International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 4.3, pp. 62-68

Section 4.3.2. Monoclinic system

E. F. Bertauta

aLaboratoire de Cristallographie, CNRS, Grenoble, France

4.3.2. Monoclinic system

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4.3.2.1. Historical note and arrangement of the tables

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In IT (1935)[link] only the b axis was considered as the unique axis. In IT (1952)[link] two choices were given: the c-axis setting was called the `first setting' and the b-axis setting was designated the `second setting'.

To avoid the presence of two standard space-group symbols side by side, in the present tables only one standard short symbol has been chosen, that conforming to the long-lasting tradition of the b-axis unique (cf. Sections 2.2.4[link] and 2.2.16[link] ). However, for reasons of rigour and completeness, in Table 4.3.2.1[link] the full symbols are given not only for the c-axis and the b-axis settings but also for the a-axis setting. Thus, Table 4.3.2.1[link] has six columns which in pairs refer to these three settings. In the headline, the unique axis of each setting is underlined.

Table 4.3.2.1| top | pdf |
Index of symbols for space groups for various settings and cells

TRICLINIC SYSTEM

No. of space groupSchoenflies symbolHermann–Mauguin symbol for all settings of the same unit cell
1 [C^{1}_{1}] P1
2 [C^{1}_{i}] [P\bar{1}]

MONOCLINIC SYSTEM

No. of space groupSchoenflies symbolStandard short Hermann–Mauguin symbolExtended Hermann–Mauguin symbols for various settings and cell choices 
abc[{\bf c}{\bar{\underline{\bf b}}}{\bf a}]    Unique axis b
  abc[{\bf ba}\bar{\underline{\bf c}}]  Unique axis c
    abc[{\bar{\underline{\bf a}}}{\bf cb}]Unique axis a
3 [C_{2}^{1}] P2 P121 P121 P112 P112 P211 P211  
4 [C_{2}^{2}] [P2_{1}] [P12_{1}1] [P12_{1}1] [P112_{1}] [P112_{1}] [P2_{1}11] [P2_{1}11]  
5 [C_{2}^{3}] C2 [\!\matrix{C1 21\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{A1 21\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{A11 2\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{B11 2\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{B 211\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{C 211\hfill\cr 2_{1}\hfill\cr}] Cell choice 1
      [\!\matrix{A1 21\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{C1 21\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{B11 2\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{A11 2\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{C 211\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{B 211\hfill\cr 2_{1}\hfill\cr}] Cell choice 2
      [\!\matrix{I1 21\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{I1 21\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{I11 2\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{I11 2\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{I 211\hfill\cr 2_{1}\hfill\cr}] [\!\matrix{I 211\hfill\cr 2_{1}\hfill\cr}] Cell choice 3
6 [C_{s}^{1}] Pm P1m1 P1m1 P11m P11m Pm11 Pm11  
7 [C_{s}^{2}] Pc P1c1 P1a1 P11a P11b Pb11 Pc11 Cell choice 1
      P1n1 P1n1 P11n P11n Pn11 Pn11 Cell choice 2
      P1a1 P1c1 P11b P11a Pc11 Pb11 Cell choice 3
8 [C_{s}^{3}] Cm [\!\matrix{C1m1\hfill\cr a\hfill\cr}] [\!\matrix{A1m1\hfill\cr c\hfill\cr}] [\!\matrix{A11m\hfill\cr b\hfill\cr}] [\!\matrix{B11m\hfill\cr a\hfill\cr}] [\!\matrix{Bm11\hfill\cr c\hfill\cr}] [\!\matrix{Cm11\hfill\cr b\hfill\cr}] Cell choice 1
      [\!\matrix{A1m1\hfill\cr c\hfill\cr}] [\!\matrix{C1m1\hfill\cr a\hfill\cr}] [\!\matrix{B11m\hfill\cr a\hfill\cr}] [\!\matrix{A11m\hfill\cr b\hfill\cr}] [\!\matrix{Cm11\hfill\cr b\hfill\cr}] [\!\matrix{Bm11\hfill\cr c\hfill\cr}] Cell choice 2
      [\!\matrix{I1m1\hfill\cr n\hfill\cr}] [\!\matrix{I1m1\hfill\cr n\hfill\cr}] [\!\matrix{I11m\hfill\cr n\hfill\cr}] [\!\matrix{I11m\hfill\cr n\hfill\cr}] [\!\matrix{Im11\hfill\cr n\hfill\cr}] [\!\matrix{Im11\hfill\cr n\hfill\cr}] Cell choice 3
9 [C_{s}^{4}] Cc [\!\matrix{C1c1\hfill\cr n\hfill\cr}] [\!\matrix{A1a1\hfill\cr n\hfill\cr}] [\!\matrix{A11a\hfill\cr n\hfill\cr}] [\!\matrix{B11b\hfill\cr n\hfill\cr}] [\!\matrix{Bb11\hfill\cr n\hfill\cr}] [\!\matrix{Cc11\hfill\cr n\hfill\cr}] Cell choice 1
      [\!\matrix{A1n1\hfill\cr a\hfill\cr}] [\!\matrix{C1n1\hfill\cr c\hfill\cr}] [\!\matrix{B11n\hfill\cr b\hfill\cr}] [\!\matrix{A11n\hfill\cr a\hfill\cr}] [\!\matrix{Cn11\hfill\cr c\hfill\cr}] [\!\matrix{Bn11\hfill\cr b\hfill\cr}] Cell choice 2
      [\!\matrix{I1a1\hfill\cr c\hfill\cr}] [\!\matrix{I1c1\hfill\cr a\hfill\cr}] [\!\matrix{I11b\hfill\cr a\hfill\cr}] [\!\matrix{I11a\hfill\cr b\hfill\cr}] [\!\matrix{Ic11\hfill\cr b\hfill\cr}] [\!\matrix{Ib11\hfill\cr c\hfill\cr}] Cell choice 3
10 [C_{2h}^{1}] P2/m [P1\displaystyle{2 \over m}1] [P1\displaystyle{2 \over m}1] [P11\displaystyle{2 \over m}] [P11\displaystyle{2 \over m}] [P\displaystyle{2 \over m}11] [P\displaystyle{2 \over m}11]  
11 [C_{2h}^{2}] [P2_{1}/m] [P1\displaystyle\displaystyle{2_{1} \over m}1] [P1\displaystyle\displaystyle{2_{1} \over m}1] [P11\displaystyle\displaystyle{2_{1} \over m}] [P11\displaystyle{2_{1} \over m}] [P\displaystyle{2_{1} \over m}11] [P\displaystyle{2_{1} \over m}11]  
12 [C_{2h}^{3}] C2/m [\!\matrix{C1\displaystyle{2 \over m}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{A1\displaystyle{2 \over m}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] [\!\matrix{A11\displaystyle{2 \over m}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] [\!\matrix{B11\displaystyle{2 \over m}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{B\displaystyle{2 \over m}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] [\!\matrix{C\displaystyle{2 \over m}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] Cell choice 1
      [\!\matrix{A1\displaystyle{2 \over m}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] [\!\matrix{C1\displaystyle{2 \over m}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{B11\displaystyle{2 \over m}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{A11\displaystyle{2 \over m}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] [\!\matrix{C\displaystyle{2 \over m}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] [\!\matrix{B\displaystyle{2 \over m}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] Cell choice 2
      [\!\matrix{I1\displaystyle{2 \over m}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{I1\displaystyle{2 \over m}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{I11\displaystyle{2 \over m}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{I11\displaystyle{2 \over m}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{I\displaystyle{2 \over m}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{I\displaystyle{2 \over m}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] Cell choice 3
13 [C_{2h}^{4}] P2/c [P1\displaystyle{2 \over c}1] [P1\displaystyle{2 \over a}1] [P11\displaystyle{2 \over a}] [P11\displaystyle{2 \over b}] [P\displaystyle{2 \over b}11] [P\displaystyle{2 \over c}11] Cell choice 1
      [P1\displaystyle{2 \over n}1] [P1\displaystyle{2 \over n}1] [P11\displaystyle{2 \over n}] [P11\displaystyle{2 \over n}] [P\displaystyle{2 \over n}11] [P\displaystyle{2 \over n}11] Cell choice 2
      [P1\displaystyle{2 \over a}1] [P1\displaystyle{2 \over c}1] [P11\displaystyle{2 \over b}] [P11\displaystyle{2 \over a}] [P\displaystyle{2 \over c}11] [P\displaystyle{2 \over b}11] Cell choice 3
14 [C_{2h}^{5}] [P2_{1}/c] [P1\displaystyle{2_{1} \over c}1] [P1\displaystyle{2_{1} \over a}1] [P11\displaystyle{2_{1} \over a}] [P11\displaystyle{2_{1} \over b}] [P\displaystyle{2_{1} \over b}11] [P\displaystyle{2_{1} \over c}11] Cell choice 1
      [P1\displaystyle{2_{1} \over n}1] [P1\displaystyle{2_{1} \over n}1] [P11\displaystyle{2_{1} \over n}] [P11\displaystyle{2_{1} \over n}] [P\displaystyle{2_{1} \over n}11] [P\displaystyle{2_{1} \over n}11] Cell choice 2
      [P1\displaystyle{2_{1} \over a}1] [P1\displaystyle{2_{1} \over c}1] [P11\displaystyle{2_{1} \over b}] [P11\displaystyle{2_{1} \over a}] [P\displaystyle{2_{1} \over c}11] [P\displaystyle{2_{1} \over b}11] Cell choice 3
15 [C_{2h}^{6}] C2/c [\!\matrix{C1\displaystyle{2 \over c}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{A1\displaystyle{2 \over a}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{A11\displaystyle{2 \over a}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{B11\displaystyle{2 \over b}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{B\displaystyle{2 \over b}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] [\!\matrix{C\displaystyle{2 \over c}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over n}\hfill\cr}] Cell choice 1
      [\!\matrix{A1\displaystyle{2 \over n}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{C1\displaystyle{2 \over n}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] [\!\matrix{B11\displaystyle{2 \over n}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] [\!\matrix{A11\displaystyle{2 \over n}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{C\displaystyle{2 \over n}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] [\!\matrix{B\displaystyle{2 \over n}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] Cell choice 2
      [\!\matrix{I1\displaystyle{2 \over a}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] [\!\matrix{I1\displaystyle{2 \over c}1\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{I11\displaystyle{2 \over b}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over a}\hfill\cr}] [\!\matrix{I11\displaystyle{2 \over a}\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] [\!\matrix{I\displaystyle{2 \over c}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over b}\hfill\cr}] [\!\matrix{I\displaystyle{2 \over b}11\hfill\cr\noalign{\vskip 6pt} \displaystyle{2_{1} \over c}\hfill\cr}] Cell choice 3

