subgroup relation between their symmetry groups. The local symmetry of a common subgroup of the two end symmetries is supposed to describe approximately the symmetry constraints of the local transient states taking place during the transformation [see e.g. Capillas et al. (2007) and the references therein].
In some cases these intermediate configurations can even be stabilized and appear as stable intermediate phases in the phase diagram. The program COMMONSUBS can be useful in both types of searches.
The problem of the determination of the supergroups of a given space group is of rather general interest. For several applications it is not sufficient to know only the spacegroup types of the supergroups of a given group; it is instead necessary to have available all different supergroups that are isomorphic to and are of the same index [i]. In the literature there are few papers treating the supergroups of space groups in detail (Koch, 1984; Wondratschek & Aroyo, 2001). In IT A one finds only listings of minimal supergroups of space groups which, in addition, are not explicit: they only provide for each space group the list of those spacegroup types in which occurs as a maximal subgroup (cf. Section 2.1.6 ). It is not trivial to determine all supergroups if only the types of the minimal supergroups are known. The Bilbao Crystallographic Server offers two basic programs that solve this problem for a given finite index [i] (Ivantchev et al., 2002): (i) MINSUP, which gives all minimal supergroups of index 2, 3 and 4 of a given space group, and (ii) SUPERGROUPS, which calculates all different supergroups of a given spacegroup type and a given index.
As in the case of subgroups, we have developed two complementary programs that involve the calculation of supergroups of space groups: (i) the program CELLSUPER, for calculating the supergroups of a space group for a given index, and (ii) COMMONSUPER for the computation of common supergroups of two or more space groups.
In analogy to the case of minimal supergroups (cf. Section 2.1.7 ), the determination of all supergroups of a given spacegroup type and index [i] of a space group can be done by inverting the data for the subgroups of of index [i]. In the following we outline the basic arguments of this procedure.
Let be a space group and be one of its subgroups of index [i]. Then all subgroups of of the same index [i] and isomorphic to can be calculated by a dedicated module of SUBGROUPGRAPH (cf. Section 1.7.3.1.1). The number of subgroups with index [i] is finite for any space group. Therefore, such a list is always finite. Let be a member of this list. We are looking for all supergroups of index [i] that are isomorphic to , . Then the supergroups are also affine equivalent to , i.e. there must be a mapping such that , where is the group of all reversible affine mappings. Different supergroups are obtained if .
As in the case of minimal supergroups (cf. Section 2.1.7 ), there are two cases to be distinguished:
Summarizing: Any supergroup of , may be found by the following procedure: (i) determine all subgroups of the same index and distribute them into classes of conjugate subgroups with respect to . From each class of conjugate subgroups, choose a representative , specified by ;^{3} (ii) apply to the group in order to obtain the group ; and (iii) test whether is already among the determined supergroups of . If it is not, then is a new supergroup of and further supergroups may be generated by the coset representatives of the decomposition of relative to as explained above.
The procedure described above for the determination of supergroups is also applied to the determination of minimal supergroups of (cf. Section 2.1.7) . In this case, the distinct maximal subgroups , representatives of the classes of conjugate subgroups with respect to , are retrieved directly from the maximalsubgroup database of the server.
Input to MINSUP and SUPERGROUPS:
Output of MINSUP and SUPERGROUPS:
For the two supergroup programs the results contain:
From the considerations given above it should have become clear that the aim of the presented procedure and the supergroup programs is to solve the following `purely' grouptheoretical problem: Given a group–subgroup pair of space groups , determine all supergroups of isomorphic to . The procedure does not include any preliminary checks on the compatibility of the metric of the studied space group with that of a supergroup. Depending on the particular case some of the supergroups obtained are not space groups but just affine groups isomorphic to space groups (see Koch, 1984). As an example consider the cubic supergroups of , No. 19: only if the three basis vectors of have equal length can one speak of supergroups of the cubic spacegroup type , No. 198. However, for each group there exist affine analogues of as supergroups.
Example 1.7.3.2.1
Here we consider supergroups of with . As an example we consider the group–supergroup pair with , No. 16, and the supergroup , No. 89, of index [i] = 2. Further, we suppose that the group has specialized cell metrics specified as a = b = c.
