A computer program for the determination of supergroups of space groups is described.

The International Tables Symmetry Database (https://symmdb.iucr.org/), part of International Tables for Crystallography, is a collection of individual databases of crystallographic space-group and point-group information. Programs are provided to access and interactively visualize the data, and also allow new data to be calculated `on the fly', facilitating 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.

The intrinsic, hyperbolic crystallography of the two-periodic, genus-two HCB and SQL surfaces is presented. All discrete groups containing the translations of the Euclidean embeddings of these surfaces are derived and examples of applications are given.

For single and double, ordinary and grey layer groups G, all the irreducible (half-)integer (co-)representations D^(μ)(G) are derived as well as the allowed (co-)representations of the little groups. Band representations for all types of layer groups are induced from the irreducible representations of the site-symmetry groups and decomposed into the irreducible components. Together with many other important characteristics of the layer groups, these results are derived with the program code POLSym and presented at the new web site https://nanolab.group/layer/.

The practical generation of all Bravais and quasicrystal symmetries from a finite group is described.

A periodic-like table of the 230 space groups, a sweeping overview in the spirit of the periodic table of the chemical elements, is presented. It organizes the 73 symmorphic space groups along two non-orthogonal axes of point group symmetry and general position multiplicity that separates the crystal systems in discrete color fields. The remaining non-symmorphic groups, for reasons of graphical economy, are enumerated as `isotopes' of their parent symmorphic groups.

A novel method for calculation of asymmetric units has been discovered. This development can simplify many crystallographic programs and speed up fast Fourier transform calculations.

A general theoretical framework based on group–subgroup and group–supergroup relations is proposed to describe and derive interpenetrating nets.

Sub- and supergroups of space groups are important tools for the prediction and investigation of continuous phase transitions. A new method is presented which facilitates the search for sub- and particularly supergroups of space groups. A new kind of phase transition is discussed.

Distinguishing between actual and apparent pseudosymmetry in electron backscatter diffraction patterns is nearly impossible, even for simulated patterns. However, the resulting lattice is always a superlattice as long as the signal is not a superposition of multiple patterns.