ORTHORHOMBIC SYSTEM

No. of space groupSchoenflies symbolStandard full Hermann–Mauguin symbol
abc
Extended Hermann–Mauguin symbols for the six settings of the same unit cell
abc (standard)[{\bf b a} {\bar{\bf c}}]cab[{\bar{\bf c}}\bf{ b a}]bca[{\bf a}\bar{\bf c}{\bf b}]
16 [D_{2}^{1}] P222 P222 P222 P222 P222 P222 P222
17 [D_{2}^{2}] [P222_{1}] [P222_{1}] [P222_{1}] [P2_{1}22] [P2_{1}22] [P22_{1}2] [P22_{1}2]
18 [D_{2}^{3}] [P2_{1}2_{1}2] [P2_{1}2_{1}2] [P2_{1}2_{1}2] [P22_{1}2_{1}] [P22_{1}2_{1}] [P2_{1}22_{1}] [P2_{1}22_{1}]
19 [D_{2}^{4}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}] [P2_{1}2_{1}2_{1}]
20 [D_{2}^{5}] [C222_{1}] [\!\matrix{C222_{1}\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{C222_{1}\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{A2_{1}22\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{A2_{1}22\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{B22_{1}2\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{B22_{1}2\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}]
21 [D_{2}^{6}] C222 [\!\matrix{C222\hfill\cr 2_{1}2_{1}2\hfill\cr}] [\!\matrix{C222\hfill\cr 2_{1}2_{1}2\hfill\cr}] [\!\matrix{A222\hfill\cr 22_{1}2_{1}\hfill\cr}] [\!\matrix{A222\hfill\cr 22_{1}2_{1}\hfill\cr}] [\!\matrix{B222\hfill\cr 2_{1}22_{1}\hfill\cr}] [\!\matrix{B222\hfill\cr 2_{1}22_{1}\hfill\cr}]
22 [D_{2}^{7}] F222 [\!\matrix{F222\hfill\cr 2_{1}2_{1}2\hfill\cr 22_{1}2_{1}\hfill\cr 2_{1}22_{1}\hfill\cr}] [\!\matrix{F222\hfill\cr 2_{1}2_{1}2\hfill\cr 2_{1}22_{1}\hfill\cr 22_{1}2_{1}\hfill\cr}] [\!\matrix{F222\hfill\cr 22_{1}2_{1}\hfill\cr 2_{1}22_{1}\hfill\cr 2_{1}2_{1}2\hfill\cr}] [\!\matrix{F222\hfill\cr 22_{1}2_{1}\hfill\cr 2_{1}2_{1}2\hfill\cr 2_{1}22_{1}\hfill\cr}] [\!\matrix{F222\hfill\cr 2_{1}22_{1}\hfill\cr 2_{1}2_{1}2\hfill\cr 22_{1}2_{1}\hfill\cr}] [\!\matrix{F222\hfill\cr 2_{1}22_{1}\hfill\cr 22_{1}2_{1}\hfill\cr 2_{1}2_{1}2\hfill\cr}]
23 [D_{2}^{8}] I222 [\!\matrix{I222\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{I222\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{I222\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{I222\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{I222\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}] [\!\matrix{I222\hfill\cr 2_{1}2_{1}2_{1}\hfill\cr}]
24 [D_{2}^{9}] [I2_{1}2_{1}2_{1}] [\!\matrix{I2_{1}2_{1}2_{1}\hfill\cr 222\hfill\cr}] [\!\matrix{I2_{1}2_{1}2_{1}\hfill\cr 222\hfill\cr}] [\!\matrix{I2_{1}2_{1}2_{1}\hfill\cr 222\hfill\cr}] [\!\matrix{I2_{1}2_{1}2_{1}\hfill\cr 222\hfill\cr}] [\!\matrix{I2_{1}2_{1}2_{1}\hfill\cr 222\hfill\cr}] [\!\matrix{I2_{1}2_{1}2_{1}\hfill\cr 222\hfill\cr}]
25 [C_{2v}^{1}] Pmm2 Pmm2 Pmm2 P2mm P2mm Pm2m Pm2m
26 [C_{2v}^{2}] [Pmc2_{1}] [Pmc2_{1}] [Pcm2_{1}] [P2_{1}ma] [P2_{1}am] [Pb2_{1}m] [Pm2_{1}b]
27 [C_{2v}^{3}] Pcc2 Pcc2 Pcc2 P2aa P2aa Pb2b Pb2b
28 [C_{2v}^{4}] Pma2 Pma2 Pbm2 P2mb P2cm Pc2m Pm2a
29 [C_{2v}^{5}] [Pca2_{1}] [Pca2_{1}] [Pbc2_{1}] [P2_{1}ab] [P2_{1}ca] [Pc2_{1}b] [Pb2_{1}a]
30 [C_{2v}^{6}] Pnc2 Pnc2 Pcn2 P2na P2an Pb2n Pn2b
31 [C_{2v}^{7}] [Pmn2_{1}] [Pmn2_{1}] [Pnm2_{1}] [P2_{1}mn] [P2_{1}nm] [Pn2_{1}m] [Pm2_{1}n]
32 [C_{2v}^{8}] Pba2 Pba2 Pba2 P2cb P2cb Pc2a Pc2a
33 [C_{2v}^{9}] [Pna2_{1}] [Pna2_{1}] [Pbn2_{1}] [P2_{1}nb] [P2_{1}cn] [Pc2_{1}n] [Pn2_{1}a]
34 [C_{2v}^{10}] Pnn2 Pnn2 Pnn2 P2nn P2nn Pn2n Pn2n
35 [C_{2v}^{11}] Cmm2 [\!\matrix{Cmm2\hfill\cr ba2\hfill\cr}] [\!\matrix{Cmm2\hfill\cr ba2\hfill\cr}] [\!\matrix{A2mm\hfill\cr 2cb\hfill\cr}] [\!\matrix{A2mm\hfill\cr 2cb\hfill\cr}] [\!\matrix{Bm2m\hfill\cr c2a\hfill\cr}] [\!\matrix{Bm2m\hfill\cr c2a\hfill\cr}]
36 [C_{2v}^{12}] [Cmc2_{1}] [\!\matrix{Cmc2_{1}\hfill\cr bn2_{1}\hfill\cr}] [\!\matrix{Ccm2_{1}\hfill\cr na2_{1}\hfill\cr}] [\!\matrix{A2_{1}ma\hfill\cr 2_{1}cn\hfill\cr}] [\!\matrix{A2_{1}am\hfill\cr 2_{1}nb\hfill\cr}] [\!\matrix{Bb2_{1}m\hfill\cr n2_{1}a\hfill\cr}] [\!\matrix{Bm2_{1}b\hfill\cr c2_{1}n\hfill\cr}]
37 [C_{2v}^{13}] Ccc2 [\!\matrix{Ccc2\cr nn2}] [\!\matrix{Ccc2\cr nn2}] [\!\matrix{A2aa\cr 2nn}] [\!\matrix{A2aa\cr 2nn}] [\!\matrix{Bb2b\cr n2n}] [\!\matrix{Bb2b\cr n2n}]
38 [C_{2v}^{14}] Amm2 [\!\matrix{Amm2\hfill\cr nc2_{1}\hfill\cr}] [\!\matrix{Bmm2\hfill\cr cn2_{1}\hfill\cr}] [\!\matrix{B2mm\hfill\cr 2_{1}na\hfill\cr}] [\!\matrix{C2mm\hfill\cr 2_{1}an\hfill\cr}] [\!\matrix{Cm2m\hfill\cr b2_{1}n\hfill\cr}] [\!\matrix{Am2m\hfill\cr n2_{1}b\hfill\cr}]
