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, p. 24   | 1 | 2 |

Section 1.2.17.3. Layer 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

1.2.17.3. Layer groups

| top | pdf |

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]
    Tetragonal/square [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]
    Trigonal/hexagonal [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]
    Hexagonal/hexagonal [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.

     11011121314151617
    Triclinic/oblique 1 [1] [C_{1}\bar{p}] [C_{1}^{1}] [1P1] [(a/b)\cdot 1] [1p1] [p1] [p1]
      2 [2] [S_{2}\bar{p}] [C_{i}^{1}] [1P\bar{1}] [(a/b)\cdot \bar{1}] [1p\bar{1}] [p2'] [p2']
    Monoclinic/oblique 3 [8] [C_{2}\bar{p}] [C_{2}^{1}] [1P2] [(a/b):2] [1p112] [p2] [p2]
      4 [3] [C_{1h}\bar{p}\mu ] [C_{1h}^{1}] [mP1] [(a/b)\cdot m] [mp1] [p^{*}1]  
      5 [4] [C_{1h}\bar{p}\alpha ] [C_{1h}^{2}] [aP1] [(a/b)\cdot \bar{b}] [bp1] [p_{b'}'1] [p_{b}'1]
      6 [12] [C_{2h}\bar{p}\mu ] [C_{2h}^{1}] [mP2] [(a/b)\cdot m:2] [mp112] [p^{*}2]  
      7 [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 8 [9] [D_{1}\bar{p}1] [C_{2}^{2}] [1P12] [(a:b)\cdot 2] [1p12] [p1m'1] [pm']
      9 [10] [D_{1}\bar{p}2] [C_{2}^{3}] [1P12_{1}] [(a:b)\cdot 2_{1}] [1p12_{1}] [p1g'1] [pg']
      10 [11] [D_{1}\bar{c}1] [C_{2}^{4}] [1C12] [\left({{a+b}\over2}/a:b\right)\cdot 2] [1c12] [c1m'1] [cm']
      11 [5] [C_{1v}\bar{p}\mu ] [C_{1h}^{3}] [1P1m] [(a:b):m] [1p1m] [p11m] [pm]
      12 [6] [C_{1v}\bar{p}\beta ] [C_{1h}^{4}] [1P1g] [(a:b):\bar{a}] [1p1a] [p11g] [pg]
      13 [7] [C_{1v}\bar{c}\mu ] [C_{1h}^{5}] [1C1m] [\left({{a+b}\over2}/a:b\right):m] [1c1m] [c11m] [cm]
      14 [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 [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]
      16 [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 [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]
      18 [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 19 [33] [D_{2}\bar{p}11] [V^{1}] [1P222] [(a:b):2:2] [1p222] [p2m'm'] [pm'm']
      20 [34] [D_{2}\bar{p}12] [V^{3}] [1P222_{1}] [(a:b):2:2_{1}] [1p22_{1}2] [p2g'm'] [pm'g']
      21 [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']
      22 [36] [D_{2}\bar{c}11] [V^{4}] [1C222] [\left({{a+b}\over2}/a:b\right):2:2] [1c222] [c2m'm'] [cm'm']
      23 [19] [C_{2v}\bar{p}\mu \mu ] [C_{2v}^{1}] [1P2mm] [(a:b):2\cdot m] [1pmm2] [p2mm] [pmm]
      24 [20] [C_{2v}\bar{p}\mu \alpha ] [C_{2v}^{2}] [1P2mg] [(a:b):2\cdot \bar{b}] [1pma2] [p2mg] [pmg]
      25 [21] [C_{2v}\bar{p}\beta \alpha ] [C_{2v}^{10}] [1P2gg] [(a:b):\bar{a}:\bar{b}] [1pba2] [p2gg] [pgg]
      26 [22] [C_{2v}\bar{c}\mu \mu ] [C_{2v}^{3}] [1C2mm] [\left({{a+b}\over2}/a:b\right):m\cdot 2] [1cmm2] [c2mm] [cmm]
      27 [23] [D_{1h}\bar{p}\mu \mu ] [C_{2v}^{4}] [mP12m] [(a:b)\cdot m\cdot 2] [mpm2] [p^{*}1m1]  
      28 [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]
      29 [24] [D_{1h}\bar{p}\mu \beta ] [C_{2v}^{7}] [mP12_{1}g] [(a:b)\cdot m\cdot 2_{1}] [mpb2_{1}] [p^{*}1g1]  
      30 [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]
      31 [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]
