Bilbao Crystallographic Server SUBGROUPGRAPH Description

The program

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

• Given the space groups G and H with unspecified index, the program returns the G-H lattice of maximal subgroups containing the possible intermediate groups Z_i.
• If the index of H in G is specified the program determines all possible chains of maximal subgroups relating G and H, the different subgroups H_j of G with the given index, and their distribution into classes of conjugate subgroups with respect to G.

The method

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 Z_i 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 (P_j, p_j), where each matrix corresponds to a subgroup H_j isomorphic to H. Some of these subgroups could coincide. To find the different H_j of G, the program transforms the elements of the subgroup H in the basis of the group G using the different matrices (P_j, p_j), and compares the elements of the subgroups H_j 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 H_j 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 H_j 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

• As an input the program needs the specification of the space groups G and H.
  The groups G and H can be specified either by their sequential numbers, as listed in the International Tables for Crystallography, vol.A (ITA), or can be chosen from the list with the 230 space groups. For the monoclinic and the rhombohedral space groups as well as for the centrosymmetrical groups listed with respect to two origins in ITA, the settings used are as follows: unique axis b setting for the monoclinic groups, hexagonal axes setting for the rhombohedral groups, and origin 2 choice for the centrosymmetrical groups.
• If the index of H in G is specified then the program determines the chains of maximal subgroups relating these groups and classifies the isomorphic subgroups H_j into classes of conjugate subgroups. If the index is not specified than the result is the lattice of all maximal subgroups Z_i that relate G and H.

Output Information

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

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

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 H_j of G

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 Z_i 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 H_j. 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 H_j 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.