International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2006). Vol. E, ch. 1.2, pp. 22-27   | 1 | 2 |

Section 1.2.17. Symbols

V. Kopskýa and D. B. Litvinb*

aDepartment 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 bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610-6009, USA
Correspondence e-mail:  u3c@psu.edu

1.2.17. Symbols

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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[link]). Specific criteria following from this general criterion are as follows:

  • (1) The symbols of subperiodic groups are to be of the Hermann–Mauguin (international) type. This is the type of symbol used for space groups in IT A (2005[link]).

  • (2) A symbol of a subperiodic group is to consist of a letter indicating the lattice centring type followed by a set of characters indicating symmetry elements. This is the format of the Hermann–Mauguin (international) space-group symbols in IT A (2005[link]).

  • (3) The sets of symmetry directions and their sequences in the symbols of the subperiodic groups are those of the corresponding space groups. Layer and rod groups are three-dimensional subperiodic groups of the three-dimensional space groups, and frieze groups are two-dimensional subperiodic groups of the two-dimensional space groups. Consequently, the symmetry directions and sequence of the characters indicating symmetry elements in layer and rod groups are those of the three-dimensional space groups; in frieze groups, they are those of the two-dimensional space groups, see Table 1.2.4.1[link] above and Table 2.2.4.1[link] of IT A (2005[link]). Layer groups appear as subgroups of three-dimensional space groups, as factor groups of three-dimensional reducible space groups (Kopský, 1986[link], 1988[link], 1989a[link],b[link], 1993[link]; Fuksa & Kopský, 1993[link]) and as the symmetries of planes which transect a crystal of a given three-dimensional space-group symmetry. For example, the layer group pmm2 is a subgroup of the three-dimensional space group Pmm2; is isomorphic to the factor group Pmm2/Tz of the three-dimensional space group Pmm2, where Tz is the translational subgroup of all translations along the z axis; and is the symmetry of the plane transecting a crystal of three-dimensional space-group symmetry Pmm2, perpendicular to the z axis, at [z = 0]. In these examples, the symbols for the three-dimensional space group and the related subperiodic layer group differ only in the letter indicating the lattice type.

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[link] , 3[link] and 4[link] .

1.2.17.1. Frieze groups

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A list of sets of symbols for the frieze groups is given in Table 1.2.17.1[link]. The information provided in this table is as follows:

  • Columns 1 and 2: sequential numbering and symbols used in Part 2[link] .

    Table 1.2.17.1| top | pdf |
    Frieze-group symbols

     1234567891011
    Oblique 1 [\scr p]1 r1 r1 r111 [(a)] t 1 p[1](1)1 r1 [\scr p]1
    2 [\scr p]211 r[\bar{1}'] r112 r112 [(a):2] [t:2] 5 p[2](1)1 r2 [\scr p]112
    Rectangular 3 [\scr p]1m1 r[\bar{1}] r1m rm11 [(a):m] [t:m] 3 p[1](1)m r1m [\scr p]m11
    4 [\scr p]11m r11′ rm r1m1 [(a)\cdot m] [t\cdot m] 2 p[1](m)1 r11m [\scr p]1m1
    5 [\scr p]11g r21 rg r1c1 [(a)\cdot\bar{a}] [t\cdot a] 4 p[1](c)1 r11g [\scr p]1a1
    6 [\scr p]2mm r[\bar{1}]1′ rmm2 rmm2 [(a):2\cdot m] [t:2\cdot m] 6 p[2](m)m r2mm [\scr p]mm2
    7 [\scr p]2mg r2[\bar{1}] rgm2 rmc2 [(a):2\cdot\bar{a}] [t:2\cdot a] 7 p[2](c)m r2mg [\scr p]ma2
  • Columns 3, 4 and 5: symbols listed by Opechowski (1986[link]).

  • Column 6: symbols listed by Shubnikov & Koptsik (1974[link]).

  • Column 7: symbols listed by Vainshtein (1981[link]).

  • Columns 8 and 9: sequential numbering and symbols listed by Bohm & Dornberger-Schiff (1967[link]).

  • Column 10: symbols listed by Lockwood & Macmillan (1978[link]).

  • Column 11: symbols listed by Shubnikov & Koptsik (1974[link]).

Sets of symbols which are of a non-Hermann–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 two-dimensional space groups. The sets of symbols in columns 3, 4, 5 and 10 do not use a lower-case script [{\scr p}] to denote a one-dimensional 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 two-dimensional space groups (IT A , 2005[link]). A letter identical with a basis-vector 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 three-dimensional glide plane (IT A , 2005[link]).

Columns 2 and 3 show the isomorphism between frieze groups and one-dimensional magnetic space groups. The one-dimensional space groups are denoted by [{\scr p}1] and [{\scr p}\bar{1}]. The list of symbols in column 3, on replacing r with [{\scr p}], is the list of one-dimensional magnetic space groups. The isomorphism between these two sets of groups interexchanges the elements [\bar{1}] and 1′ of the one-dimensional magnetic space groups and, respectively, the elements [m_x] and [m_y], mirror lines perpendicular to the [10] and [01] directions, of the frieze groups.

1.2.17.2. Rod groups

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A list of sets of symbols for the rod groups is given in Table 1.2.17.2[link]. The information provided in the columns of this table is as follows:

  • Columns 1 and 2: sequential numbering and symbols used in Part 3[link] .