      32 [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]
      33 [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]
      34 [29] [D_{1h}\bar{p}\upsilon \beta ] [C_{2v}^{12}] [nP12g] [(a:b)\cdot ab\cdot 2] [npb2] [c'1m'1] [p_{c}'1m']
      35 [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]  
      36 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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]
    Tetragonal/square 49 [58] [C_{4}\bar{p}] [C_{4}^{1}] [1P4] [(a:a):4] [1p4] [p4] [p4]
      50 [57] [S_{4}\bar{p}] [S_{4}^{1}] [1P\bar{4}] [(a:a):\bar{4}] [1p\bar{4}] [p4'] [p4']
      51 [61] [C_{4h}\bar{p}\mu ] [C_{4h}^{1}] [mP4] [(a:a):4:m] [mp4] [p^{*}4]  
      52 [62] [C_{4h}\bar{p}\upsilon ] [C_{4h}^{2}] [nP4] [(a:a):4:ab] [np4] [c'4] [p'4]
      53 [67] [D_{4}\bar{p}11] [D_{4}^{1}] [1P422] [(a:a):4:2] [1p422] [p4m'm'] [p4m'm']
      54 [68] [D_{4}\bar{p}21] [D_{4}^{2}] [1P42_{1}2] [(a:a):4:2_{1}] [1p42_{1}2] [p4g'm'] [p4g'm']
      55 [59] [C_{4v}\bar{p}\mu \mu ] [C_{4v}^{1}] [1P4mm] [(a:a):4\cdot m] [1p4mm] [p4mm] [p4mm]
      56 [60] [C_{4v}\bar{p}\beta \mu ] [C_{4v}^{2}] [1P4gm] [(a:a):4\odot b] [1p4bm] [p4gm] [p4gm]
      57 [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]
      58 [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]
      59 [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']
      60 [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']
      61 [69] [D_{4h}\bar{p}\mu \mu \mu ] [D_{4h}^{1}] [mP4mm] [(a:a)\cdot m:4\cdot m] [mp42/m2/m] [p^{*}4mm]  
      62 [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]
      63 [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]  
      64 [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]
    Trigonal/hexagonal 65 [49] [C_{3}\bar{c}] [C_{3}^{1}] [1P3] [(a/a):3] [1p3] [p3] [p3]
      66 [50] [S_{6}\bar{p}] [C_{3i}^{1}] [1P\bar{3}] [(a/a):\bar{3}] [1p\bar{3}] [p6'] [p6']
      67 [54] [D_{3}\bar{c}1] [D_{3}^{1}] [1P312] [(a/a):2:3] [1p312] [p3m'1] [p3m'1]
      68 [53] [D_{3}\bar{h}1] [D_{3}^{2}] [1P321] [(a/a)\cdot 2:3] [1p321] [p31m'] [p31m']
      69 [51] [C_{3v}\bar{c}\mu ] [C_{3v}^{2}] [1P3m1] [(a/a):m\cdot 3] [1p3m1] [p3m1] [p3m1]
      70 [52] [C_{3v}\bar{h}\mu ] [C_{3v}^{1}] [1P31m] [(a/a)\cdot m\cdot 3] [1p31m] [p31m] [p31m]
      71 [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]
      72 [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']
    Hexagonal/hexagonal 73 [76] [C_{6}\bar{c}] [C_{6}^{1}] [1P6] [(a/a):6] [1p6] [p6] [p6]
      74 [73] [C_{3h}\bar{c}\mu ] [C_{3h}^{1}] [mP3] [(a/a):3:m] [mp3] [p^{*}3]  
      75 [78] [C_{6h}\bar{c}\mu ] [C_{6h}^{1}] [mP6] [(a/a)\cdot m:6] [mp6] [p^{*}6]  
      76 [79] [D_{6}\bar{c}11] [D_{6}^{1}] [1P622] [(a/a)\cdot 2:6] [1p622] [p6m'm'] [p6m'm']
      77 [77] [C_{6v}\bar{c}\mu \mu ] [C_{6v}^{1}] [1P6mm] [(a/a):m\cdot 6] [1p6mm] [p6mm] [p6mm]
      78 [74] [D_{3h}\bar{c}\mu \mu ] [D_{3h}^{1}] [mP3m2] [(a/a):m\cdot 3:m] [mp3m2] [p^{*}3m1]  
      79 [75] [D_{3h}\bar{h}\mu \mu ] [D_{3h}^{2}] [mP32m] [(a/a)\cdot m:3\cdot m] [mp32m] [p^{*}31m]  
      80 [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.