39 [C_{2v}^{15}] Aem2 [\!\matrix{Aem2\hfill\cr ec2_{1}\hfill\cr}] [\!\matrix{Bme2\hfill\cr ce2_{1}\hfill\cr}] [\!\matrix{B2em\hfill\cr 2_{1}ea\hfill\cr}] [\!\matrix{C2me\hfill\cr 2_{1}ae\hfill\cr}] [\!\matrix{Cm2e\hfill\cr b2_{1}e\hfill\cr}] [\!\matrix{Ae2m\hfill\cr e2_{1}b\hfill\cr}]
40 [C_{2v}^{16}] Ama2 [\!\matrix{Ama2\hfill\cr nn2_{1}\hfill\cr}] [\!\matrix{Bbm2\hfill\cr nn2_{1}\hfill\cr}] [\!\matrix{B2mb\hfill\cr 2_{1}nn\hfill\cr}] [\!\matrix{C2cm\hfill\cr 2_{1}nn\hfill\cr}] [\!\matrix{Cc2m\hfill\cr n2_{1}n\hfill\cr}] [\!\matrix{Am2a\hfill\cr n2_{1}n\hfill\cr}]
41 [C_{2v}^{17}] Aea2 [\!\matrix{Aea2\hfill\cr en2_{1}\hfill\cr}] [\!\matrix{Bbe2\hfill\cr ne2_{1}\hfill\cr}] [\!\matrix{B2eb\hfill\cr 2_{1}en\hfill\cr}] [\!\matrix{C2ce\hfill\cr 2_{1}ne\hfill\cr}] [\!\matrix{Cc2e\hfill\cr n2_{1}e\hfill\cr}] [\!\matrix{Ae2a\hfill\cr e2_{1}n\hfill\cr}]
42 [C_{2v}^{18}] Fmm2 [\!\matrix{Fmm2\hfill\cr ba2\hfill\cr nc2_{1}\hfill\cr cn2_{1}\hfill\cr}] [\!\matrix{Fmm2\hfill\cr ba2\hfill\cr cn2_{1}\hfill\cr nc2_{1}\hfill\cr}] [\!\matrix{F2mm\hfill\cr 2cb\hfill\cr 2_{1}na\hfill\cr 2_{1}an\hfill\cr}] [\!\matrix{F2mm\hfill\cr 2cb\hfill\cr 2_{1}an\hfill\cr 2_{1}na\hfill\cr}] [\!\matrix{Fm2m\hfill\cr c2a\hfill\cr b2_{1}n\hfill\cr n2_{1}b\hfill\cr}] [\!\matrix{Fm2m\hfill\cr c2a\hfill\cr n2_{1}b\hfill\cr b2_{1}n\hfill\cr}]
43 [C_{2v}^{19}] Fdd2 [\!\matrix{Fdd2\hfill\cr dd2_{1}\hfill\cr}] [\!\matrix{Fdd2\hfill\cr dd2_{1}\hfill\cr}] [\!\matrix{F2dd\hfill\cr 2_{1}dd\hfill\cr}] [\!\matrix{F2dd\hfill\cr 2_{1}dd\hfill\cr}] [\!\matrix{Fd2d\hfill\cr d2_{1}d\hfill\cr}] [\!\matrix{Fd2d\hfill\cr d2_{1}d\hfill\cr}]
44 [C_{2v}^{20}] Imm2 [\!\matrix{Imm2\hfill\cr nn2_{1}\hfill\cr}] [\!\matrix{Imm2\hfill\cr nn2_{1}\hfill\cr}] [\!\matrix{I2mm\hfill\cr 2_{1}nn\hfill\cr}] [\!\matrix{I2mm\hfill\cr 2_{1}nn\hfill\cr}] [\!\matrix{Im2m\hfill\cr n2_{1}n\hfill\cr}] [\!\matrix{Im2m\hfill\cr n2_{1}n\hfill\cr}]
45 [C_{2v}^{21}] Iba2 [\!\matrix{Iba2\hfill\cr cc2_{1}\hfill\cr}] [\!\matrix{Iba2\hfill\cr cc2_{1}\hfill\cr}] [\!\matrix{I2cb\hfill\cr 2_{1}aa\hfill\cr}] [\!\matrix{I2cb\hfill\cr 2_{1}aa\hfill\cr}] [\!\matrix{Ic2a\hfill\cr b2_{1}b\hfill\cr}] [\!\matrix{Ic2a\hfill\cr b2_{1}b\hfill\cr}]
46 [C_{2v}^{22}] Ima2 [\!\matrix{Ima2\hfill\cr nc2_{1}\hfill\cr}] [\!\matrix{Ibm2\hfill\cr cn2_{1}\hfill\cr}] [\!\matrix{I2mb\hfill\cr 2_{1}na\hfill\cr}] [\!\matrix{I2cm\hfill\cr 2_{1}an\hfill\cr}] [\!\matrix{Ic2m\hfill\cr b2_{1}n\hfill\cr}] [\!\matrix{Im2a\hfill\cr n2_{1}b\hfill\cr}]
47 [D_{2h}^{1}] [P\displaystyle{2 \over m}\displaystyle{2 \over m}\displaystyle{2 \over m}] Pmmm Pmmm Pmmm Pmmm Pmmm Pmmm
48 [D_{2h}^{2}] [P\displaystyle{2 \over n}\displaystyle{2 \over n}\displaystyle{2 \over n}] Pnnn Pnnn Pnnn Pnnn Pnnn Pnnn
49 [D_{2h}^{3}] [P\displaystyle{2 \over c}\displaystyle{2 \over c}\displaystyle{2 \over m}] Pccm Pccm Pmaa Pmaa Pbmb Pbmb
50 [D_{2h}^{4}] [P\displaystyle{2 \over b}\displaystyle{2 \over a}\displaystyle{2 \over n}] Pban Pban Pncb Pncb Pcna Pcna
51 [D_{2h}^{5}] [P\displaystyle{2_{1} \over m}\displaystyle{2 \over m}\displaystyle{2 \over a}] Pmma Pmmb Pbmm Pcmm Pmcm Pmam
52 [D_{2h}^{6}] [P\displaystyle{2 \over n}\displaystyle{2_{1} \over n}\displaystyle{2 \over a}] Pnna Pnnb Pbnn Pcnn Pncn Pnan
53 [D_{2h}^{7}] [P\displaystyle{2 \over m}\displaystyle{2 \over n}\displaystyle{2_{1} \over a}] Pmna Pnmb Pbmn Pcnm Pncm Pman
54 [D_{2h}^{8}] [P\displaystyle{2_{1} \over c}\displaystyle{2 \over c}\displaystyle{2 \over a}] Pcca Pccb Pbaa Pcaa Pbcb Pbab
55 [D_{2h}^{9}] [P\displaystyle{2_{1} \over b}\displaystyle{2_{1} \over a}\displaystyle{2 \over m}] Pbam Pbam Pmcb Pmcb Pcma Pcma
56 [D_{2h}^{10}] [P\displaystyle{2_{1} \over c}\displaystyle{2_{1} \over c}\displaystyle{2 \over n}] Pccn Pccn Pnaa Pnaa Pbnb Pbnb
57 [D_{2h}^{11}] [P\displaystyle{2 \over b}\displaystyle{2_{1} \over c}\displaystyle{2_{1} \over m}] Pbcm Pcam Pmca Pmab Pbma Pcmb
58 [D_{2h}^{12}] [P\displaystyle{2_{1} \over n}\displaystyle{2_{1} \over n}\displaystyle{2 \over m}] Pnnm Pnnm Pmnn Pmnn Pnmn Pnmn
59 [D_{2h}^{13}] [P\displaystyle{2_{1} \over m}\displaystyle{2_{1} \over m}\displaystyle{2 \over n}] Pmmn Pmmn Pnmm Pnmm Pmnm Pmnm
60 [D_{2h}^{14}] [P\displaystyle{2_{1} \over b}\displaystyle{2 \over c}\displaystyle{2_{1} \over n}] Pbcn Pcan Pnca Pnab Pbna Pcnb
