International Tables for Crystallography

Symmetry database

doi: 10.1107/97809553602060000112


This database is a unique resource that provides more extensive symmetry information than can be found in the print editions of International Tables for Crystallography Volumes A and A1. This additional symmetry information can either be retrieved directly from the database or simple calculations can be carried out using the database to generate specific data 'on the fly'. Use of the symmetry database thus facilitates the in-depth study of group-subgroup relations, domain structures and twinning, families of related crystal structures and phase transitions, and the prediction of new crystal structures.

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About the symmetry database

In the symmetry database, the data in Volume A are extended to include the generators, general positions and Wyckoff positions of all 230 space groups, including the 530 settings for the monoclinic and orthorhombic space groups listed in Table of Volume A. If data are not available for a particular setting directly, an arbitrary basis transformation can be specified and the data will be transformed to this new basis.

The maximal subgroup data given in Volume A1 are extended in the symmetry database to subgroups of arbitrary index for all the space groups, and series of isomorphic subgroups are available in a parametric form for indices up to 50 for orthorhombic, tetragonal, trigonal and hexagonal space groups and for indices up to 27 and 125 for cubic space groups. Interactive contracted and complete graphs of chains of maximal subgroups, including basis transformations and origin shifts for each step, can also be generated. In addition, data for supergroups of arbitrary index of all the space groups are provided. In contrast to Volume A1, where only space-group types of supergroups are indicated, in the symmetry database each supergroup is listed individually and specified by the transformation matrix that relates the conventional bases of the group and the supergroup. The subgroup and supergroup data can be transformed to the basis of the group. Left and right coset decomposition calculations can be carried out, and Wyckoff-position splittings can be obtained along with the relations between the coordinates of the positions within the group and subgroup.

Help can be obtained throughout the database by clicking on the help symbols help provided.