TheThe server is built on a core of databases, and contains different shells. The innermost one is formed by simple retrieval tools which serve as an interface to the databases and permit to obtain the stored symmetry information for space groups and layer groups. The

The Bilbao Crystallographic Server is a web site with crystallographic programs and databases freely available on-line (http://www.cryst.ehu.es). The server gives access to general information related to crystallographic symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones etc.). Apart from the simple tools for retrieving the stored data, there are programs for the analysis of group-subgroup relations between space groups (subgroups and supergroups, Wyckoff-position splitting schemes etc.). There are also software packages studying specific problems of solid-state physics, structural chemistry and crystallography. This article reports on the programs treating representations of point and space groups. There are tools for the construction of irreducible representations, for the study of the correlations between representations of group-subgroup pairs of space groups and for the decompositions of Kronecker products of representations.

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The Brillouin-zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) offers k-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknell et al. [Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and special k vectors. Guides to the k-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.

The Bilbao Crystallographic Server (www.cryst.ehu.es) is a free web site with an access to crystallographic data of space and point groups, magnetic space groups, subperiodic groups, their representations and group-subgroup relations [1]. Wide range of complex solid-state physics and structure-chemistry aspects of materials studies are facilitated by the specialized software provided by the server. The server offers a set of structure-utility programs including basic tools for transformations between different structure descriptions or transformations compatible with a specific symmetry reduction. There is an online tool (COMPSTRU) for a quantitative analysis of the similarity of two structure models, also helpful for the recognition of identical or nearly identical atomic arrangements of different compounds. The program STRUCTURE RELATIONS for the analysis of structure relations between two phases of the same compound with group-subgroup related space groups is of great utility for the construction of family trees of homeotypic crystal structures, known as Baernighausen trees. The program AMPLIMODES performs the decomposition of the global distortion into symmetry-mode contributions and classifies the correlated atomic displacements separating the so-called primary modes (fundamental for the phase stability), from the weaker distortions of limited relevance for the transition mechanism [2]. The server also offers online tools for the evaluation of the pseudosymmetry of a given structure with respect to a supergroup of its space group, which could serve as a powerful method for the prediction of new ferroic materials. Of special interest is the program SUBGROUPS which extends further the capabilities of the server in the symmetry characterization of distorted structures by providing their possible subgroup symmetries given the relationship of the distorted to the parent undistorted lattice. Recently implemented computational tools and databases in the server allow the systematic application of symmetry arguments in the study of magnetic structures [3]. There is an online access to basic crystallographic data of magnetic space groups in different settings (MGENPOS, MWYCKPOS), to the systematic absences for non-polarized neutron magnetic diffraction and also to the symmetry-adapted forms of the corresponding structure factors (MAGNEXT). The user can identify a magnetic space group from its symmetry operations given in an arbitrary setting (IDENTIFY MAGNETIC GROUP), derive the possible magnetic space groups for a given set of propagation vectors (MAXMAGN, k-SUBGROUPSMAG) or generate a magnetic structure model complying with a chosen magnetic space group (MAGMODELIZE). The server offers an access to a database of more than 400 published magnetic structures (MAGNDATA) described using magnetic space groups for commensurate structures, and magnetic superspace groups for incommensurate structures. The presentation of the databases and programs offered by the Bilbao Crystallographic Server will be accompanied by case stu...