61 [D_{2h}^{15}] [P\displaystyle{2_{1} \over b}\displaystyle{2_{1} \over c}\displaystyle{2_{1} \over a}] Pbca Pcab Pbca Pcab Pbca Pcab
62 [D_{2h}^{16}] [P\displaystyle{2_{1} \over n}\displaystyle{2_{1} \over m}\displaystyle{2_{1} \over a}] Pnma Pmnb Pbnm Pcmn Pmcn Pnam
63 [D_{2h}^{17}] [C\displaystyle{2 \over m}\displaystyle{2 \over c}\displaystyle{2_{1} \over m}] [\!\matrix{Cmcm\hfill\cr bnn\hfill\cr}] [\!\matrix{Ccmm\hfill\cr nan\hfill\cr}] [\!\matrix{Amma\hfill\cr ncn\hfill\cr}] [\!\matrix{Amam\hfill\cr nnb\hfill\cr}] [\!\matrix{Bbmm\hfill\cr nna\hfill\cr}] [\!\matrix{Bmmb\hfill\cr cnn\hfill\cr}]
64 [D_{2h}^{18}] [C\displaystyle{2 \over m}\displaystyle{2 \over c}\displaystyle{2_{1} \over e}] [\!\matrix{Cmce\hfill\cr bne\hfill\cr}] [\!\matrix{Ccme\hfill\cr nae\hfill\cr}] [\!\matrix{Aema\hfill\cr ecn\hfill\cr}] [\!\matrix{Aeam\hfill\cr enb\hfill\cr}] [\!\matrix{Bbem\hfill\cr nea\hfill\cr}] [\!\matrix{Bmeb\hfill\cr cen\hfill\cr}]
65 [D_{2h}^{19}] [C\displaystyle{2 \over m}\displaystyle{2 \over m}\displaystyle{2 \over m}] [\!\matrix{Cmmm\hfill\cr ban\hfill\cr}] [\!\matrix{Cmmm\hfill\cr ban\hfill\cr}] [\!\matrix{Ammm\hfill\cr ncb\hfill\cr}] [\!\matrix{Ammm\hfill\cr ncb\hfill\cr}] [\!\matrix{Bmmm\hfill\cr cna\hfill\cr}] [\!\matrix{Bmmm\hfill\cr cna\hfill\cr}]
66 [D_{2h}^{20}] [C\displaystyle{2 \over c}\displaystyle{2 \over c}\displaystyle{2 \over m}] [\!\matrix{Cccm\hfill\cr nnn\hfill\cr}] [\!\matrix{Cccm\hfill\cr nnn\hfill\cr}] [\!\matrix{Amaa\hfill\cr nnn\hfill\cr}] [\!\matrix{Amaa\hfill\cr nnn\hfill\cr}] [\!\matrix{Bbmb\hfill\cr nnn\hfill\cr}] [\!\matrix{Bbmb\hfill\cr nnn\hfill\cr}]
67 [D_{2h}^{21}] [C\displaystyle{2 \over m}\displaystyle{2 \over m}\displaystyle{2 \over e}] [\!\matrix{Cmme\hfill\cr bae\hfill\cr}] [\!\matrix{Cmme\hfill\cr bae\hfill\cr}] [\!\matrix{Aemm\hfill\cr ecb\hfill\cr}] [\!\matrix{Aemm\hfill\cr ecb\hfill\cr}] [\!\matrix{Bmem\hfill\cr cea\hfill\cr}] [\!\matrix{Bmem\hfill\cr cea\hfill\cr}]
68 [D_{2h}^{22}] [C\displaystyle{2 \over c}\displaystyle{2 \over c}\displaystyle{2 \over e}] [\!\matrix{Ccce\hfill\cr nne\hfill\cr}] [\!\matrix{Ccce\hfill\cr nne\hfill\cr}] [\!\matrix{Aeaa\hfill\cr enn\hfill\cr}] [\!\matrix{Aeaa\hfill\cr enn\hfill\cr}] [\!\matrix{Bbeb\hfill\cr nen\hfill\cr}] [\!\matrix{Bbeb\hfill\cr nen\hfill\cr}]
69 [D_{2h}^{23}] [F\displaystyle{2 \over m}\displaystyle{2 \over m}\displaystyle{2 \over m}] [\!\matrix{Fmmm\hfill\cr ban\hfill\cr ncb\hfill\cr cna\hfill\cr}] [\!\matrix{Fmmm\hfill\cr ban\hfill\cr cna\hfill\cr ncb\hfill\cr}] [\!\matrix{Fmmm\hfill\cr ncb\hfill\cr cna\hfill\cr ban\hfill\cr}] [\!\matrix{Fmmm\hfill\cr ncb\hfill\cr ban\hfill\cr cna\hfill\cr}] [\!\matrix{Fmmm\hfill\cr cna\hfill\cr ban\hfill\cr ncb\hfill\cr}] [\!\matrix{Fmmm\hfill\cr cna\hfill\cr ncb\hfill\cr ban\hfill\cr}]
70 [D_{2h}^{24}] [F\displaystyle{2 \over d}\displaystyle{2 \over d}\displaystyle{2 \over d}] Fddd Fddd Fddd Fddd Fddd Fddd
71 [D_{2h}^{25}] [I\displaystyle{2 \over m}\displaystyle{2 \over m}\displaystyle{2 \over m}] [\!\matrix{I\;mmm\hfill\cr nnn\hfill\cr}] [\!\matrix{I\;mmm\hfill\cr nnn\hfill\cr}] [\!\matrix{I\;mmm\hfill\cr nnn\hfill\cr}] [\!\matrix{I\;mmm\hfill\cr nnn\hfill\cr}] [\!\matrix{I\;mmm\hfill\cr nnn\hfill\cr}] [\!\matrix{I\;mmm\hfill\cr nnn\hfill\cr}]
72 [D_{2h}^{26}] [I\displaystyle{2 \over b}\displaystyle{2 \over a}\displaystyle{2 \over m}] [\!\matrix{I\;bam\hfill\cr ccn\hfill\cr}] [\!\matrix{I\;bam\hfill\cr ccn\hfill\cr}] [\!\matrix{I\;mcb\hfill\cr naa\hfill\cr}] [\!\matrix{I\;mcb\hfill\cr naa\hfill\cr}] [\!\matrix{I\;cma\hfill\cr bnb\hfill\cr}] [\!\matrix{I\;cma\hfill\cr bnb\hfill\cr}]
73 [D_{2h}^{27}] [I\displaystyle{2_{1} \over b}\displaystyle{2_{1} \over c}\displaystyle{2_{1} \over a}] [\!\matrix{I\;bca\hfill\cr cab\hfill\cr}] [\!\matrix{I\;cab\hfill\cr bca\hfill\cr}] [\!\matrix{I\;bca\hfill\cr cab\hfill\cr}] [\!\matrix{I\;cab\hfill\cr bca\hfill\cr}] [\!\matrix{I\;bca\hfill\cr cab\hfill\cr}] [\!\matrix{I\;cab\hfill\cr bca\hfill\cr}]
74 [D_{2h}^{28}] [I\displaystyle{2_{1} \over m}\displaystyle{2_{1} \over m}\displaystyle{2_{1} \over a}] [\!\matrix{I\;mma\hfill\cr nnb\hfill\cr}] [\!\matrix{I\;mmb\hfill\cr nna\hfill\cr}] [\!\matrix{I\;bmm\hfill\cr cnn\hfill\cr}] [\!\matrix{I\;cmm\hfill\cr bnn\hfill\cr}] [\!\matrix{I\;mcm\hfill\cr nan\hfill\cr}] [\!\matrix{I\;mam\hfill\cr ncn\hfill\cr}]

