a computer program for analysis of group-subgroup relations between space groups
By S. Ivantchev,
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
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.
- 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 Zi.
- 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
Hj of G with the given index, and their distribution into
classes of conjugate subgroups with respect to G.
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).
- 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 Hj into classes of conjugate subgroups. If the index is not
specified than the result is the lattice of all maximal subgroups
Zi that relate G and H.
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
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
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
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.