In the subgroup data of there is only one entry for the subgroup of index 2. We are interested in all supergroups of index 2 of the group . The affine normalizer of coincides with its Euclidean normalizer and it has the translations , and the inversion as additional generators (cf. IT A, Table 15.2.1.4 ). The Euclidean normalizer of with a = b = c coincides with its affine normalizer. It corresponds to the cubic group with the additional generating translations , and (cf. IT A, Table 15.2.1.3 ). The decomposition of with respect to the intersection of the two normalizers contains six cosets, i.e. the group has six supergroups isomorphic to each other. The different supergroups, as calculated by MINSUP, are listed in Table 1.7.3.2. They are distinguished by their transformation matrix–column pairs and the coset representatives of the decomposition of with respect to . The existence of the six different supergroups becomes obvious if we consider the type and location of the symmetry elements corresponding to the listed coset representatives of the different supergroups (Table 1.7.3.2). Owing to the specialized metrics of , the fourfold axis of can be chosen along any of the three orthorhombic axes. Accordingly, the six supergroups are distributed into three pairs. Comparison of the spacegroup diagrams of and (Fig. 1.7.3.5) shows that the two supergroups for each orientation of the fourfold axis correspond exactly to the two possible locations of the fourfold axis in the orthorhombic cell.

Example 1.7.3.2.2
Here we consider the supergroups of with . Consider the group , No. 62, and its minimal supergroups of type , No. 63, of index 2. The group has two maximal subgroups: specified by , and with . The Euclidean and affine normalizers of and are identical and correspond to the group with the additional translations , and . Accordingly, the application of the normalizer procedure to any of the two group–subgroup pairs will not generate further equivalent supergroups. The first pair gives rise to the minimal supergroup . The second supergroup can only be obtained considering the second group–subgroup pair. Both supergroups are related by a cyclic rotation of the three axes which is not in the normalizer of (or ).
The number of supergroups of a space group of a finite index is not always finite. This is the case of a space group whose normalizer contains continuous translations in one, two or three independent directions (see IT A, Part 15 ). As typical examples one can consider the infinitely many centrosymmetric supergroups of the polar groups: there are no restrictions on the location of the additional inversion centre on the polar axis. For such group–supergroup pairs there could be up to three parameters r, s and t in the originshift column of the transformation matrix and in the translational part of the coset representatives. The parameters can have any value and each value corresponds to a different supergroup of the same spacegroup type.
The program CELLSUPER is an application similar to CELLSUB (cf. Section 1.7.3.1.4): in this case, the search is for the spacegroup types of supergroups of of a given maximum lattice index . The algorithm is similar to that of CELLSUB: using the data for the index and spacegroup types of minimal supergroups, the program constructs a tree of minimal supergroups starting from and imposing the condition . The input data of CELLSUPER coincide with those of CELLSUB with the only difference being that they are referred to the lowsymmetry group . The output data include:
The program COMMONSUPER calculates the spacegroup types of common supergroups of two space groups and for a given maximal lattice index . The procedure used is analogous to the one implemented in the program COMMONSUBS (cf. Section 1.7.3.1.5). The two sets of supergroups of and of are determined by the program CELLSUPER. The intersection of the sets of supergroups gives the set of the spacegroup types of the common supergroups of and with . A relation between the indices and is obtained by imposing the structural requirement of equal numbers of formula units in the (primitive) unit cell of the common supergroup obtained from the numbers of the formula units and of and : The program COMMONSUPER selects and lists those supergroups of the set whose indices and satisfy the above condition.
The input data for COMMONSUPER include the specification of and , the numbers of formula units per conventional unit cell, and the maximum lattice index . The output data of COMMONSUPER are:
Example 1.7.3.2.3
The program COMMONSUPER is useful in the search for structural relationships between structures whose symmetry groups and are not group–subgroup related. The derivation of the two structures as different distortions from a basic structure is a clear manifestation of such relationships. The symmetry group of the basic structure is a common supergroup of and . Consider the ternary intermetallic compound CeAuGe. At 8.7 GPa a firstorder phase transition is observed from a hexagonal arrangement (space group , No. 186, two formula units per unit cell, ) into an orthorhombic highpressure modification of symmetry , No. 62, (Brouskov et al., 2005). There is no group–subgroup relation between the symmetry groups of the high and lowpressure structures. For the program finds two common supergroups of , and , : (i) the group with and , and (ii) , with and . The common basic structure of the AlB_{2} type, proposed by Brouskov et al. (2005), corresponds to the common supergroup found by COMMONSUPER.
References
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