    Table 1.2.17.2| top | pdf |
    Rod-group symbols

     12 3456789
    Triclinic [1] [{\scr p}1] [1] [P(11)1] [1] [(a)\cdot 1] [p1] [r1] [1P1]
    [2] [{\scr p}\bar{1}] [2] [P(\bar{1}\bar{1})\bar{1}] [7] [(a)\cdot \bar{1}] [p\bar{1}] [r\bar{1}] [1P\bar{1}]
    Monoclinic/inclined [3] [{\scr p}211] [6] [P(12)1] [2] [(a):2] [p112] [r112] [1P2]
    [4] [{\scr p}m11] [3] [P(1m)1] [22] [(a)\cdot m] [p11m] [r1m1] [mP1]
    [5] [{\scr p}c11] [5] [P(1c)1] [24] [(a)\cdot \bar{a}] [p11a] [r1c1] [gP1]
    [6] [{\scr p}2/m11] [9] [P(12/m)1] [25] [(a):2:m] [p112/m] [r12/m1] [mP2]
    [7] [{\scr p}2/c11] [12] [P(12/c)1] [28] [(a):2:\bar{a}] [p112/a] [r12/c1] [gP2]
    Monoclinic/orthogonal [8] [{\scr p}112] [7] [P(11)2] [3] [(a)\cdot 2] [p211] [r211] [2P1]
    [9] [{\scr p}112_{1}] [8] [P(11)2_{1}] [8] [(a)\cdot 2_{1}] [p2_{1}] [r2_{1}] [2_{1}P1]
    [10] [{\scr p}11m] [4] [P(11)m] [23] [(a):m] [pm11] [rm11] [1Pm]
    [11] [{\scr p}112/m] [10] [P(11)2/m] [26] [(a)\cdot 2:m] [p2/m11] [r2/m11] [2Pm]
    [12] [{\scr p}112_{1}/m] [11] [P(11)2_{1}/m] [27] [(a)\cdot 2_{1}:m] [p2_{1}/m11] [r2_{1}/m11] [2_{1}Pm]
    Orthorhombic [13] [{\scr p}222] [18] [P(22)2] [61] [(a)\cdot 2:2] [p222] [r222] [2P22]
    [14] [{\scr p}222_{1}] [19] [P(22)2_{1}] [62] [(a)\cdot 2_{1}:2] [p2_{1}22] [r2_{1}22] [2_{1}P22]
    [15] [{\scr p}mm2] [13] [P(mm)2] [34] [(a)\cdot 2\cdot m] [p2mm] [r2mm] [2mmP1]
    [16] [{\scr p}cc2] [16] [P(cc)2] [35] [(a)\cdot 2\cdot \bar{a}] [p2aa] [r2cc] [2ggP1]
    [17] [{\scr p}mc2_{1}] [15] [P(mc)2_{1}] [36] [(a)\cdot 2_{1}\cdot m] [p2_{1}ma] [r2_{1}mc] [2_{1}mgP1]
    [18] [{\scr p}2mm] [14] [P(2m)m] [33] [(a):2\cdot m] [pmma] [rmm2] [mPm2]
    [19] [{\scr p}2cm] [17] [P(2c)m] [37] [(a):2\cdot \bar{a}] [pma2] [rmc2] [gPm2]
    [20] [{\scr p}mmm] [20] [P(2/m2/m)2/m] [46] [(a)\cdot m\cdot 2:m] [pmmm] [r2/m2/m2/m] [mmPm]
    [21] [{\scr p}ccm] [21] [P(2/c2/c)2/m] [47] [(a)\cdot \bar{a}\cdot 2:m] [pmaa] [r2/m2/c2/c] [ggPm]
    [22] [{\scr p}mcm] [22] [P(2/m2/c)2_{1}/m] [48] [(a)\cdot m\cdot 2_{1}:m] [pmma] [r2_{1}/m2/m2/c] [mgPm]
    Tetragonal [23] [{\scr p}4] [26] [P4(11)] [5] [(a)\cdot 4] [p4] [r4] [4P1]
    [24] [{\scr p}4_{1}] [27] [P4_{1}(11)] [11] [(a)\cdot 4_{1}] [p4_{1}] [r4_{1}] [4_{1}P1]
    [25] [{\scr p}4_{2}] [28] [P4_{2}(11)] [12] [(a)\cdot 4_{2}] [p4_{2}] [r4_{2}] [4_{2}P1]
    [26] [{\scr p}4_{3}] [29] [P4_{3}(11)] [13] [(a)\cdot 4_{3}] [p4_{3}] [r4_{3}] [4_{3}P1]
    [27] [{\scr p}\bar{4}] [23] [P\bar{4}(11)] [20] [(a)\cdot \bar{4}] [p\bar{4}] [r\bar{4}] [1P\bar{4}]
    [28] [{\scr p}4/m] [30] [P4/m(11) ] [29] [(a)\cdot 4:m] [p4/m] [r4/m] [4Pm]
    [29] [{\scr p}4_{2}/m] [31] [P4_{2}/m(11)] [30] [(a)\cdot 4_{2}:m] [p4_{2}/m] [r4_{2}/m] [4_{2}Pm]
    [30] [{\scr p}422] [35] [P4(22)] [66] [(a)\cdot 4:2] [p422] [r422] [4P22]
    [31] [{\scr p}4_{1}22] [36] [P4_{1}(22)] [67] [(a)\cdot 4_{1}:2] [p4_{1}22] [r4_{1}22] [4_{1}P22]
    [32] [{\scr p}4_{2}22] [37] [P4_{2}(22)] [68] [(a)\cdot 4_{2}:2] [p4_{2}22] [r4_{2}22] [4_{2}P22]
    [33] [{\scr p}4_{3}22] [38] [P4_{3}(22)] [69] [(a)\cdot 4_{3}:2] [p4_{3}22] [r4_{3}22] [4_{3}P22]
    [34] [{\scr p}4mm] [32] [P4(mm)] [40] [(a)\cdot 4\cdot m] [p4mm] [r4mm] [4mmP1]
    [35] [{\scr p}4_{2}cm] [33] [P4_{2}(cm)] [42] [(a)\cdot 4_{2}\cdot m] [p4_{2}ma] [r4_{2}mc] [4_{2}mgP1]
    [36] [{\scr p}4cc] [34] [P4(cc)] [41] [(a)\cdot 4\cdot \bar{a}] [p4aa] [r4cc] [4ggP1]
    [37] [{\scr p}\bar{4}2m] [24] [P\bar{4}(2m)] [49] [(a)\cdot \bar{4}\cdot m] [p\bar{4}2m] [r\bar{4}m2] [mP\bar{4}2]
    [38] [{\scr p}\bar{4}2c] [25] [P\bar{4}(2c)] [50] [(a)\cdot \bar{4}\cdot \bar{a}] [p\bar{4}2a] [r\bar{4}c2] [gP\bar{4}2]
    [39] [{\scr p}4/mmm] [39] [P4/m(2/m2/m)] [53] [(a)\cdot m\cdot 4:m] [p4/mmm] [r4/m2/m2/m] [4mmPm]
    [40] [{\scr p}4/mmc] [40] [P4/m(2/c2/c)] [54] [(a)\cdot \bar{a}\cdot 4:m] [p4/maa] [r4/m2/c2/c] [4ggPm]
    [41] [{\scr p}4_{2}/mmc] [41] [P4_{2}/m(2/m2/c)] [55] [(a)\cdot m\cdot 4_{2}:m] [p4_{2}/mma] [r4_{2}/m2/m2/c] [4_{2}mgPm]
    Trigonal [42] [{\scr p}3] [42] [P3(11)] [4] [(a)\cdot 3] [p3] [r3] [3P1]
    [43] [{\scr p}3_{1}] [43] [P3_{1}(11)] [9] [(a)\cdot 3_{1}] [p3_{1}] [r3_{1}] [3_{1}P1]
    [44] [{\scr p}3_{2}] [44] [P3_{2}(11)] [10] [(a)\cdot 3_{2}] [p3_{2}] [r3_{2}] [3_{2}P1]
    [45] [{\scr p}\bar{3}] [45] [P\bar{3}(11)] [19] [(a)\cdot \bar{6}] [p\bar{3}] [r\bar{3}] [3P\bar{1}]
    [46] [{\scr p}312] [48] [P3(21)] [63] [(a)\cdot 3:2] [p32] [r32] [3P2]
    [47] [{\scr p}3_{1}12] [49] [P3_{1}(21)] [64] [(a)\cdot 3_{1}:2] [p3_{1}2] [r3_{1}2] [3_{1}P2]
    [48] [{\scr p}3_{2}12] [50] [P3_{2}(21)] [65] [(a)\cdot 3_{2}:2] [p3_{2}2] [r3_{2}2] [3_{2}P2]
    [49] [{\scr p}3m1] [46] [P3(m1)] [38] [(a)\cdot 3\cdot m] [p3m] [r3m] [3mP1]
    [50] [{\scr p}3c1] [47] [P3(c1)] [39] [(a)\cdot 3\cdot \bar{a}] [p3a] [r3c] [3gP1]
    [51] [{\scr p}\bar{3}1m] [51] [P\bar{3}(m1)] [59] [(a)\cdot \bar{6}\cdot m] [p\bar{3}m] [r\bar{3}2/m] [3mP\bar{1}2]
    [52] [{\scr p}\bar{3}1c] [52] [P\bar{3}(c1)] [60] [(a)\cdot \bar{6}\cdot \bar{a}] [p\bar{3}a] [r\bar{3}2/c] [3gP\bar{1}2]
    Hexagonal [53] [{\scr p}6] [56] [P6(11)] [6] [(a)\cdot 6] [p6] [r6] [6P1]
    [54] [{\scr p}6_{1}] [57] [P6_{1}(11)] [14] [(a)\cdot 6_{1}] [p6_{1}] [r6_{1}] [6_{1}P1]
    [55] [{\scr p}6_{2}] [59] [P6_{2}(11)] [15] [(a)\cdot 6_{2}] [p6_{2}] [r6_{2}] [6_{2}P1]
    [56] [{\scr p}6_{3}] [61] [P6_{3}(11)] [16] [(a)\cdot 6_{3}] [p6_{3}] [r6_{3}] [6_{3}P1]
    [57] [{\scr p}6_{4}] [60] [P6_{4}(11)] [17] [(a)\cdot 6_{4}] [p6_{4}] [r6_{4}] [6_{4}P1]
    [58] [{\scr p}6_{5}] [58] [P6_{5}(11)] [18] [(a)\cdot 6_{5}] [p6_{5}] [r6_{5}] [6_{5}P1]
    [59] [{\scr p}\bar{6}] [53] [P\bar{6}(11)] [21] [(a)\cdot 3:m] [p\bar{6}] [r\bar{6}] [3Pm]
    [60] [{\scr p}6/m] [62] [P6/m(11)] [31] [(a)\cdot 6:m] [p6/m] [r6/m] [6Pm]
    [61] [{\scr p}6_{3}/m] [63] [P6_{3}/m(11)] [32] [(a)\cdot 6_{3}:m] [p6_{3}/m] [r6_{3}/m] [6_{3}Pm]
    [62] [{\scr p}622] [67] [P6(22)] [70] [(a)\cdot 6:2] [p622] [r622] [6P22]
    [63] [{\scr p}6_{1}22] [68] [P6_{1}(22)] [71] [(a)\cdot 6_{1}:2] [p6_{1}22] [r6_{1}22] [6_{1}P22]
    [64] [{\scr p}6_{2}22] [70] [P6_{2}(22)] [72] [(a)\cdot 6_{2}:2] [p6_{2}22] [r6_{2}22] [6_{2}P22]
    [65] [{\scr p}6_{3}22] [72] [P6_{3}(22)] [73] [(a)\cdot 6_{3}:2] [p6_{3}22] [r6_{3}22] [6_{3}P22]
    [66] [{\scr p}6_{4}22] [71] [P6_{4}(22)] [74] [(a)\cdot 6_{4}:2] [p6_{4}22] [r6_{4}22] [6_{4}P22]
    [67] [{\scr p}6_{5}22] [69] [P6_{5}(22)] [75] [(a)\cdot 6_{5}:2] [p6_{5}22] [r6_{5}22] [6_{5}P22]
    [68] [{\scr p}6mm] [64] [P6(mm)] [43] [(a)\cdot 6\cdot m] [p6mm] [r6mm] [6mmP1]
    [69] [{\scr p}6cc] [65] [P6(cc)] [44] [(a)\cdot 6\cdot \bar{a}] [p6aa] [r6cc] [6ggP1]
    [70] [{\scr p}6_{3}mc] [66] [P6_{3}(cm)] [45] [(a)\cdot 6_{3}\cdot m] [p6_{3}ma] [r6_{3}mc] [6_{3}mgP1]
    [71] [{\scr p}\bar{6}m2] [54] [P\bar{6}(m2)] [51] [(a)\cdot m\cdot 3:m] [p\bar{6}m2] [r\bar{6}m2] [3mPm2]
    [72] [{\scr p}\bar{6}c2] [55] [P\bar{6}(c2)] [52] [(a)\cdot \bar{a}\cdot 3:m] [p\bar{6}a2] [r\bar{6}c2] [3gPm2]
    [73] [{\scr p}6/mmm] [73] [P6/m(2/m2/m)] [56] [(a)\cdot m\cdot 6:m] [p6/mmm] [r6/m2/m2/m] [6mmPm]
    [74] [{\scr p}6/mcc] [74] [P6/m(2/c2/c)] [57] [(a)\cdot \bar{a}\cdot 6:m] [p6/maa] [r6/m2/c2/c] [6ggPm]
    [75] [{\scr p}6_{3}/mmc] [75] [P6_{3}/m(2/c2/m)] [58] [(a)\cdot m\cdot 6_{3}:m] [p6_{3}/mma] [r6_{3}/m2/m2/c] [6_{3}mgPm]
  • Columns 3 and 4: sequential numbering and symbols listed by Bohm & Dornberger-Schiff (1966[link], 1967[link]).