     11819202122232425
    Triclinic/oblique 1 [p1] [47]     [p1]      
      2 [p2'] [1] [p2'] [p2^{-}] [p2'] [p2[2]_{1}] [2'11] [p2/p1]
    Monoclinic/oblique 3 [p2] [48]     [p2]      
      4 [p1'] [64]     [p11']      
      5 [p_{b}'1] [2] [pt'] [pt^{-}] [p_{2b}1] [p1[2]] [b11] [p1/p1]
      6 [p21'] [65]     [p21']      
      7 [p_{b}'2] [3] [p2t'] [p2t^{-}] [p_{2b}2] [p2[2]_{2}] [2/b11] [p2/p2]
    Monoclinic/rectangular 8 [pm'] [4] [pm'] [pm^{-}] [pm'] [pm[2]_{4}] [12'1] [pm/p1]
      9 [pg'] [5] [pg'] [pg^{-}] [pg'] [pg[2]_{1}] [112_{1}'] [pg/p1]
      10 [cm'] [6 ] [cm'] [cm^{-}] [cm'] [cm[2]_{1}] [c112'] [cm/p1]
      11 [pm] [49]     [pm]      
      12 [pg] [50]     [pg]      
      13 [cm] [51 ]     [cm]      
      14 [pmm'] [14] [pmm'] [pmm^{-}] [pm'm] [pmm[2]_{2}] [2'2'2] [pmm/pm]
      15 [pmg'] [17] [pmg'] [pmg^{-}] [pmg'] [pmg[2]_{4}] [2'2_{1}'2] [pmg/pm]
      16 [pgg'] [18] [pgg'] [pgg^{-}] [pgg'] [pgg[2]_{1}] [2'2_{1}'2_{1}] [pgg/pg]
      17 [pm'g] [16] [pm'g] [pm^{-}g] [pm'g] [pmg[2]_{2}] [2'2_{1}2'] [pmg/pg]
      18 [cmm'] [21] [cmm'] [cmm^{-}] [cmm'] [cmm[2]_{2}] [c2'22'] [cmm/cm]
    Orthorhombic/rectangular 19 [pm'm'] [15] [pm'm'] [pm^{-}m^{-}] [pm'm'] [pmm[2]_{5}] [22'2'] [pmm/p2]
      20 [pm'g'] [20] [pm'g'] [pm^{-}g^{-}] [pm'g'] [pmg[2]_{5}] [22'2_{1}'] [pmg/p2]
      21 [pg'g'] [19] [pg'g'] [pg^{-}g^{-}] [pg'g'] [pgg[2]_{2}] [22_{1}'2_{1}'] [pgg/p2]
      22 [cm'm'] [22] [cm'm'] [cm^{-}m^{-}] [cm'm'] [cmm[2]_{4}] [c22'2'] [cmm/p2]
      23 [pmm2] [52]     [pmm]      
      24 [pmg2] [53]     [pmg]      
      25 [pgg2] [54]     [pgg]      
      26 [cmm2] [55]     [cmm]      
      27 [pm1'] [66]     [pm1']      
      28 [p_{b}'m] [7] [pm+t'] [pm+t^{-}] [p_{2b}m] [pm[2]_{3}] [b12] [pm/pm(m)]
      29 [pg1'] [67]     [pg1']      
      30 [p_{b}'g] [8] [pg+t'] [pg+t^{-}] [p_{2b}m'] [pm[2]_{1}] [b12_{1}] [pm/pg]
      31 [p_{b}'1m] [9] [pm+m'] [pm+m^{-}] [p_{2a}m] [pm[2]_{5}] [b'1m] [pm/pm(m')]
      32 [p_{c}'m] [11] [pm+g'] [pm+g^{-}] [c_{p}m] [cm[2]_{3}] [n12] [cm/pm]
      33 [p_{b}'1g] [10] [pg+g'] [pg+g^{-}] [p_{2a}g] [pg[2]_{2}] [b2_{1}1] [pg/pg]
      34 [p_{c}'g] [12] [pg+m'] [pg+m^{-}] [c_{p}m'] [cm[2]_{2}] [n12_{1}] [cm/pg]
      35 [cm1'] [68]     [cm1']      
      36 [c'm] [13] [cm+m'] [cm+m^{-}] [p_{c}m] [pm[2]_{2}] [ca12] [pm/cm]
      37 [pmm21'] [69]     [pmm1']      
      38 [p_{b}'gm] [25] [pg,m+m'] [pg,m+m^{-}] [p_{2a}mm'] [pmm[2]_{4}] [a2_{1}2] [pmm/pmg]
      39 [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]
      40 [pmg21'] [70]     [pmg1']      
      41 [p_{b}'mm] [23] [pm,m+m'] [pm,m+m^{-}] [p_{2a}mm] [pmm[2]_{1}] [a22] [pmm/pmm]
