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

International Tables for Crystallography (2010). Vol. E, ch. 1.2, pp. 22-24   | 1 | 2 |

Section 1.2.17.2. Rod groups

V. Kopskýa and D. B. Litvinb*

aFreelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and bDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail:  u3c@psu.edu

1.2.17.2. 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.

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).]
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.
Opechowski, W. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: North Holland.
Shubnikov, A. V. & Koptsik, V. A. (1974). Symmetry in Science and Art. New York: Plenum.








































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