  • Columns 5, 6 and 7: sequential numbering and two sets of symbols listed by Shubnikov & Koptsik (1974[link]).

  • Column 8: symbols listed by Opechowski (1986[link]).

  • Column 9: symbols listed by Niggli (Chapuis, 1966[link]).

Sets of symbols which are of a non-Hermann–Mauguin (international) type are the set of symbols in column 6 and the Niggli-type set of symbols in column 9. The set of symbols in column 8 does not use the lower-case script letter [{\scr p}], as does IT A (2005[link]), to denote a one-dimensional 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 three-dimensional space groups. The set of symbols in column 4 have the characters indicating symmetry elements along non-lattice directions enclosed in parentheses, and do not use a lower-case script letter to denote the one-dimensional lattice. Lastly, the set of symbols in column 4, without the parentheses and with the one-dimensional lattice denoted by a lower-case script [{\scr p}], are identical with the symbols in Part 3[link] , or in some cases are the second setting of rod groups whose symbols are given in Part 3[link] . These second-setting symbols are included in the symmetry diagrams of the rod groups.

1.2.17.3. Layer groups

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A list of sets of symbols for the layer groups is given in Table 1.2.17.3[link]. The information provided in the columns of this table is as follows:

  • Columns 1 and 2: sequential numbering and symbols used in Part 4[link] .

    Table 1.2.17.3| top | pdf |
    Layer-group symbols

    (a) Columns 1–9.

