There are three systems for space group designations (see table in detail):
i) All groups are numbered from 1 to 230 in the order of increasing point symmetry of the corresponding Bravais lattice (syngonies from triclinic to cubic).
ii) The more informative Schoenflies symbols is applied to space groups.
iii) The most detailed information on a space group symbol is contained in socalled international designations (see below). In the notations of space groups, the first (uppercase) letter of a space group symbol denotes the lattice type which is primitive (P), singleface centred (A, B, or C), allface centred (F), bodycentred (I), or rhombohedrally centred (R). However, for rhombohedral space groups, a primitive unit cell may also be used, but the symbol R is still selected in order to distinguish these space groups from the primitive trigonal space groups based on hexagonal axes.
HermannMauguin (HM) System is usually referred to as the International System since it has been used most frequently in crystallographic analysis. Some conventional "rules" for the symbols (names) of the space groups are used with the HM notation:
i) The symmetry elements are ordered in the same way as in the symbol of corresponding point group.
ii) The symbols for symmetry elements of space groups are more diverse than those for point groups, because in addition to rotation axes and mirror planes, a space group may contain more complex symmetry elements, i.e. screw axes (combination of rotation and translation) and glide planes (combination of mirror reflection and translation). Therefore, many different space groups can correspond to the same point group.
iii) If two or more axes have the same direction, only the axis with higher symmetry is listed in the symbols of space groups, for instance, space group P4_{2}/m2_{l}/n2/m is normally simplified as P4_{2}/mnm.
iv) For some space groups, secondary directions are the directions of the symmetry elements along unit cell translations b and c, while the tertiary directions correspond to the direction between unit cell translations b and c. For instance, symbols P6m2 and P62m represent two different space groups. For the space groups with oddorder axes 3 and 3, the perpendicular symmetry elements can go along unit cell translations b and c (e.g. P321) or between them (e.g. P312).