      42 [p_{c}'mg] [28] [pm+g',g+m'] [pm+g^{-},g+m^{-}] [c_{p}mm'] [cmm[2]_{3}] [n22_{1}] [cmm/pmg]
      43 [p_{b}'gg] [26] [pg,g+g'] [pg,g+g^{-}] [p_{2b}m'g] [pmg[2]_{3}] [a2_{1}2_{1}] [pmg/pgg]
      44 [pgg21'] [71]     [pgg1']      
      45 [p_{b}'mg] [24] [pm,g+g'] [pm,g+g^{-}] [p_{2b}mg] [pmg[2]_{1}] [b2_{1}2] [pmg/pmg]
      46 [p_{c}'mm] [27] [pm+g',m+g'] [pm+g^{-},m+g^{-}] [c_{p}mm] [cmm[2]_{5}] [n22] [cmm/pmm]
      47 [cmm21'] [72]     [cmm1']      
      48 [c'mm] [30] [cm+m',m+m'] [cm+m^{-},m+m^{-}] [p_{c}mm] [pmm[2]_{3}] [ca22] [pmm/cmm]
    Tetragonal/square 49 [p4] [56]     [p4]      
      50 [p4'] [31] [p4'] [p4^{-}] [p4'] [p4[2]_{2}] [4'11] [p4/p2]
      51 [p41'] [73]     [p41']      
      52 [p_{c}'4] [32] [p4t'] [p4t^{-}] [p_{p}4] [p4[2]_{1}] [4/n11] [p4/p4]
      53 [p4m'm'] [35] [p4m'm'] [p4m^{-}m^{-}] [p4m'] [pm4[2]_{2}] [42'2'] [p4m/p4]
      54 [p4g'm'] [38] [p4g'm'] [p4g^{-}m^{-}] [p4g'] [p4g[2]_{1}] [42_{1}'2'] [p4g/p4]
      55 [p4mm] [57]     [p4m]      
      56 [p4gm] [58]     [p4g]      
      57 [p4'm'm] [34] [p4'm'm] [p4^{-}m^{-}m] [p4'm'] [p4m[2]_{3}] [4'2'2] [p4m/cmm]
      58 [p4'g'm] [37] [p4'g'm] [p4^{-}g^{-}m] [p4'g'] [p4g[2]_{2}] [4'2_{1}'2] [p4g/cmm]
      59 [p4'mm'] [33] [p4'mm'] [p4^{-}mm^{-}] [p4'm] [p4m[2]_{4}] [4'22'] [p4m/pmm]
      60 [p4'gm'] [36] [p4'gm'] [p4^{-}gm^{-}] [p4'g] [p4g[2]_{3}] [4'2_{1}2'] [p4g/pgg]
      61 [p4mm1'] [74]     [p4m1']      
      62 [p_{c}'4gm] [40] [p4g+m',m+m'] [p4g+m^{-},m+m^{-}] [p_{p}4m'] [p4m[2]_{1}] [4/n2_{1}2] [p4m/p4g]
      63 [p4gm1'] [75]     [p4g1']      
      64 [p_{c}'4mm] [39] [p4m+g',m+m'] [p4m+g^{-},m+m^{-}] [p_{p}4m] [p4m[2]_{5}] [4/n22] [p4m/p4m]
    Trigonal/hexagonal 65 [p3] [59]     [p3]      
      66 [p6'] [43] [p6'] [p6^{-}] [p6'] [p6[2]] [6'] [p6/p3]
      67 [p3m'] [41] [p3m'1] [p3m^{-}1] [p3m'1] [p3m1[2]] [312'] [p3m1/p3]
      68 [p31m'] [42] [p31m'] [p31m^{-}] [p31m'] [p31m[2]] [32'1] [p31m/p3]
      69 [p3m] [60]     [p3m1]      
      70 [p31m] [61]     [p31m]      
      71 [p6'm'm] [44] [p6'm'm] [p6^{-}m^{-}m] [p6'm'] [p6m[2]_{1}] [6'22'] [p6m/p31m]
      72 [p6'mm'] [45] [p6'mm'] [p6^{-}mm^{-}] [p6'm] [p6m[2]_{2}] [6'2'2] [p6m/p3m1]
    Hexagonal/hexagonal 73 [p6] [62]     [p6]      
      74 [p3'] [76]     [p31']      
      75 [p61'] [79]     [p61']      
      76 [p6m'm'] [46] [p6m'm'] [p6m^{-}m^{-}] [p6m'] [p6m[2]_{3}] [62'2'] [p6m/p6]
      77 [p6mm] [63]     [p6m]      
      78 [p3'm] [77]     [p3m11']      
      79 [p3'1m] [78]     [p31m1']      
      80 [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. (1957a[link],b[link]).