     123456789
    Triclinic/oblique [1] [p1] [1] [P1] [1] [P11(1)] [1] [p1] [p1]
    [2] [p\bar{1}] [2] [P\bar{1}] [2] [P\bar{1}\bar{1}(\bar{1})] [3] [p\bar{1}] [p\bar{1}]
    Monoclinic/oblique [3] [p112] [3] [P211] [9] [P11(2)] [5] [p112] [p21]
    [4] [p11m] [4] [Pm11] [4] [P11(m)] [2] [p11m] [pm1]
    [5] [p11a] [5] [Pb11] [5] [P11(b)] [4] [p11b] [pa1]
    [6] [p112/m] [6] [P2/m11] [13] [P11(2/m)] [6] [p112/m] [p2/m1]
    [7] [p112/a] [7] [P2/b11] [17] [P11(2/b)] [7] [p112/b] [p2/a1]
    Monoclinic/rectangular [8] [p211] [8] [P112] [8] [P12(1)] [14] [p121] [p12]
    [9] [p2_{1}11] [9] [P112_{1}] [10] [P12_{1}(1)] [15] [p12_{1}1] [p12_{1}]
    [10] [c211] [10] [C112] [11] [C12(1)] [16] [c121] [c12]
    [11] [pm11] [11] [P11m] [3] [P1m(1)] [8] [p1m1] [p1m]
    [12] [pb11] [12] [P11a] [5] [P1a(1)] [10] [p1a1] [p1b]
    [13] [cm11] [13] [C11m] [7] [C1m(1)] [12] [c1m1] [c1m]
    [14] [p2/m11] [14] [P112/m] [12] [P12/m(1)] [17] [p12/m1] [p12/m]
    [15] [p2_{1}/m11] [15] [P112_{1}/m] [14] [P12_{1}/m(1)] [18] [p12_{1}/m1] [p12_{1}/m]
    [16] [p2/b11] [17] [P112/a] [16] [P12/a(1)] [20] [p12/a1] [p12/b]
    [17] [p2_{1}/b11] [18] [P112_{1}/a] [18] [P12_{1}/a(1)] [21] [p12_{1}/a1] [p12_{1}/b]
    [18] [c2/m11] [16] [C112/m] [15] [C12/m(1)] [19] [c12/m1] [c12/m]
    Orthorhombic/rectangular [19] [p222] [19] [P222] [33] [P22(2)] [37] [p222] [p222]
    [20] [p2_{1}22] [20] [P222_{1}] [34] [P2_{1}2(2)] [38] [p2_{1}22] [p222_{1}]
    [21] [p2_{1}2_{1}2] [21] [P22_{1}2_{1}] [35] [P2_{1}2_{1}(2)] [39] [p2_{1}2_{1}2] [p22_{1}2_{1}]
    [22] [c222] [22] [C222] [36] [C22(2)] [40] [c222] [c222]
    [23] [pmm2] [23] [P2mm] [19] [Pmm(2)] [22] [pmm2] [p2mm]
    [24] [pma2] [28] [P2ma] [24] [Pma(2)] [24] [pbm2] [p2ma]
    [25] [pba2] [33] [P2ba] [29] [Pba(2)] [26] [pba2] [p2ba]
    [26] [cmm2] [34] [C2mm] [30] [Cmm(2)] [28] [cmm2] [c2mm]
    [27] [pm2m] [24] [Pmm2] [20] [P2m(m)] [9] [p2mm] [pm2m]
    [28] [pm2_{1}b] [26] [Pbm2_{1}] [21] [P2_{1}m(a)] [30] [p2_{1}ma] [pa2_{1}m]
    [29] [pb2_{1}m] [25] [Pm2_{1}a] [22] [P2_{1}a(m)] [11] [p2_{1}am] [pm2_{1}a]
    [30] [pb2b] [27] [Pbb2] [23] [P2a(a)] [31] [p2aa] [pa2a]
    [31] [pm2a] [29] [Pam2] [25] [P2m(b)] [32] [p2mb] [pb2m]
    [32] [pm2_{1}n] [32] [Pnm2_{1}] [28] [P2_{1}m(n)] [35] [p2_{1}mn] [pn2_{1}m]
    [33] [pb2_{1}a] [30] [Pab2_{1}] [26] [P2_{1}a(b)] [33] [p2_{1}ab] [pb2_{1}a]
    [34] [pb2n] [31] [Pnb2] [27] [P2a(n)] [34] [p2an] [pn2a]
    [35] [cm2m] [35] [Cmm2] [31] [C2m(m)] [13] [c2mm] [cm2m]
    [36] [cm2e] [36] [Cam2] [32] [Cm2(a)] [36] [c2mb] [cb2m]
    [37] [pmmm] [37] [P2/m2/m2/m] [37] [P2/m2/m(2/m)] [23] [pmmm] [p2/m2/m2/m]
    [38] [pmaa] [38] [P2/a2/m2/a] [38] [P2/m2/a(2/a)] [41] [pmaa] [p2/a2/m2/a]
    [39] [pban] [39] [P2/n2/b2/a] [39] [P2/b2/a(2/n)] [42] [pban] [p2/n2/b2/a]
    [40] [pmam] [40] [P2/m2_{1}/m2/a] [41] [P2/b2_{1}/m(2/m)] [25] [pbmm] [p2/m2_{1}/m2/a]
    [41] [pmma] [41] [P2/a2_{1}/m2/m] [40] [P2_{1}/m2/m(2/a)] [43] [pmma] [p2/a2_{1}/m2/m]
    [42] [pman] [42] [P2/n2/m2_{1}/a] [42] [P2_{1}/b2/m(2/n)] [44] [pbmn] [p2/n2/m2_{1}/a]
    [43] [pbaa] [43] [P2/a2/b2_{1}/a] [43] [P2/b2_{1}/a(2/a)] [45] [pbaa] [p2/a2/b2_{1}/a]
    [44] [pbam] [44] [P2/m2_{1}/b2_{1}/a] [44] [P2_{1}/b2_{1}/a(2/m)] [27] [pbam] [p2/m2_{1}/b2_{1}/a]
    [45] [pbma] [45] [P2/a2_{1}/b2_{1}/m] [45] [P2_{1}/m2_{1}/a(2/b)] [46] [pmab] [p2/a2_{1}/b2_{1}/m]
    [46] [pmmn] [46] [P2/n2_{1}/m2_{1}/m] [46] [P2_{1}/m2_{1}/m(2/n)] [47] [pmmn] [p2/n2_{1}/m2_{1}/m]
    [47] [cmmm] [47] [C2/m2/m2/m] [47] [C2/m2/m(2/m)] [29] [cmmm] [c2/m2/m2/m]
    [48] [cmme] [48] [C2/a2/m2/m] [48] [C2/m2/m(2/a)] [48] [cmma] [c2/a2/m2/m]
    [49] [p4] [49] [P4] [54] [P(4)11] [50] [p4] [p4]
    [50] [p\bar{4}] [50] [P\bar{4}] [49] [P(\bar{4})11] [49] [p\bar{4}] [p\bar{4}]
    [51] [p4/m] [51] [P4/m] [55] [P(4/m)11] [51] [p4/m] [p4/m]
    [52] [p4/n] [52] [P4/n] [56] [P(4/n)11] [57] [p4/n] [p4/n]
    [53] [p422] [53] [P422] [59] [P(4)22] [55] [p422] [p422]
    [54] [p42_{1}2] [54] [P42_{1}2] [60] [P(4)2_{1}2] [56] [p42_{1}2] [p42_{1}2]
    [55] [p4mm] [55] [P4mm] [57] [P(4)mm] [52] [p4mm] [p4mm]
    [56] [p4bm] [56] [P4bm] [58] [P(4)bm] [59] [p4bm] [p4bm]
    [57] [p\bar{4}2m] [57] [P\bar{4}2m] [50] [P(\bar{4})2m] [54] [p\bar{4}2m] [p\bar{4}2m]
    [58] [p\bar{4}2_{1}m] [58] [P\bar{4}2_{1}m] [51] [P(\bar{4})2_{1}m] [60] [p\bar{4}2_{1}m] [p\bar{4}2_{1}m]
    [59] [p\bar{4}m2] [59] [P\bar{4}m2] [52] [P(\bar{4})m2] [61] [p\bar{4}m2] [p\bar{4}m2]
    [60] [p\bar{4}b2] [60] [P\bar{4}b2] [53] [P(\bar{4})b2] [64] [p\bar{4}b2] [p\bar{4}b2]
    [61] [p4/mmm] [61] [P4/m2/m2/m] [61] [P(4/m)2/m2/m] [53] [p4/mmm] [p4/m2/m2/m]
    [62] [p4/nbm] [62] [P4/n2/b2/m] [62] [P(4/n)2/b2/m] [62] [p4/nbm] [p4/n2/b2/m]
    [63] [p4/mbm] [63] [P4/m2_{1}/b2/m] [63] [P(4/m)2_{1}/b2/m] [58] [p4/mbm] [p4/m2_{1}/b2/m]
    [64] [p4/nmm] [64] [P4/n2_{1}/m2/m] [64] [P(4/n)2_{1}/m2/m] [63] [p4/nmm] [p4/n2_{1}/m2/m]
    [65] [p3] [65] [P3] [65] [P(3)11] [65] [p3] [p3]
    [66] [p\bar{3}] [66] [P\bar{3}] [66] [P(\bar{3})11] [67] [p\bar{3}] [p\bar{3}]
    [67] [p312] [67] [P312] [70] [P(3)12] [72] [p312] [p312]
    [68] [p321] [68] [P321] [69] [P(3)21] [73] [p321] [p321]
    [69] [p3m1] [69] [P3m1] [67] [P(3)m1] [68] [p3m1] [p3m1]
    [70] [p31m] [70] [P31m] [68] [P(3)1m] [70] [p31m] [p31m]
    [71] [p\bar{3}1m] [71] [P\bar{3}12/m] [72] [P(\bar{3})1m] [74] [p\bar{3}1m] [p\bar{3}12/m]
    [72] [p\bar{3}m1] [72] [P\bar{3}2/m1] [71] [P(\bar{3})m1] [75] [p\bar{3}m1] [p\bar{3}2/m1]
    [73] [p6] [73] [P6] [76] [P(6)11] [76] [p6] [p6]
    [74] [p\bar{6}] [74] [P\bar{6}] [73] [P(\bar{6})11] [66] [p\bar{6}] [p\bar{6}]
    [75] [p6/m] [75] [P6/m] [77] [P(6/m)11] [77] [p6/m] [p6/m]
    [76] [p622] [76] [P622] [79] [P(6)22] [80] [p622] [p622]
    [77] [p6mm] [77] [P6mm] [78] [P(6)mm] [78] [p6mm] [p6mm]
    [78] [p\bar{6}m2] [78] [P\bar{6}m2] [74] [P(\bar{6})m2] [69] [p\bar{6}m2] [p\bar{6}m2]
    [79] [p\bar{6}2m] [79] [P\bar{6}2m] [75] [P(\bar{6})2m] [71] [p\bar{6}2m] [p\bar{6}2m]
    [80] [p6/mmm] [80] [P6/m2/m2/m] [80] [P(6/m)2/m2/m] [79] [p6/mmm] [p6/m2/m2/m]

    (b) Columns 10–17.

