The crystallographic problem: The problem of the determination of the supergroups of a given space group is of rather general interest. It is useful for search of overlooked symmetries in crystal structure determination, or in the analysis of successive phase transitions, where the introduction of a hypothetical (supergroup) parent phase can be important. Another important application is related with the detection of pseudosymmetries in crystal structures as a method of predicting structural phase transitions at higher temperatures (Igartua, Aroyo, Perez-Mato, 1996). There are few papers treating the supergroups of space groups in some detail (Koch, 1984). The existing listings of minimal supergroups of space groups are not complete as they provide only a list of those space groups G which contain a space group H as a maximal non-isomorphic subgroup (International Tables for Crystallography, vol. A, 1992 (ITA)). This information is in general not sufficient because it does not include all the possible supergroups of H isomorphic to G. The presented program determines all possible supergroups G_i of H which belong to the space group type of G.

Method of solution: SUPERGROUPS determines all possible supergroups G_i systematically inverting the data on maximal subgroups of space groups (International Tables for Crystallography, vol. A1 (ITA1)). This data has been prepared a CIF format in computer-readable form and is used by the program to determine all possible chains of maximal subgroups between the two groups and determines the transformations relating their conventional bases for a given group-subgroup pair G>H and index n. Following the procedure based in normalizers of space groups (Koch 1984, Wondratchek 1996) the program calculates all supergroups G_i>H, .

Software environment: SUPERGROUPS is running under any Unix or Unix-like operating system (Digital Unix, HP-UX, Sun, BSD, Linux, etc.). SUPERGROUPS is written in C. Only standard library functions are used. Parts of the program related to constructing the group-subgroup chains of maximal subgroups are written in Perl and awk. No overlay structure have been applied. For graphical representation of group-subgroup lattice the system daVinci (Frohlich, Werner, 1996) is used. The program can be used without local installation from any computer with www-browser (Unix, VMS, Macintosh, DOS, Windows, etc.)

Hardware environment: The program is running in any computer with Unix operating system (Intel, Alpha, Sparc, Mips, etc.). The executable programs takes about 50 kbyte of disk space and 500 kbyte memory. For the program one needs ITA1 in CIF format (3500 kbyte of disk space).

Program specifications:

Input: (i) Groups G and H. The data for G, H and their normalizers, available in the database, is in accordance with the conventions stated in ITA. In addition, the user is given the possibility to change the normalizer of H.
(ii a) The index of the subgroup in the supergroup or
(ii b) The transformation matrix relating conventional bases of the subgroup and supergroup.

The output: (i) All supergroups G_i in the basis of the subgroup represented by the coset representatives of G_i relative to H.

For input of type (ii a) All possible group-subgroup chains of minimal supergroups for the given G, H and index n and the corresponding transformation matrices. Graphical representation of the constructed lattice of minimal supergroups is possible too.

Documentation: A description of mathematical background of the used procedures, as well as a user's manual with the description of input and output of the program is included in the package. All the documentation is available on-line at http://lcdx00.wm.lc.ehu.es/cryst

Availability: The program can be used without local installation from any computer with www-browser via Internet. The URL of the documentation and the program itself is http://lcdx00.wm.lc.ehu.es/cryst.

Keywords: Space groups, supergroups, subgroups, group-subgroup chains, group-subgroup lattice.

References:
Igartua J.M., Aroyo M.I. & Perez-Mato J.M., Phys. Rev. B 54 (1996), 12744.
Koch E., Acta Cryst. A40 (1984) 593.
Wondratchek H., private communication.
International Tables of Crystallography (1992). Vol. A, Space Group Symmetry, 3nd ed., Dordrecht, Kluwer Academic Publishers.
International Tables of Crystallography Vol. A1, Maximal Subroup of Plane and Space Groups, to be published.
Frohlich M. & Werner M., daVinci V2.0 Online Documentation, Universitat Bremen, June 1996, http://www.informatik.uni-bremen.de/ davinci