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

  • Columns 20 and 21: symbols listed by Cochran as listed, respectively, by Cochran (1952[link]) and Belov & Tarkhova (1956a[link],b[link],c[link],d[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. (1957a). Shubnikov groups. Kristallografia, 2, 315–325.
Belov, N. V., Neronova, N. N. & Smirnova, T. S. (1957b). Shubnikov groups. Sov. Phys. Crystallogr. 2, 311–322.
Belov, N. V. & Tarkhova, T. N. (1956a). Color symmetry groups. Kristallografia, 1, 4–13. [Reprinted in: Colored Symmetry. (1964). Edited by W. T. Holser. New York: Macmillan.]
Belov, N. V. & Tarkhova, T. N. (1956b). Color symmetry groups. 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. (1956c). Color symmetry groups. Kristallografia, 1, 619–620.
Belov, N. V. & Tarkhova, T. N. (1956d). Color symmetry groups. 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.
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.
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.
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.
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.
Vainshtein, B. K. (1981). Modern Crystallography I. Berlin: Springer-Verlag.
Weber, L. (1929). Die Symmetrie homogener ebener Punktsysteme. Z. Kristallogr. 70, 309–327.
Wood, E. (1964a). The 80 diperiodic groups in three dimensions. Bell Syst. Tech. J. 43, 541–559.
Wood, E. (1964b). The 80 diperiodic groups in three dimensions. Bell Telephone Technical Publications, Monograph 4680.
Woods, H. J. (1935a). The geometrical basis of pattern design. Part I. Point and line symmetry in simple figures and borders. J. Text. Inst. 26, T197–T210.
Woods, H. J. (1935b). The geometrical basis of pattern design. Part II. Nets and sateens. J. Text. Inst. 26, T293–T308.
Woods, H. J. (1935c). The geometrical basis of pattern design. Part III. Geometrical symmetry in plane patterns. J. Text. Inst. 26, T341–T357.
Woods, H. J. (1936). The geometrical basis of pattern design. Part IV. Counterchange symmetry of plane patterns. J. Text. Inst. 27, T305–T320.








































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