     1011121314151617
    Triclinic/oblique [1] [C_{1}\bar{p}] [C_{1}^{1}] [1P1] [(a/b)\cdot 1] [1p1] [p1] [p1]
    [2] [S_{2}\bar{p}] [C_{i}^{1}] [1P\bar{1}] [(a/b)\cdot \bar{1}] [1p\bar{1}] [p2'] [p2']
    Monoclinic/oblique [8] [C_{2}\bar{p}] [C_{2}^{1}] [1P2] [(a/b):2] [1p112] [p2] [p2]
    [3] [C_{1h}\bar{p}\mu ] [C_{1h}^{1}] [mP1] [(a/b)\cdot m] [mp1] [p^{*}1]  
    [4] [C_{1h}\bar{p}\alpha ] [C_{1h}^{2}] [aP1] [(a/b)\cdot \bar{b}] [bp1] [p_{b'}'1] [p_{b}'1]
    [12] [C_{2h}\bar{p}\mu ] [C_{2h}^{1}] [mP2] [(a/b)\cdot m:2] [mp112] [p^{*}2]  
    [13] [C_{2h}\bar{p}\alpha ] [C_{2h}^{2}] [aP2] [(a/b)\cdot \bar{b}:2] [bp112] [p_{b'}'2] [p_{b}'2]
    Monoclinic/rectangular [9] [D_{1}\bar{p}1] [C_{2}^{2}] [1P12] [(a:b)\cdot 2] [1p12] [p1m'1] [pm']
    [10] [D_{1}\bar{p}2] [C_{2}^{3}] [1P12_{1}] [(a:b)\cdot 2_{1}] [1p12_{1}] [p1g'1] [pg']
    [11] [D_{1}\bar{c}1] [C_{2}^{4}] [1C12] [\left({{a+b}\over2}/a:b\right)\cdot 2] [1c12] [c1m'1] [cm']
    [5] [C_{1v}\bar{p}\mu ] [C_{1h}^{3}] [1P1m] [(a:b):m] [1p1m] [p11m] [pm]
    [6] [C_{1v}\bar{p}\beta ] [C_{1h}^{4}] [1P1g] [(a:b):\bar{a}] [1p1a] [p11g] [pg]
    [7] [C_{1v}\bar{c}\mu ] [C_{1h}^{5}] [1C1m] [\left({{a+b}\over2}/a:b\right):m] [1c1m] [c11m] [cm]
    [14] [D_{1d}\bar{p}\mu 1] [C_{2h}^{3}] [1P12/m] [(a:b)\cdot 2:m] [1p12/m] [p2'm'm] [pm'm]
    [15] [D_{1d}\bar{p}\mu 2] [C_{2h}^{5}] [1P12_{1}/m] [(a:b)\cdot 2_{1}:m] [1p12_{1}/m] [p2'g'm] [pg'm]
    [18] [D_{1d}\bar{p}\beta 2] [C_{2h}^{6}] [1P12/g] [(a:b)\cdot 2\cdot \bar{a}] [1p12_{1}/a] [p2'g'g] [pg'g]
    [17] [D_{1d}\bar{p}\beta 1] [C_{2h}^{4}] [1P12_{1}/g] [(a:b)\cdot 2_{1}:\bar{a}] [1p12/a] [p2'm'g] [pm'g]
    [16] [D_{1d}\bar{c}\mu 1] [C_{2h}^{7}] [1C12/m] [\left({{a+b}\over2}/a:b\right)\cdot 2:m] [1c12/m] [c2'm'm] [cm'm]
    Orthorhombic/rectangular [33] [D_{2}\bar{p}11] [V^{1}] [1P222] [(a:b):2:2] [1p222] [p2m'm'] [pm'm']
    [34] [D_{2}\bar{p}12] [V^{3}] [1P222_{1}] [(a:b):2:2_{1}] [1p22_{1}2] [p2g'm'] [pm'g']
    [35] [D_{2}\bar{p}22] [V^{2}] [1P22_{1}2_{1}] [(a:b)\cdot 2_{1}:2_{1}] [1p2_{1}2_{1}2] [p2g'g'] [pg'g']
    [36] [D_{2}\bar{c}11] [V^{4}] [1C222] [\left({{a+b}\over2}/a:b\right):2:2] [1c222] [c2m'm'] [cm'm']
    [19] [C_{2v}\bar{p}\mu \mu ] [C_{2v}^{1}] [1P2mm] [(a:b):2\cdot m] [1pmm2] [p2mm] [pmm]
    [20] [C_{2v}\bar{p}\mu \alpha ] [C_{2v}^{2}] [1P2mg] [(a:b):2\cdot \bar{b}] [1pma2] [p2mg] [pmg]
    [21] [C_{2v}\bar{p}\beta \alpha ] [C_{2v}^{10}] [1P2gg] [(a:b):\bar{a}:\bar{b}] [1pba2] [p2gg] [pgg]
    [22] [C_{2v}\bar{c}\mu \mu ] [C_{2v}^{3}] [1C2mm] [\left({{a+b}\over2}/a:b\right):m\cdot 2] [1cmm2] [c2mm] [cmm]
    [23] [D_{1h}\bar{p}\mu \mu ] [C_{2v}^{4}] [mP12m] [(a:b)\cdot m\cdot 2] [mpm2] [p^{*}1m1]  
    [25] [D_{1h}\bar{p}\beta \mu ] [C_{2v}^{5}] [aP12_{1}m] [(a:b):m\cdot 2_{1}] [bpm2_{1}] [p_{b'}'1m1] [p_{a}'1m]
    [24] [D_{1h}\bar{p}\mu \beta ] [C_{2v}^{7}] [mP12_{1}g] [(a:b)\cdot m\cdot 2_{1}] [mpb2_{1}] [p^{*}1g1]  
    [26] [D_{1h}\bar{p}\beta \beta ] [C_{2v}^{6}] [aP12g] [(a:b)\cdot \bar{a}\cdot 2] [bpb2] [p_{b'}'1m'1] [p_{a}'1g]
    [27] [D_{1h}\bar{p}\alpha \mu ] [C_{2v}^{11}] [bP12m] [(a:b)\cdot \bar{b}\cdot 2] [apm2] [p_{a'}'1m1] [p_{b}'1m]
    [30] [D_{1h}\bar{p}\upsilon \mu ] [C_{2v}^{13}] [nP12_{1}m] [(a:b)\cdot ab\cdot 2_{1}] [npm2_{1}] [c'1m1] [p_{c}'1m]
    [28] [D_{1h}\bar{p}\alpha \beta ] [C_{2v}^{14}] [bP12_{1}g] [(a:b)\cdot \bar{b}:\bar{a}] [apb2_{1}] [p_{a'}'1g1] [p_{b}'1g]
    [29] [D_{1h}\bar{p}\upsilon \beta ] [C_{2v}^{12}] [nP12g] [(a:b)\cdot ab\cdot 2] [npb2] [c'1m'1] [p_{c}'1m']
    [31] [D_{1h}\bar{c}\mu \mu ] [C_{2v}^{8}] [mC12m] [\left({{a+b}\over2}/a:b\right)\cdot m\cdot 2] [mcm2] [c^{*}1m1]  
    [32] [D_{1h}\bar{c}\alpha \mu ] [C_{2v}^{9}] [aC12m] [\left({{a+b}\over2}/a:b\right)\cdot \bar{b}\cdot 2] [acm2] [p_{a'b'}'1m1] [c'1m]
    [37] [D_{2h}\bar{p}\mu \mu \mu ] [V_{h}^{1}] [mP2mm] [(a:b)\cdot m:2\cdot m] [mp2/m2/m2] [p^{*}2mm]  
    [38] [D_{2h}\bar{p}\alpha \mu \alpha ] [V_{h}^{5}] [aP2mg] [(a:b)\cdot \bar{a}:2\cdot \bar{a}] [ip2/m2/a2] [p_{a'}'2mg] [p_{a}'mg]
    [39] [D_{2h}\bar{p}\upsilon \beta \alpha ] [V_{h}^{6}] [nP2gg] [(a:b)\cdot ab:2\cdot a] [np2/b2/a2] [c'2m'm'] [p_{c}'m'm']