TETRAGONAL SYSTEM

Note: The glide planes g, [g_{1}] and [g_{2}] have the glide components [g({1 \over 2}, {1 \over 2}, 0)], [g_{1}({1 \over 4}, {1 \over 4}, 0)] and [g_{2}({1 \over 4}, {1 \over 4}, {1 \over 2})]. For the glide plane symbol `e', see the Foreword to the Fourth Edition (IT 1995) and Section 1.3.2, Note (x)[link] .

No. of space groupSchoen-flies symbolHermann–Mauguin symbols for standard cell P or IMultiple cell C or F
ShortExtendedShortExtended
75 [C_{4}^{1}] P4   C4  
76 [C_{4}^{2}] [P4_{1}]   [C4_{1}]  
77 [C_{4}^{3}] [P4_{2}]   [C4_{2}]  
78 [C_{4}^{4}] [P4_{3}]   [C4_{3}]  
79 [C_{4}^{5}] I 4 [\!\matrix{I4\hfill\cr4_{2}\hfill\cr}] F4 [\!\matrix{F4\hfill\cr 4_{2}\hfill\cr}]
80 [C_{4}^{6}] [I\;4_{1}] [\!\matrix{I4_{1}\hfill\cr 4_{3}\hfill\cr}] [F4_{1}] [\!\matrix{F4_{1}\hfill\cr 4_{3}\hfill\cr}]
81 [S_{4}^{1}] [P\bar{4}]   [C\bar{4}]  
82 [S_{4}^{2}] [I \bar{4}]   [F\bar{4}]  
83 [C_{4h}^{1}] [P4/m]   [C4/m] [\!\matrix{C4_{2}/m\hfill\cr /}n\hfill\cr}]
84 [C_{4h}^{2}] [P4_{2}/m]   [C4_{2}/m] [\!\matrix{C4_{2}/m\hfill\cr /}n\hfill\cr}]
85 [C_{4h}^{3}] [P4/n]   [C4/a] [\!\matrix{C4/a\hfill\cr b\hfill\cr}]
86 [C_{4h}^{4}] [P4_{2}/n]   [C4_{2}/a] [\!\matrix{C4_{2}/a\hfill\cr /}b\hfill\cr}]
87 [C_{4h}^{5}] [I\;4/m] [\!\matrix{I4/m\hfill\cr 4_{2}/n\hfill\cr}] [F4/m] [\!\matrix{F4/m\hfill\cr 4_{2}/a\hfill\cr}]
88 [C_{4h}^{6}] [I\;4_{1}/a] [\!\matrix{I4_{1}/a\hfill\cr 4_{3}/b\hfill\cr}] [F4_{1}/d] [\!\matrix{F4_{1}/d\hfill\cr 4_{3}/d\hfill\cr}]
89 [D_{4}^{1}] P422 [\!\matrix{P422\hfill\cr 2_{1}\hfill\cr}] C422 [\!\matrix{C422\hfill\cr 2_{1}\hfill\cr}]
90 [D_{4}^{2}] [P42_{1}2] [\!\matrix{P42_{1}2\hfill\cr }2_{1}\hfill\cr}] [C422_{1}] [\!\matrix{C422_{1}\hfill\cr 2_{1}\hfill\cr}]
91 [D_{4}^{3}] [P4_{1}22] [\!\matrix{P4_{1}22\hfill\cr 2}2_{1}\hfill\cr}] [C4_{1}22] [\!\matrix{C4_{1}22\hfill\cr 2_{1}\hfill\cr}]
92 [D_{4}^{4}] [P4_{1}2_{1}2] [\!\matrix{P4_{1}2_{1}2\hfill\cr 2_{1}}2_{1}\hfill\cr}] [C4_{1}22_{1}] [\!\matrix{C4_{1}22_{1}\hfill\cr }2_{1}\hfill\cr}]
93 [D_{4}^{5}] [P4_{2}22] [\!\matrix{P4_{2}22\hfill\cr 2}2_{1}\hfill\cr}] [C4_{2}22] [\!\matrix{C4_{2}22\hfill\cr }2_{1}\hfill\cr}]
94 [D_{4}^{6}] [P4_{2}2_{1}2] [\!\matrix{P4_{2}2_{1}2\hfill\cr 2_{1}}2_{1}\hfill\cr}] [C4_{2}22_{1}] [\!\matrix{C4_{2}22_{1}\hfill\cr }2_{1}\hfill\cr}]
95 [D_{4}^{7}] [P4_{3}22] [\!\matrix{P4_{3}22\hfill\cr 2}2_{1}\hfill\cr}] [C4_{3}22] [\!\matrix{C4_{3}22\hfill\cr }2_{1}\hfill\cr}]
96 [D_{4}^{8}] [P4_{3}2_{1}2] [\!\matrix{P4_{3}2_{1}2\hfill\cr 2_{1}}2_{1}\hfill\cr}] [C4_{3}22_{1}] [\!\matrix{C4_{3}22_{1}\hfill\cr }2_{1}\hfill\cr}]
97 [D_{4}^{9}] I 422 [\!\matrix{I\;422\hfill\cr 4_{2}2_{1}2_{1}\hfill\cr}] F422 [\!\matrix{F422\hfill\cr 4_{2}2_{1}2_{1}\hfill\cr}]
98 [D_{4}^{10}] [I 4_{1}22] [\!\matrix{I\;4_{1}22\hfill\cr 4_{3}2_{1}2_{1}\hfill\cr}] [F4_{1}22] [\!\matrix{F4_{1}22\hfill\cr 4_{3}2_{1}2_{1}\hfill\cr}]
99 [C_{4v}^{1}] P4mm [\!\matrix{P4mm\hfill\cr g\hfill\cr}] C4mm [\!\matrix{C4mm\hfill\cr b\hfill\cr}]
100 [C_{4v}^{2}] P4bm [\!\matrix{P4bm\hfill\cr g\hfill\cr}] [C4mg_{1}] [\!\matrix{C4mg_{1}\hfill\cr b\hfill\cr}]
101 [C_{4v}^{3}] [P4_{2}cm] [\!\matrix{P4_{2}cm\hfill\cr c}g\hfill\cr}] [C4_{2}mc] [\!\matrix{C4_{2}mc\hfill\cr }b\hfill\cr}]
102 [C_{4v}^{4}] [P4_{2}nm] [\!\matrix{P4_{2}nm\hfill\cr n}g\hfill\cr}] [C4_{2}mg_{2}] [\!\matrix{C4_{2}mg_{2}\hfill\cr }b\hfill\cr}]
103 [C_{4v}^{5}] P4cc [\!\matrix{P4cc\hfill\cr n\hfill\cr}] C4cc [\!\matrix{C4cc\hfill\cr n\hfill\cr}]
104 [C_{4v}^{6}] P4nc [\!\matrix{P4nc\hfill\cr n\hfill\cr}] [C4cg_{2}] [\!\matrix{C4cg_{2}\hfill\cr n\hfill\cr}]
105 [C_{4v}^{7}] [P4_{2}mc] [\!\matrix{P4_{2}mc\hfill\cr m}n\hfill\cr}] [C4_{2}cm] [\!\matrix{C4_{2}cm\hfill\cr }n\hfill\cr}]
106 [C_{4v}^{8}] [P4_{2}bc] [\!\matrix{P4_{2}bc\hfill\cr b}n\hfill\cr}] [C4_{2}cg_{1}] [\!\matrix{C4_{2}cg_{1}\hfill\cr }n\hfill\cr}]
107 [C_{4v}^{9}] I 4mm [\!\matrix{I\;4mm\hfill\cr 4_{2}ne\hfill\cr}] F4mm [\!\matrix{F4mm\hfill\cr 4_{2}eg_{2}\hfill\cr}]
108 [C_{4v}^{10}] I 4cm [\!\matrix{I\;4ce\hfill\cr 4_{2}bm\hfill\cr}] F4mc [\!\matrix{F4ec\hfill\cr 4_{2}mg_{1}\hfill\cr}]
109 [C_{4v}^{11}] [I\;4_{1}md] [\!\matrix{I\;4_{1}md\hfill\cr 4_{1}nd\hfill\cr}] [F4_{1}dm] [\!\matrix{F4_{1}dm\hfill\cr 4_{3}dg_{2}\hfill\cr}]
110 [C_{4v}^{12}] [I\;4_{1}cd] [\!\matrix{I\;4_{1}cd\hfill\cr 4_{3}bd\hfill\cr}] [F4_{1}dc] [\!\matrix{F4_{1}dc\hfill\cr 4_{3}dg_{1}\hfill\cr}]
111 [D_{2d}^{1}] [P\bar{4}2m] [\!\matrix{P\bar{4}2m\hfill\cr 2}g\hfill\cr}] [C\bar{4}m2] [\!\matrix{C\bar{4}m2\hfill\cr }b\hfill\cr}]
112 [D_{2d}^{2}] [P\bar{4}2c] [\!\matrix{P\bar{4}2c\hfill\cr 2}n\hfill\cr}] [C\bar{4}c2] [\!\matrix{C\bar{4}c2\hfill\cr }n\hfill\cr}]
