The program SYM (1982) for the derivation of the symmetry operations from the space-group symbols is improved. In the new program the symbol is checked for validity. In addition to the symmetry operations a group multiplication table is computed and seminvariance conditions are evaluated. Different settings of the 230 space-group types described by 541 different symbols can be generated using the origins of the International Tables or origins fixed by the user. Computer language is FORTRAN IV.
The set of origins available in the description of space groups is investigated with respect to equivalence. Privileged origins may be derived employing two different criteria: (i) A unique origin can be found for each short international space-group symbol by the application of simple rules (origin related to the symbol). (ii) The first standard settings of the International Tables (1952) are chosen such that the set of equivalent origins is a subset of the origins of their affine or Euclidean normalizers respectively. A list of the affine normalizers for space groups with higher than monoclinic symmetry is presented.
From the short Hermann-Mauguin space-group symbol a set of generating operators can be derived. The matrix description of the operators depends on three free parameters related to the origin of the setting. Simple rules allow the specification of an origin, the origin of the symbol. The use of any other origin is notated by appending a shift vector from the symbol origin to the new one selected.