    [40] [D_{2h}\bar{p}\mu \mu \alpha ] [V_{h}^{3}] [mP2mg] [(a:b)\cdot m:2\cdot \bar{b}] [np2_{1}/m2/a2] [p^{*}2mg]  
    [41] [D_{2h}\bar{p}\alpha\mu\mu] [V_{h}^{9}] [aP2mm] [(a:b)\cdot \bar{a}:2\cdot m] [ap2_{1}/m2/m2] [p_{a'}'2mm] [p_{b}'mm]
    [42] [D_{2h}\bar{p}\upsilon \mu \alpha ] [V_{h}^{11}] [nP2mg] [(a:b)\cdot ab:2\cdot b] [np2/m2_{1}/a2] [c'2mm'] [p_{c}'m'm]
    [43] [D_{2h}\bar{p}\alpha \beta \alpha ] [V_{h}^{10}] [aP2gg] [(a:b)\cdot \bar{a}\cdot 2:\bar{b}] [ap2/b2_{1}/a2] [p_{a'}'2gg] [p_{b}'gg]
    [44] [D_{2h}\bar{p}\mu \beta \alpha ] [V_{h}^{2}] [mP2gg] [(a:b)\cdot m:\bar{a}:\bar{b}] [np2_{1}/b2_{1}/a2] [p^{*}2gg]  
    [45] [D_{2h}\bar{p}\alpha \beta \mu ] [V_{h}^{7}] [aP2gm] [(a:b)\cdot \bar{b}:2\cdot \bar{a}] [ap2_{1}/b2_{1}/m2] [p_{a'}'2gm] [p_{b}'mg]
    [46] [D_{2h}\bar{p}\upsilon \mu \mu ] [V_{h}^{8}] [nP2mm] [(a:b)\cdot ab:2\cdot m] [np2_{1}/m2_{1}/m2] [c'2mm] [p_{c}'mm]
    [47] [D_{2h}\bar{c}\mu \mu \mu ] [V_{h}^{4}] [mC2mm] [\left({{a+b}\over2}/a:b\right)\cdot m:2\cdot m] [mc2/m2/m2] [c^{*}2mm]  
    [48] [D_{2h}\bar{c}\alpha \mu \mu ] [V_{h}^{12}] [aC2mm] [\left({{a+b}\over2}/a:b\right)\cdot \bar{a}:2\cdot m] [ac2/m2/m2] [p_{a'b'}'2mm] [c'mm]
    [58] [C_{4}\bar{p}] [C_{4}^{1}] [1P4] [(a:a):4] [1p4] [p4] [p4]
    [57] [S_{4}\bar{p}] [S_{4}^{1}] [1P\bar{4}] [(a:a):\bar{4}] [1p\bar{4}] [p4'] [p4']
    [61] [C_{4h}\bar{p}\mu ] [C_{4h}^{1}] [mP4] [(a:a):4:m] [mp4] [p^{*}4]  
    [62] [C_{4h}\bar{p}\upsilon ] [C_{4h}^{2}] [nP4] [(a:a):4:ab] [np4] [c'4] [p'4]
    [67] [D_{4}\bar{p}11] [D_{4}^{1}] [1P422] [(a:a):4:2] [1p422] [p4m'm'] [p4m'm']
    [68] [D_{4}\bar{p}21] [D_{4}^{2}] [1P42_{1}2] [(a:a):4:2_{1}] [1p42_{1}2] [p4g'm'] [p4g'm']
    [59] [C_{4v}\bar{p}\mu \mu ] [C_{4v}^{1}] [1P4mm] [(a:a):4\cdot m] [1p4mm] [p4mm] [p4mm]
    [60] [C_{4v}\bar{p}\beta \mu ] [C_{4v}^{2}] [1P4gm] [(a:a):4\odot b] [1p4bm] [p4gm] [p4gm]
    [63] [D_{2d}\bar{p}\mu 1] [V_{d}^{1}] [1P\bar{4}2m] [(a:a):\bar{4}:2] [1p\bar{4}2m] [p4'm'm] [p4'm'm]
    [64] [D_{2d}\bar{p}\mu 2] [V_{d}^{2}] [1P\bar{4}2_{1}m] [(a:a):\bar{4}\odot 2_{1}] [1p\bar{4}2_{1}m] [p4'g'm] [p4'g'm]
    [65] [D_{2d}\bar{c}\mu 1] [V_{d}^{3}] [1P\bar{4}m2] [(a:a):\bar{4}\cdot m] [1p\bar{4}m2] [p4'mm'] [p4'mm']
    [66] [D_{2d}\bar{c}\beta 1] [V_{d}^{4}] [1P\bar{4}g2] [(a:a):\bar{4}\odot \bar{b}] [1p\bar{4}b2] [p4'gm'] [p4'gm']
    [69] [D_{4h}\bar{p}\mu \mu \mu ] [D_{4h}^{1}] [mP4mm] [(a:a)\cdot m:4\cdot m] [mp42/m2/m] [p^{*}4mm]  
    [70] [D_{4h}\bar{p}\upsilon \beta \mu ] [D_{4h}^{2}] [nP4gm] [(a:a):ab:4\odot b] [np42/b2/m] [c'4m'm] [p'4gm]
    [71] [D_{4h}\bar{p}\mu \beta \mu ] [D_{4h}^{3}] [mP4gm] [(a:a)\cdot m:4\odot b] [mp42_{1}/b2/m] [p^{*}4gm]  
    [72] [D_{4h}\bar{p}\upsilon \mu \mu ] [D_{4h}^{4}] [nP4mm] [(a:a)\cdot ab:4\cdot m] [np42_{1}/m2/m] [c'4mm] [p'4mm]
    [49] [C_{3}\bar{c}] [C_{3}^{1}] [1P3] [(a/a):3] [1p3] [p3] [p3]
    [50] [S_{6}\bar{p}] [C_{3i}^{1}] [1P\bar{3}] [(a/a):\bar{3}] [1p\bar{3}] [p6'] [p6']
    [54] [D_{3}\bar{c}1] [D_{3}^{1}] [1P312] [(a/a):2:3] [1p312] [p3m'1] [p3m'1]
    [53] [D_{3}\bar{h}1] [D_{3}^{2}] [1P321] [(a/a)\cdot 2:3] [1p321] [p31m'] [p31m']
    [51] [C_{3v}\bar{c}\mu ] [C_{3v}^{2}] [1P3m1] [(a/a):m\cdot 3] [1p3m1] [p3m1] [p3m1]
    [52] [C_{3v}\bar{h}\mu ] [C_{3v}^{1}] [1P31m] [(a/a)\cdot m\cdot 3] [1p31m] [p31m] [p31m]
    [55] [D_{3d}\bar{c}\mu 1] [D_{3d}^{2}] [1P\bar{3}1m] [(a/a)\cdot m\cdot \bar{6}] [1p\bar{3}12/m] [p6'm'm] [p6'm'm]
    [56] [D_{3d}\bar{h}\mu 1] [D_{3d}^{1}] [1P\bar{3}m1] [(a/a):m\cdot \bar{6}] [1p\bar{3}2/m1] [p6'mm'] [p6'mm']
    [76] [C_{6}\bar{c}] [C_{6}^{1}] [1P6] [(a/a):6] [1p6] [p6] [p6]
    [73] [C_{3h}\bar{c}\mu ] [C_{3h}^{1}] [mP3] [(a/a):3:m] [mp3] [p^{*}3]  
    [78] [C_{6h}\bar{c}\mu ] [C_{6h}^{1}] [mP6] [(a/a)\cdot m:6] [mp6] [p^{*}6]  
    [79] [D_{6}\bar{c}11] [D_{6}^{1}] [1P622] [(a/a)\cdot 2:6] [1p622] [p6m'm'] [p6m'm']
    [77] [C_{6v}\bar{c}\mu \mu ] [C_{6v}^{1}] [1P6mm] [(a/a):m\cdot 6] [1p6mm] [p6mm] [p6mm]
    [74] [D_{3h}\bar{c}\mu \mu ] [D_{3h}^{1}] [mP3m2] [(a/a):m\cdot 3:m] [mp3m2] [p^{*}3m1]  
    [75] [D_{3h}\bar{h}\mu \mu ] [D_{3h}^{2}] [mP32m] [(a/a)\cdot m:3\cdot m] [mp32m] [p^{*}31m]  
    [80] [D_{6h}\bar{c}\mu \mu \mu ] [D_{6h}^{1}] [mP6mm] [(a/a)\cdot m:6\cdot m] [mp6mm] [p^{*}6mm]  

