Bilbao Crystallographic Server SUBGROUPGRAPH Description


a computer program for analysis of group-subgroup relations between space groups

By S. Ivantchev, E. Kroumova, G. Madariaga, J. M. Pérez-Mato, M. I. Aroyo
Departamentos de Física de la Materia Condensada y Física Aplicada II, Universidad del País Vasco, Apdo 644, 48080 Bilbao, Spain

The program

If two space groups G and H form a group-subgroup pair G > H, it is always possible to represent their relation by a chain of intermediate maximal subgroups Zi:
G > Z1 > ... > Zn > H.
For a specified index of H in G there are, in general, a number of possible chains relating both groups, and a number of different subgroups Hj isomorphic to H.

The program SUBGROUPGRAPH analyzes the group-subgroup relations between space groups. Its results can be summarized as follows:

In addition, the group-subgroup lattice/chain are represented as graphs with vertices corresponding to the space groups involved.

The method

The program is based on the data for the maximal subgroups of index 2, 3, and 4 of the space groups (International Tables for Crystallography, vol A1). This data is transformed into a graph with 230 vertices corresponding to the 230 space groups. If two vertices 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 space-group type are distinguished by the corresponding transformation matrices which describe the relations between the conventional bases 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 G > H leads to a reduction of the general graph to a subgraph with G as the top vertex and H as the bottom one. In addition, the G > H subgraph contains all possible groups Zi which appear as intermediate maximal subgroups between G and H. The number of the vertices is further reduced if the index of H in G is specified.

Different chains of maximal subgroups for the group-subgroup pair G > H are obtained following the possible paths connecting the top of the graph (the group G) with the bottom (the group H). Each group - maximal subgroup pair determines one step from this chain. The index of H in G equals the product of the indices for each one of the intermediate edges. The transformation matrices, relating the conventional bases of G and H, are obtained by multiplying the matrices of each step of the chain. Thus, for each chain with a given index, there is a set of transformation matrices (Pj, pj), where each matrix corresponds to a subgroup Hj isomorphic to H. Some of these subgroups could coincide. To find the different Hj of G, the program transforms the elements of the subgroup H in the basis of the group G using the different matrices (Pj, pj), and compares the elements of the subgroups Hj in the group basis. Two subgroups that are characterized by different transformation matrices are considered identical if their elements, transformed to the basis of the group G, coincide.

The different subgroups Hj are further distributed into classes of conjugate subgroups with respect to G by checking directly their conjugation relations with elements of G.

The distribution of the subgroups Hj into classes of conjugate subgroups obtained by this method can be compared with the corresponding results obtained by the application of the normalizer procedure described by Koch (Koch E., (1984) Acta Cryst. A 40, 593-600).

Input Information

Output Information

Lattice of maximal subgroups relating G and H with non specified index

When the index of the H in G is not specified, the program returns as a result the list of the possible intermediate space groups Zi relating G and H. The list is given in the form of a table which rows correspond to the intermediate space groups Zi, specified by their Hermann-Mauguin symbols. In addition, the table contains the maximal subgroups of Zi, specified by their ITA-numbers and the corresponding indices given in brackets.

This list is represented also as a graph. Each space group in the list corresponds to one vertex in the graph, and its maximal subgroups are the neighbors (successors) of this vertex. Group-subgroup relations in both directions (for example Pm-3 > Fm-3 and Fm-3 > Pm-3) are represented by vertices connected with two lines. If one vertex is connected to itself (a loop edge) then the corresponding space group has a maximal subgroup of the same type.

Chains of maximal subgroups relating G and H with given index

If the index of the subgroup H in the group G is specified the program returns a list with all of the possible chains of maximal subgroups relating G and H with this index. (Please note, that for the moment the program has no access to the data on maximal isomorphic subgroups with indices higher than four.) The indices corresponding to each step in the chain are given in brackets. The number of different transformation matrices as well as a link to the list with these matrices are given for each of the possible chains.

The graphical representation contains the intermediate groups that connect G and H with the specified index. This graph is a subgraph of the lattice of maximal subgroups with unspecified index.

Classification of the different subgroups Hj of G

Once the index of H in G is given, and the chains relating these groups with the corresponding index are obtained, the different subgroups are calculated and distributed into classes of conjugate subgroups.

Each class is represented in a table which contains the chains and the transformation matrices used to obtain the subgroups in this class. There is also a link to a list of the elements of the subgroups transformed to the basis of the group.

The graph in this case contains the same space group types Zi as the graph of the previous step but the different isomorphic subgroups are represented by different vertices. At the bottom of the graph are given all isomorphic subgroups Hj. Their labels are formed by the symbol of the subgroup followed by a number given in parenthesis which specifies the class of conjugate subgroups to which the subgroup Hj belongs.

Note that for group-subgroup pairs with high indices, where a lot of intermediate maximal subgroups occur, the resulting graph could be very complicated and difficult to overview.

Bilbao Crystallographic Server
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