113 [D_{2d}^{3}] [P\bar{4}2_{1}m] [\!\matrix{P\bar{4}2_{1}m\hfill\cr 2_{1}}g\hfill\cr}] [C\bar{4}m2_{1}] [\!\matrix{C\bar{4}m2_{1}\hfill\cr }b\hfill\cr}]
114 [D_{2d}^{4}] [P\bar{4}2_{1}c] [\!\matrix{P\bar{4}2_{1}c\hfill\cr 2_{1}}n\hfill\cr}] [C\bar{4}c2_{1}] [\!\matrix{C\bar{4}c2_{1}\hfill\cr }n\hfill\cr}]
115 [D_{2d}^{5}] [P\bar{4}m2] [\!\matrix{P\bar{4}m2\hfill\cr m}2_{1}\hfill\cr}] [C\bar{4}2m] [\!\matrix{C\bar{4}2m\hfill\cr }2_{1}\hfill\cr}]
116 [D_{2d}^{6}] [P\bar{4}c2] [\!\matrix{P\bar{4}c2\hfill\cr c}2_{1}\hfill\cr}] [C\bar{4}2c] [\!\matrix{C\bar{4}2c\hfill\cr }2_{1}\hfill\cr}]
117 [D_{2d}^{7}] [P\bar{4}b2] [\!\matrix{P\bar{4}b2\hfill\cr b}2_{1}\hfill\cr}] [C\bar{4}2g_{1}] [\!\matrix{C\bar{4}2g_{1}\hfill\cr }2_{1}\hfill\cr}]
118 [D_{2d}^{8}] [P\bar{4}n2] [\!\matrix{P\bar{4}n2\hfill\cr n}2_{1}\hfill\cr}] [C\bar{4}2g_{2}] [\!\matrix{C\bar{4}2g_{2}\hfill\cr }2_{1}\hfill\cr}]
119 [D_{2d}^{9}] [I\bar{4}m2] [\!\matrix{I\;\bar{4}m2\hfill\cr }n2_{1}\hfill\cr}] [F\bar{4}2m] [\!\matrix{F\bar{4}2m\hfill\cr }2_{1}g_{2}\hfill\cr}]
120 [D_{2d}^{10}] [I\;\bar{4}c2] [\!\matrix{I\;\bar{4}c2\hfill\cr }b2_{1}\hfill\cr}] [F\bar{4}2c] [\!\matrix{F\bar{4}2c\hfill\cr }2_{1}n\hfill\cr}]
121 [D_{2d}^{11}] [I\;\bar{4}2m] [\!\matrix{I\;\bar{4}2m\hfill\cr }2_{1}e\hfill\cr}] [F\bar{4}m2] [\!\matrix{F\bar{4}m2\hfill\cr }e2_{1}\hfill\cr}]
122 [D_{2d}^{12}] [I\;\bar{4}2d] [\!\matrix{I\;\bar{4}2d\hfill\cr }2_{1}d\hfill\cr}] [F\bar{4}d2] [\!\matrix{F\bar{4}d2\hfill\cr }d2_{1}\hfill\cr}]
123 [D_{4h}^{1}] [P4/mmm] [\!\matrix{P4/m &2/m &2/m\cr &&2_{1}/g\cr}] [C4/mmm] [\!\matrix{C4/mmm\hfill\cr nb\hfill\cr}]
124 [D_{4h}^{2}] [P4/mcc] [\!\matrix{P4/m &2/c &2/c\cr &&{\hbox to 4pt{}}2_{1}/n\cr}] [C4/mcc] [\!\matrix{C4/mcc\hfill\cr nn\hfill\cr}]
125 [D_{4h}^{3}] [P4/nbm] [\!\matrix{P4/n &2/b &2/m\cr &&{\hbox to 2pt{}}2_{1}/g\cr}] [C4/amg_{1}] [\!\matrix{C4/amg_{1}\hfill\cr bb\hfill\cr}]
126 [D_{4h}^{4}] [P4/nnc] [\!\matrix{P4/n &2/n &2/c\cr &&{\hbox to 4pt{}}2_{1}/n\cr}] [C4/acg_{2}] [\!\matrix{C4/acg_{2}\hfill\cr bn\hfill\cr}]
127 [D_{4h}^{5}] [P4/mbm] [\!\matrix{P4/m &2_{1}/b &2/m\cr &&{\hbox to 2pt{}}2_{1}/g\cr}] [C4/mmg_{1}] [\!\matrix{C4/mmg_{1}\hfill\cr nb\hfill\cr}]
128 [D_{4h}^{6}] [P4/mnc] [\!\matrix{P4/m &2_{1}/n &2/c\cr &&{\hbox to 4pt{}}2_{1}/n\cr}] [C4/mcg_{2}] [\!\matrix{C4/mcg_{2}\hfill\cr nn\hfill\cr}]
129 [D_{4h}^{7}] [P4/nmm] [\!\matrix{P4/n &2_{1}/m &2/m\cr &&{\hbox to 2pt{}}2_{1}/g\cr}] [C4/amm] [\!\matrix{C4amm\hfill\cr bb\hfill\cr}]
130 [D_{4h}^{8}] [P4/ncc] [\!\matrix{P4/n &2_{1}/c &2/c\cr &&{\hbox to 4pt{}}2_{1}/n\cr}] [C4/acc] [\!\matrix{C4/acc\hfill\cr bn\hfill\cr}]
131 [D_{4h}^{9}] [P4_{2}/mmc] [\!\matrix{P4_{2}/m2/m &2/c\hfill\cr &2_{1}/n\hfill\cr}] [C4_{2}/mcm] [\!\matrix{C4_{2}/mcm\hfill\cr /}nn\hfill\cr}]
132 [D_{4h}^{10}] [P4_{2}/mcm] [\!\matrix{P4_{2}/m2/c &2/m\hfill\cr &2_{1}/g\hfill\cr}] [C4_{2}/mmc] [\!\matrix{C4_{2}/mmc\hfill\cr /}nb\hfill\cr}]
133 [D_{4h}^{11}] [P4_{2}/nbc] [\!\matrix{P4_{2}/n\;2/b &2/c\hfill\cr &2_{1}/n\hfill\cr}] [C4_{2}/acg_{1}] [\!\matrix{C4_{2}/acg_{1}\hfill\cr /}bn\hfill\cr}]
134 [D_{4h}^{12}] [P4_{2}/nnm] [\!\matrix{P4_{2}/n\;2/n &2/m\hfill\cr &2_{1}/g\hfill\cr}] [C4_{2}/amg_{2}] [\!\matrix{C4_{2}/amg_{2}\hfill\cr /}bb\hfill\cr}]
135 [D_{4h}^{13}] [P4_{2}/mbc] [\!\matrix{P4_{2}/m2_{1}/b &2/c\hfill\cr &2_{1}/n\hfill\cr}] [C4_{2}/mcg_{1}] [\!\matrix{C4_{2}/mcg_{1}\hfill\cr /}nn\hfill\cr}]
136 [D_{4h}^{14}] [P4_{2}/mnm] [\!\matrix{P4_{2}/m2_{1}/n &2/m\hfill\cr &2_{1}/g\hfill\cr}] [C4_{2}/mmg_{2}] [\!\matrix{C4_{2}/mmg_{2}\hfill\cr /}nb\hfill\cr}]
137 [D_{4h}^{15}] [P4_{2}/nmc] [\!\matrix{P4_{2}/n &2_{1}/m\;\;\;\; 2/c\hfill\cr &/m}{\hbox to 3pt{}}2_{1}/n\hfill\cr}] [C4_{2}/acm] [\!\matrix{C4_{2}/acm\hfill\cr /}bn\hfill\cr}]
138 [D_{4h}^{16}] [P4_{2}/ncm] [\!\matrix{P4_{2}/n &2_{1}/c &2/m\hfill\cr &&2_{1}/g\hfill\cr}] [C4_{2}/amc] [\!\matrix{C4_{2}/amc\hfill\cr /}bb\hfill\cr}]
139 [D_{4h}^{17}] [I\;4/mmm] [\!\matrix{I\;4/m &2/m &2/m\hfill\cr 4_{2}/n &2_{1}/n &2_{1}/e\hfill\cr}] [F4/mmm] [\!\matrix{F4/mmm\hfill\cr 4_{2}/aeg_{2}\hfill\cr}]
140 [D_{4h}^{18}] [I4/mcm] [\!\matrix{I\;4/m &2/c &2/e\hfill\cr 4_{2}/n &2_{1}/b &2_{1}/m\hfill\cr}] [F4/mmc] [\!\matrix{F4/mec\hfill\cr 4_{2}/amg_{1}\hfill\cr}]
141 [D_{4h}^{19}] [I\;4_{1}/amd] [\!\matrix{I\;4_{1}/a &2/m &2/d\hfill\cr 4_{3}/b &2_{1}/n &2_{1}/d\hfill\cr}] [F4_{1}/ddm] [\!\matrix{F4_{1}/ddm\hfill\cr 4_{3}/ddg_{2}\hfill\cr}]
142 [D_{4h}^{20}] [I\;4_{1}/acd] [\!\matrix{I\;4_{1}/a &2/c &2/d\hfill\cr 4_{3}/b &2_{1}/b &2_{1}/d\hfill\cr}] [F4_{1}/ddc] [\!\matrix{F4_{1}/ddc\hfill\cr 4_{3}/ddg_{1}\hfill\cr}]