    (c) Columns 18–25.

     1819202122232425
    Triclinic/oblique [p1] [47]     [p1]      
    [p2'] [1] [p2'] [p2^{-}] [p2'] [p2[2]_{1}] [2'11] [p2/p1]
    Monoclinic/oblique [p2] [48]     [p2]      
    [p1'] [64]     [p11']      
    [p_{b}'1] [2] [pt'] [pt^{-}] [p_{2b}1] [p1[2]] [b11] [p1/p1]
    [p21'] [65]     [p21']      
    [p_{b}'2] [3] [p2t'] [p2t^{-}] [p_{2b}2] [p2[2]_{2}] [2/b11] [p2/p2]
    Monoclinic/rectangular [pm'] [4] [pm'] [pm^{-}] [pm'] [pm[2]_{4}] [12'1] [pm/p1]
    [pg'] [5] [pg'] [pg^{-}] [pg'] [pg[2]_{1}] [112_{1}'] [pg/p1]
    [cm'] [6 ] [cm'] [cm^{-}] [cm'] [cm[2]_{1}] [c112'] [cm/p1]
    [pm] [49]     [pm]      
    [pg] [50]     [pg]      
    [cm] [51 ]     [cm]      
    [pmm'] [14] [pmm'] [pmm^{-}] [pm'm] [pmm[2]_{2}] [2'2'2] [pmm/pm]
    [pmg'] [17] [pmg'] [pmg^{-}] [pmg'] [pmg[2]_{4}] [2'2_{1}'2] [pmg/pm]
    [pgg'] [18] [pgg'] [pgg^{-}] [pgg'] [pgg[2]_{1}] [2'2_{1}'2_{1}] [pgg/pg]
    [pm'g] [16] [pm'g] [pm^{-}g] [pm'g] [pmg[2]_{2}] [2'2_{1}2'] [pmg/pg]
    [cmm'] [21] [cmm'] [cmm^{-}] [cmm'] [cmm[2]_{2}] [c2'22'] [cmm/cm]
    Orthorhombic/rectangular [pm'm'] [15] [pm'm'] [pm^{-}m^{-}] [pm'm'] [pmm[2]_{5}] [22'2'] [pmm/p2]
    [pm'g'] [20] [pm'g'] [pm^{-}g^{-}] [pm'g'] [pmg[2]_{5}] [22'2_{1}'] [pmg/p2]
    [pg'g'] [19] [pg'g'] [pg^{-}g^{-}] [pg'g'] [pgg[2]_{2}] [22_{1}'2_{1}'] [pgg/p2]
    [cm'm'] [22] [cm'm'] [cm^{-}m^{-}] [cm'm'] [cmm[2]_{4}] [c22'2'] [cmm/p2]
    [pmm2] [52]     [pmm]      
    [pmg2] [53]     [pmg]      
    [pgg2] [54]     [pgg]      
    [cmm2] [55]     [cmm]      
    [pm1'] [66]     [pm1']      
    [p_{b}'m] [7] [pm+t'] [pm+t^{-}] [p_{2b}m] [pm[2]_{3}] [b12] [pm/pm(m)]
    [pg1'] [67]     [pg1']      
    [p_{b}'g] [8] [pg+t'] [pg+t^{-}] [p_{2b}m'] [pm[2]_{1}] [b12_{1}] [pm/pg]
    [p_{b}'1m] [9] [pm+m'] [pm+m^{-}] [p_{2a}m] [pm[2]_{5}] [b'1m] [pm/pm(m')]
    [p_{c}'m] [11] [pm+g'] [pm+g^{-}] [c_{p}m] [cm[2]_{3}] [n12] [cm/pm]
    [p_{b}'1g] [10] [pg+g'] [pg+g^{-}] [p_{2a}g] [pg[2]_{2}] [b2_{1}1] [pg/pg]
    [p_{c}'g] [12] [pg+m'] [pg+m^{-}] [c_{p}m'] [cm[2]_{2}] [n12_{1}] [cm/pg]
    [cm1'] [68]     [cm1']      
    [c'm] [13] [cm+m'] [cm+m^{-}] [p_{c}m] [pm[2]_{2}] [ca12] [pm/cm]
    [pmm21'] [69]     [pmm1']      
    [p_{b}'gm] [25] [pg,m+m'] [pg,m+m^{-}] [p_{2a}mm'] [pmm[2]_{4}] [a2_{1}2] [pmm/pmg]
    [p_{c}'gg] [29] [pg+m',g+m'] [pg+m^{-},g+m^{-}] [c_{p}m'm'] [cmm[2]_{1}] [n2_{1}2_{1}] [cmm/pgg]
    [pmg21'] [70]     [pmg1']      
    [p_{b}'mm] [23] [pm,m+m'] [pm,m+m^{-}] [p_{2a}mm] [pmm[2]_{1}] [a22] [pmm/pmm]
    [p_{c}'mg] [28] [pm+g',g+m'] [pm+g^{-},g+m^{-}] [c_{p}mm'] [cmm[2]_{3}] [n22_{1}] [cmm/pmg]
    [p_{b}'gg] [26] [pg,g+g'] [pg,g+g^{-}] [p_{2b}m'g] [pmg[2]_{3}] [a2_{1}2_{1}] [pmg/pgg]
    [pgg21'] [71]     [pgg1']      
    [p_{b}'mg] [24] [pm,g+g'] [pm,g+g^{-}] [p_{2b}mg] [pmg[2]_{1}] [b2_{1}2] [pmg/pmg]
    [p_{c}'mm] [27] [pm+g',m+g'] [pm+g^{-},m+g^{-}] [c_{p}mm] [cmm[2]_{5}] [n22] [cmm/pmm]
    [cmm21'] [72]     [cmm1']      
    [c'mm] [30] [cm+m',m+m'] [cm+m^{-},m+m^{-}] [p_{c}mm] [pmm[2]_{3}] [ca22] [pmm/cmm]
    [p4] [56]     [p4]      
    [p4'] [31] [p4'] [p4^{-}] [p4'] [p4[2]_{2}] [4'11] [p4/p2]
    [p41'] [73]     [p41']      
    [p_{c}'4] [32] [p4t'] [p4t^{-}] [p_{p}4] [p4[2]_{1}] [4/n11] [p4/p4]
    [p4m'm'] [35] [p4m'm'] [p4m^{-}m^{-}] [p4m'] [pm4[2]_{2}] [42'2'] [p4m/p4]
    [p4g'm'] [38] [p4g'm'] [p4g^{-}m^{-}] [p4g'] [p4g[2]_{1}] [42_{1}'2'] [p4g/p4]
    [p4mm] [57]     [p4m]      
    [p4gm] [58]     [p4g]      
    [p4'm'm] [34] [p4'm'm] [p4^{-}m^{-}m] [p4'm'] [p4m[2]_{3}] [4'2'2] [p4m/cmm]
    [p4'g'm] [37] [p4'g'm] [p4^{-}g^{-}m] [p4'g'] [p4g[2]_{2}] [4'2_{1}'2] [p4g/cmm]
    [p4'mm'] [33] [p4'mm'] [p4^{-}mm^{-}] [p4'm] [p4m[2]_{4}] [4'22'] [p4m/pmm]
    [p4'gm'] [36] [p4'gm'] [p4^{-}gm^{-}] [p4'g] [p4g[2]_{3}] [4'2_{1}2'] [p4g/pgg]
    [p4mm1'] [74]     [p4m1']      
    [p_{c}'4gm] [40] [p4g+m',m+m'] [p4g+m^{-},m+m^{-}] [p_{p}4m'] [p4m[2]_{1}] [4/n2_{1}2] [p4m/p4g]
    [p4gm1'] [75]     [p4g1']      
    [p_{c}'4mm] [39] [p4m+g',m+m'] [p4m+g^{-},m+m^{-}] [p_{p}4m] [p4m[2]_{5}] [4/n22] [p4m/p4m]
    [p3] [59]     [p3]      
    [p6'] [43] [p6'] [p6^{-}] [p6'] [p6[2]] [6'] [p6/p3]
    [p3m'] [41] [p3m'1] [p3m^{-}1] [p3m'1] [p3m1[2]] [312'] [p3m1/p3]
    [p31m'] [42] [p31m'] [p31m^{-}] [p31m'] [p31m[2]] [32'1] [p31m/p3]
    [p3m] [60]     [p3m1]      
    [p31m] [61]     [p31m]      
    [p6'm'm] [44] [p6'm'm] [p6^{-}m^{-}m] [p6'm'] [p6m[2]_{1}] [6'22'] [p6m/p31m]
    [p6'mm'] [45] [p6'mm'] [p6^{-}mm^{-}] [p6'm] [p6m[2]_{2}] [6'2'2] [p6m/p3m1]
    [p6] [62]     [p6]      
    [p3'] [76]     [p31']      
    [p61'] [79]     [p61']      
    [p6m'm'] [46] [p6m'm'] [p6m^{-}m^{-}] [p6m'] [p6m[2]_{3}] [62'2'] [p6m/p6]
    [p6mm] [63]     [p6m]      
    [p3'm] [77]     [p3m11']      
    [p3'1m] [78]     [p31m1']      
    [p6mm1'] [80]     [p6m1']      
  • Columns 3 and 4: sequential numbering and symbols listed by Wood (1964a[link],b[link]) and Litvin & Wike (1991[link]).

