International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. E, ch. 1.2, pp. 2227
Section 1.2.17. Symbols^{a}Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^{b}Department of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 196106009, USA 
The following general criterion was used in selecting the sets of symbols for the subperiodic groups: consistency with the symbols used for the space groups given in IT A (2005). Specific criteria following from this general criterion are as follows:
A survey of sets of symbols that have been used for the subperiodic groups is given below. Considering these sets of symbols in relation to the above criteria leads to the sets of symbols for subperiodic groups used in Parts 2 , 3 and 4 .
A list of sets of symbols for the frieze groups is given in Table 1.2.17.1. The information provided in this table is as follows:

Sets of symbols which are of a nonHermann–Mauguin (international) type are the set of symbols of the `black and white' symmetry type (column 3) and the sets of symbols in columns 6 and 7. The sets of symbols in columns 4, 5 and 11 do not follow the sequence of symmetry directions used for twodimensional space groups. The sets of symbols in columns 3, 4, 5 and 10 do not use a lowercase script to denote a onedimensional lattice. The set of symbols in column 9 uses parentheses and square brackets to denote specific symmetry directions. The symbol g is used in Part 1 to denote a glide line, a standard symbol for twodimensional space groups (IT A , 2005). A letter identical with a basisvector symbol, e.g. a or c, is not used to denote a glide line, as is done in the symbols of columns 5, 6, 7, 9 and 11, as such a letter is a standard notation for a threedimensional glide plane (IT A , 2005).
Columns 2 and 3 show the isomorphism between frieze groups and onedimensional magnetic space groups. The onedimensional space groups are denoted by and . The list of symbols in column 3, on replacing r with , is the list of onedimensional magnetic space groups. The isomorphism between these two sets of groups interexchanges the elements and 1′ of the onedimensional magnetic space groups and, respectively, the elements and , mirror lines perpendicular to the [10] and [01] directions, of the frieze groups.
A list of sets of symbols for the rod groups is given in Table 1.2.17.2. The information provided in the columns of this table is as follows:

Sets of symbols which are of a nonHermann–Mauguin (international) type are the set of symbols in column 6 and the Nigglitype set of symbols in column 9. The set of symbols in column 8 does not use the lowercase script letter , as does IT A (2005), to denote a onedimensional lattice. The order of the characters indicating symmetry elements in the set of symbols in column 7 does not follow the sequence of symmetry directions used for threedimensional space groups. The set of symbols in column 4 have the characters indicating symmetry elements along nonlattice directions enclosed in parentheses, and do not use a lowercase script letter to denote the onedimensional lattice. Lastly, the set of symbols in column 4, without the parentheses and with the onedimensional lattice denoted by a lowercase script , are identical with the symbols in Part 3 , or in some cases are the second setting of rod groups whose symbols are given in Part 3 . These secondsetting symbols are included in the symmetry diagrams of the rod groups.
A list of sets of symbols for the layer groups is given in Table 1.2.17.3. The information provided in the columns of this table is as follows:

There is also a notation for layer groups, introduced by Janovec (1981), in which all elements in the group symbol which change the direction of the normal to the plane containing the translations are underlined, e.g. p4/m. However, we know of no listing of all layergroup types in this notation.
Sets of symbols which are of a nonHermann–Mauguin (international) type are the sets of symbols of the Schoenflies type (columns 11 and 12) and symbols of the `black and white' symmetry type (columns 16, 17, 18, 20, 21, 22, 24 and 25). Additional nonHermann–Mauguin (international) type sets of symbols are those in columns 14 and 23.
Sets of symbols which do not begin with a letter indicating the lattice centring type are the sets of symbols of the Niggli type (columns 13 and 15). The order of the characters indicating symmetry elements in the sets of symbols in columns 4 and 9 does not follow the sequence of symmetry directions used for threedimensional space groups. The set of symbols in column 6 uses parentheses to denote a symmetry direction which is not a lattice direction. In addition, the set of symbols in column 6 uses uppercase letters to denote the twodimensional lattice of the layer group, where as in IT A (2005) uppercase letters denote threedimensional lattices.
The symbols in column 8 are either identical with or, in some monoclinic and orthorhombic cases, are the secondsetting or alternativecellchoice symbols of the layer groups whose symbols are given in Part 4 . These secondsetting and alternativecellchoice symbols are included in the symmetry diagrams of the layer groups.
The isomorphism between layer groups and twodimensional magnetic space groups can be seen in Table 1.2.17.3. The set of symbols which we use for layer groups is given in column 2. The sets of symbols in columns 16, 17 and 22 are sets of symbols for the twodimensional magnetic space groups. The basic relationship between these two sets of groups is the interexchanging of the magnetic symmetry element 1′ and the layer symmetry element m_{z}. A detailed discussion of the relationship between these two sets of groups has been given by Opechowski (1986).
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