TRIGONAL SYSTEM

No. of space groupSchoen-flies symbolHermann-Mauguin symbols for standard cell P or RTriple cell H
ShortFullExtended
143 [C_{3}^{1}] P3     H3
144 [C_{3}^{2}] [P3_{1}]     [H3_{1}]
145 [C_{3}^{3}] [P3_{2}]     [H3_{2}]
146 [C_{3}^{4}] R3   [\!\matrix{R3\hfill\cr 3_{1,2}\hfill\cr}]  
147 [C_{3i}^{1}] [P\bar{3}]     [H\bar{3}]
148 [C_{3i}^{2}] [R\bar{3}]   [\!\matrix{R\bar{3}\hfill\cr 3_{1,2}\hfill\cr}]  
149 [D_{3}^{1}] P312   [\!\matrix{P312\hfill\cr 2_{1}\hfill\cr}] H321
150 [D_{3}^{2}] P321   [\!\matrix{P321\hfill\cr 2_{1}\hfill\cr}] H312
151 [D_{3}^{3}] [P3_{1}12]   [\!\matrix{P3_{1}12\hfill\cr 1}2_{1}\hfill\cr}] [H3_{1}21]
152 [D_{3}^{4}] [P3_{1}21]   [\!\matrix{P3_{1}21\hfill\cr }2_{1}\hfill\cr}] [H3_{1}12]
153 [D_{3}^{5}] [P3_{2}12]   [\!\matrix{P3_{2}12\hfill\cr 1}2_{1}\hfill\cr}] [H3_{2}21]
154 [D_{3}^{6}] [P3_{2}21]   [\!\matrix{P3_{2}21\hfill\cr }2_{1}\hfill\cr}] [H3_{2}12]
155 [D_{3}^{7}] R32   [\!\matrix{R3}2\hfill\cr 3_{1,2}2_{1}\hfill\cr}]  
156 [C_{3v}^{1}] P3m1   [\!\matrix{P3m1\hfill\cr b\hfill\cr}] H31m
157 [C_{3v}^{2}] P31m   [\!\matrix{P31m\hfill\cr a\hfill\cr}] H3m1
158 [C_{3v}^{3}] P3c1   [\!\matrix{P3c1\hfill\cr n\hfill\cr}] H31c
159 [C_{3v}^{4}] P31c   [\!\matrix{P31c\hfill\cr n\hfill\cr}] H3c1
160 [C_{3v}^{5}] R3m   [\!\matrix{R3} m\hfill\cr 3_{1,2} b\hfill\cr}]  
161 [C_{3v}^{6}] R3c   [\!\matrix{R3} c\hfill\cr 3_{1,2} n\hfill\cr}]  
162 [D_{3d}^{1}] [P\bar{3}1m] [P\bar{3}12/m] [\!\matrix{P\bar{3}12/m\hfill\cr 2_{1}/a\hfill\cr}] [H\bar{3}m1]
163 [D_{3d}^{2}] [P\bar{3}1c] [P\bar{3}12/c] [\!\matrix{P\bar{3}12/c\hfill\cr 2_{1}/n\hfill\cr}] [H\bar{3}c1]
164 [D_{3d}^{3}] [P\bar{3}m1] [P\bar{3}2/m1] [\!\matrix{P\bar{3}2/m1\hfill\cr 2_{1}/b\hfill\cr}] [H\bar{3}1m]
165 [D_{3d}^{4}] [P\bar{3}c1] [P\bar{3}2/c1] [\!\matrix{P\bar{3}2/c1\hfill\cr 2_{1}/n\hfill\cr}] [H\bar{3}1c]
166 [D_{3d}^{5}] [R\bar{3}m] [R\bar{3}2/m] [\!\matrix{R\bar{3}} 2/m\hfill\cr 3_{1,2} 2_{1}/b\cr}]  
167 [D_{3d}^{6}] [R\bar{3}c] [R\bar{3}2/c] [\!\matrix{R\bar{3}} 2/c\hfill\cr  3_{1,2} 2_{1}/n\hfill\cr}]  

HEXAGONAL SYSTEM

No. of space groupSchoen-flies symbolHermann–Mauguin symbols for standard cell PTriple cell H
ShortFullExtended
168 [C_{6}^{1}] P6     H6
169 [C_{6}^{2}] [P6_{1}]     [H6_{1}]
170 [C_{6}^{3}] [P6_{5}]     [H6_{5}]
171 [C_{6}^{4}] [P6_{2}]     [H6_{2}]
172 [C_{6}^{5}] [P6_{4}]     [H6_{4}]
173 [C_{6}^{6}] [P6_{3}]     [H6_{3}]
174 [C_{3h}^{1}] [P\bar{6}]     [H\bar{6}]
175 [C_{6h}^{1}] P6/m     H6/m
176 [C_{6h}^{2}] [P6_{3}/m]     [H6_{3}/m]
177 [D_{6}^{1}] P622   [\!\matrix{P622\hfill\cr 2_{1}2_{1}\hfill\cr}] H622
178 [D_{6}^{2}] [P6_{1}22]   [\!\matrix{P6_{1}22\hfill\cr }2_{1}2_{1}\hfill\cr}] [H6_{1}22]
179 [D_{6}^{3}] [P6_{5}22]   [\!\matrix{P6_{5}22\hfill\cr }2_{1} 2_{1}\hfill\cr}] [H6_{5}22]
180 [D_{6}^{4}] [P6_{2}22]   [\!\matrix{P6_{2}22\hfill\cr }2_{1}2_{1}\hfill\cr}] [H6_{2}22]
181 [D_{6}^{5}] [P6_{4}22]   [\!\matrix{P6_{4}22\hfill\cr }2_{1}2_{1}\hfill\cr}] [H6_{4}22]
182 [D_{6}^{6}] [P6_{3}22]   [\!\matrix{P6_{3}22\hfill\cr }2_{1}2_{1}\hfill\cr}] [H6_{3}22]
183 [C_{6v}^{1}] P6mm   [\!\matrix{P6mm\hfill\crb\; a\hfill\cr}] H6mm
184 [C_{6v}^{2}] P6cc   [\!\matrix{P6 c c\hfill\cr nn\hfill\cr}] H6cc
185 [C_{6v}^{3}] [P6_{3}cm]   [\!\matrix{P6_{3} c m\hfill\cr }na\hfill\cr}] [H6_{3}mc]
186 [C_{6v}^{4}] [P6_{3}mc]   [\!\matrix{P6_{3} m c\hfill\cr }b\;n\hfill\cr}] [H6_{3}cm]
187 [D_{3h}^{1}] [P\bar{6}m2]   [\!\matrix{P\bar{6}m 2\hfill\cr b\;2_{1}\hfill\cr}] [H\bar{6}2m]
188 [D_{3h}^{2}] [P\bar{6}c2]   [\!\matrix{P\bar{6}c 2\hfill\cr n2_{1}\hfill\cr}] [H\bar{6}2c]
189 [D_{3h}^{3}] [P\bar{6}2m]   [\!\matrix{P\bar{6} 2m\hfill\cr 2_{1}a\hfill\cr}] [H\bar{6}m2]
190 [D_{3h}^{4}] [P\bar{6}2c]   [\!\matrix{P\bar{6}2 c\hfill\cr }2_{1} n\hfill\cr}] [H\bar{6}c2]
191 [D_{6h}^{1}] [P6/mmm] [P6/m\; 2/m2/m] [\!\matrix{P6/m\hfill &2/m\hfill &2/m\hfill\cr &2_{1}/b\hfill &2_{1}/a\hfill\cr}] [H6/mmm]
192 [D_{6h}^{2}] [P6/mcc] [P6/m\;2/c\;2/c] [\!\matrix{P6/m \hfill&2/c\hfill &2/c\hfill\cr &2_{1}/n\hfill &2_{1}/n\hfill\cr}] [H6/mcc]
193 [D_{6h}^{3}] [P6_{3}/mcm] [P6_{3}/m\;2/c\;2/m] [\!\matrix{P6_{3}/m\hfill &2/c\hfill &2/m\hfill\cr &2_{1}/b\hfill &2_{1}/a\hfill\cr}] [H6_{3}/mmc]
194 [D_{6h}^{4}] [P6_{3}/mmc] [P6_{3}/m2/m2/c] [\!\matrix{P6_{3}/m &2/m &2/c\hfill\cr &2_{1}/b &2_{1}/n\hfill\cr}] [H6_{3}/mcm]

CUBIC SYSTEM

Note: The glide planes g, [g_{1}] and [g_{2}] have the glide components [g({1 \over 2}, {1 \over 2}, 0)], [g_{1}({1 \over 4}, {1 \over 4}, 0)] and [g_{2}({1 \over 4}, {1 \over 4}, {1 \over 2})].