  • Columns 5 and 6: sequential numbering and symbols listed by Bohm & Dornberger-Schiff (1966[link], 1967[link]).

  • Columns 7 and 8: sequential numbering and symbols listed by Shubnikov & Koptsik (1974[link]) and Vainshtein (1981[link]).

  • Column 9: symbols listed by Holser (1958[link]).

  • Column 10: sequential numbering listed by Weber (1929[link]).

  • Column 11: symbols listed by Hermann (1929a[link],b[link]).

  • Column 12: symbols listed by Alexander & Herrmann (1929a[link],b[link]).

  • Column 13: symbols listed by Niggli (Wood, 1964a[link],b[link]).

  • Column 14: symbols listed by Shubnikov & Koptsik (1974[link]).

  • Columns 15 and 16: symbols listed by Aroyo & Wondratschek (1987[link]).

  • Column 17: symbols listed by Belov et al. (1957[link]).

  • Columns 18 and 19: symbols and sequential numbering listed by Belov & Tarkhova (1956a[link],b[link]).

  • Columns 20 and 21: symbols listed by Cochran as listed, respectively, by Cochran (1952[link]) and Belov & Tarkhova (1956a[link],b[link]).

  • Column 22: symbols listed by Opechowski (1986[link]).

  • Column 23: symbols listed by Grunbaum & Shephard (1987[link]).

  • Column 24: symbols listed by Woods (1935a[link],b[link],c[link], 1936[link]).

  • Column 25: symbols listed by Coxeter (1986[link]).

There is also a notation for layer groups, introduced by Janovec (1981[link]), 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 layer-group types in this notation.

Sets of symbols which are of a non-Hermann–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 non-Hermann–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 three-dimensional 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 upper-case letters to denote the two-dimensional lattice of the layer group, where as in IT A (2005[link]) upper-case letters denote three-dimensional lattices.

The symbols in column 8 are either identical with or, in some monoclinic and orthorhombic cases, are the second-setting or alternative-cell-choice symbols of the layer groups whose symbols are given in Part 4[link] . These second-setting and alternative-cell-choice symbols are included in the symmetry diagrams of the layer groups.

The isomorphism between layer groups and two-dimensional magnetic space groups can be seen in Table 1.2.17.3[link]. 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 two-dimensional 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 mz. A detailed discussion of the relationship between these two sets of groups has been given by Opechowski (1986[link]).

References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]
Alexander, E. & Herrmann, K. (1929a). Zur Theorie der flussigen Kristalle. Z. Kristallogr. 69, 285–299.
Alexander, E. & Herrmann, K. (1929b). Die 80 zweidimensionalen Raumgruppen. Z. Kristallogr. 70, 328–345, 460.
Aroyo, M. I. & Wondratschek, H. (1987). Private communication.
Belov, N. V., Neronova, N. N. & Smirnova, T. S. (1957). Shubnikov groups. Kristallografia, 2, 315–325. (Sov. Phys. Crystallogr. 2, 311–322.)
Belov, N. V. & Tarkhova, T. N. (1956a). Color symmetry groups. Kristallografia, 1, 4–13. (Sov. Phys. Crystallogr. 1, 5–11.) [Reprinted in: Colored Symmetry. (1964). Edited by W. T. Holser. New York: Macmillan.]
Belov, N. V. & Tarkhova, T. N. (1956b). Color symmetry groups. Kristallografia, 1, 619–620. (Sov. Phys. Crystallogr. 1, 487–488.)
Bohm, J. & Dornberger-Schiff, K. (1966). The nomenclature of crystallographic symmetry groups. Acta Cryst. 21, 1004–1007.
Bohm, J. & Dornberger-Schiff, K. (1967). Geometrical symbols for all crystallographic symmetry groups up to three dimensions. Acta Cryst. 23, 913–933.
Chapuis, G. (1966). Anwendung der Raumgruppenmatrizen auf die ein- und zweifach periodischen Symmetriegruppen in drei Dimensionen. Diplomarbeit, University of Zurich, Switzerland.
Cochran, W. (1952). The symmetry of real periodic two-dimensional functions. Acta Cryst. 5, 630–633.
Coxeter, H. S. M. (1986). Coloured symmetry. In M. C. Escher: Art and science, edited by H. S. M. Coxeter, pp. 15–33. Amsterdam: North-Holland.
Fuksa, J. & Kopský, V. (1993). Layer and rod classes of reducible space groups. I. Z-reducible cases. Acta Cryst. A49, 280–287.
Grunbaum, G. & Shephard, G. C. (1987). Tilings and patterns. New York: Freeman.
Hermann, C. (1929a). Zur systematischen Strukturtheorie. III. Ketten- und Netzgruppen. Z. Kristallogr. 69, 259–270.
Hermann, C. (1929b). Zur systematischen Struckturtheorie. IV. Untergruppen. Z. Kristallogr. 69, 533–555.
Holser, W. T. (1958). Point groups and plane groups in a two-sided plane and their subgroups. Z. Kristallogr. 110, 266–281.
Janovec, V. (1981). Symmetry and structure of domain walls. Ferroelectrics, 35, 105–110.
Kopský, V. (1986). The role of subperiodic and lower-dimensional groups in the structure of space groups. J. Phys. A, 19, L181–L184.
Kopský, V. (1988). Reducible space groups. Lecture Notes in Physics, 313, 352–356. Proceedings of the 16th International Colloquium on Group-Theoretical Methods in Physics, Varna, 1987. Berlin: Springer Verlag.
Kopský, V. (1989a). Subperiodic groups as factor groups of reducible space groups. Acta Cryst. A45, 805–815.
Kopský, V. (1989b). Subperiodic classes of reducible space groups. Acta Cryst. A45, 815–823.
Kopský, V. (1993). Layer and rod classes of reducible space groups. I. Z-decomposable cases. Acta Cryst. A49, 269–280.
Litvin, D. B. & Wike, T. R. (1991). Character tables and compatability relations of the eighty layer groups and the seventeen plane groups. New York: Plenum.
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