No. of space groupSchoenflies symbolHermann–Mauguin symbols
ShortFullExtended§
195 [T^{1}] P23    
196 [T^{2}] F23   [\!\matrix{F23\hfill\cr 2\hfill\cr 2_{1}\hfill\cr 2_{1}\hfill\cr}]
197 [T^{3}] I23   [\!\matrix{I23\hfill\cr 2_{1}\hfill\cr}]
198 [T^{4}] [P2_{1}3]    
199 [T^{5}] [I2_{1}3]   [\!\matrix{I2_{1}3\hfill\cr 2\hfill\cr}]
200 [T_{h}^{1}] [Pm\bar{3}] [P2/m\;\bar{3}]  
201 [T_{h}^{2}] [Pn\bar{3}] [P2/n\;\bar{3}]  
202 [T_{h}^{3}] [Fm\bar{3}] [F/2m\;\bar{3}] [\!\matrix{F2/m\;\bar{3}\hfill\cr 2/n\hfill\cr 2_{1}/e\hfill\cr 2_{1}/e\hfill\cr}]
203 [T_{h}^{4}] [Fd\bar{3}] [F2/d\;\bar{3}] [\!\matrix{F2/d\;\bar{3}\hfill\cr 2/d\hfill\cr 2_{1}/d\hfill\cr 2_{1}/d\hfill\cr}]
204 [T_{h}^{5}] [Im\bar{3}] [I2/m\;\bar{3}] [\!\matrix{I2/m\;\bar{3}\hfill\cr 2_{1}/n\hfill\cr}]
205 [T_{h}^{6}] [Pa\bar{3}] [P2_{1}/a \bar{3}]  
206 [T_{h}^{7}] [Ia\bar{3}] [I2_{1}/a \bar{3}] [\!\matrix{I2_{1}/a\;\bar{3}\hfill\cr 2/b\hfill\cr}]
207 [O^{1}] P432   [\!\matrix{P4\quad\ 32\hfill\cr 2_{1}\hfill\cr}]
208 [O^{2}] [P4_{2}32]   [\!\matrix{P4_{2} 32\hfill\cr2_{1}\hfill\cr}]
209 [O^{3}] F432   [\!\matrix{F4 32\hfill\cr 4 \phantom {_2}2\hfill\cr 4_{2}2_{1}\hfill\cr 4_{2} 2_{1}\hfill\cr}]
210 [O^{4}] [F4_{1}32]   [\!\matrix{F4_{1} 32\hfill\cr 4_{1} {\phantom 3}2\hfill\cr 4_{3} {\phantom 3}2_{1}\hfill\cr 4_{3} {\phantom 3}2_{1}\hfill\cr}]
211 [O^{5}] I432   [\!\matrix{I4 32\hfill\cr 4_{2} 2_{1}\hfill\cr}]
212 [O^{6}] [P4_{3}32]   [\!\matrix{P4_{3} &32\hfill\cr &2_{1}\hfill\cr}]
213 [O^{7}] [P4_{1}32]   [\!\matrix{P4_{1}32\hfill\cr \phantom {P4_13}2_{1}\hfill\cr}]
214 [O^{8}] [I4_{1}32]   [\!\matrix{I4_{1} 32\hfill\cr 4_{3} {\phantom 3}2_{1}\hfill\cr}]
215 [T_{d}^{1}] [P\bar{4}3m]   [\!\matrix{P\bar{4}3m\hfill\cr 3}g\hfill\cr}]
216 [T_{d}^{2}] [F\bar{4}3m]   [\!\matrix{F\bar{4}3m\hfill\cr 3}g\hfill\cr 3}g_{2}\hfill\cr 3}g_{2}\hfill\cr}]
217 [T_{d}^{3}] [I\bar{4}3m]   [\!\matrix{I\bar{4}3m\hfill\cr 3}e\hfill\cr}]
218 [T_{d}^{4}] [P\bar{4}3n]   [\!\matrix{P\bar{4}3n\hfill\cr 3}c\hfill\cr}]
219 [T_{d}^{5}] [F\bar{4}3c]   [\!\matrix{F\bar{4}3n\hfill\cr 3}c\hfill\cr 3}g_{1}\hfill\cr 3}g_{1}\hfill\cr}]
220 [T_{d}^{6}] [I\bar{4}3d]   [\!\matrix{I\bar{4}3d\hfill\cr 3}d\hfill\cr}]
221 [O_{h}^{1}] [Pm\bar{3}m] [P4/m\;\bar{3}2/m] [\!\matrix{P4/m\;\bar{3}2/m\hfill\cr }2_{1}/g\hfill\cr}]
222 [O_{h}^{2}] [Pn\bar{3}n] [P4/n\;\bar{3}2/n] [\!\matrix{P4/n\;\bar{3}2/n\hfill\cr }2_{1}/c\hfill\cr}]
223 [O_{h}^{3}] [Pm\bar{3}n] [P4_{2}/m \bar{3}2/n] [\!\matrix{P4_{2}/m\bar{3}2/n\hfill\cr /m\bar{3}}2_{1}/c\hfill\cr}]
224 [O_{h}^{4}] [Pn\bar{3}m] [P4_{2}/n \bar{3}2/m] [\!\matrix{P4_{2}/n\bar{3}2/m\hfill\cr /n\bar{3}}2_{1}/g\hfill\cr}]
225 [O_{h}^{5}] [Fm\bar{3}m] [F4/m\;\bar{3}2/m] [\!\matrix{F4/m &\bar{3}2/m\hfill\cr 4/n &2/g\hfill\cr 4_{2}/e &2_{1}/g_{2}\hfill\cr 4_{2}/e &2_{1}/g_{2}\hfill\cr}]
226 [O_{h}^{6}] [Fm\bar{3}c] [F4/m\;\bar{3}2/c] [\!\matrix{F4/m \bar{3}2/n\hfill\cr 4/n 2/c\hfill\cr 4_{2}/e 2_{1}/g_{1}\hfill\cr 4_{2}/e 2_{1}/g_{1}\hfill\cr}]
227 [O_{h}^{7}] [Fd\bar{3}m] [F4_{1}/d \bar{3}2/m] [\!\matrix{F4_{1}/d \bar{3}2/m\hfill\cr 4_{1}/d 2/g\hfill\cr 4_{3}/d 2_{1}/g_{2}\hfill\cr 4_{3}/d 2_{1}/g_{2}\hfill\cr}]
228 [O_{h}^{8}] [Fd\bar{3}c] [F4_{1}/d \bar{3}2/c] [\!\matrix{F4_{1}/d \bar{3}2/n\hfill\cr 4_{1}/d 2/c\hfill\cr 4_{3}/d 2_{1}/g_{1}\hfill\cr 4_{3}/d 2_{1}/g_{1}\hfill\cr}]
229 [O_{h}^{9}] [Im\bar{3}m] [I4/m\;\bar{3}2/m] [\!\matrix{I4/m \bar{3}2/m\hfill\cr 4_{2}/n 2_{1}/e\hfill\cr}]
230 [O_{h}^{10}] [Ia\bar{3}d] [I4_{1}/a\;\bar{3}2/d] [\!\matrix{I4_{1}/a \bar{3}2/d\hfill\cr 4_{3}/b 2_{1}/d\cr}]
For the five space groups Aem2 (39), Aea2 (41), Cmce (64), Cmme (67) and Ccce (68), the `new' space-group symbols, containing the symbol `e' for the `double' glide plane, are given for all settings. These symbols were first introduced in the Fourth Edition of this volume (IT 1995); cf. Foreword to the Fourth Edition. For further explanations, see Section 1.3.2, Note (x)[link] and the space-group diagrams.
For space groups Cmca (64), Cmma (67) and Imma (74), the first lines of the extended symbols, as tabulated here, correspond with the symbols for the six settings in the diagrams of these space groups (Part 7). An alternative formulation which corresponds with the coordinate triplets is given in Section 4.3.3[link].
§Axes [3_{1}] and [3_{2}] parallel to axes 3 are not indicated in the extended symbols: cf. Chapter 4.1[link] . For the glide-plane symbol `e', see the Foreword to the Fourth Edition (IT 1995[link]) and Section 1.3.2, Note (x)[link] .

Additional complications arise from the presence of fractional translations due to glide planes in the primitive cell [groups [Pc] (7), [P2/c] (13), [P2_{1}/c] (14)], due to centred cells [[C2] (5), [Cm] (8), [C2/m] (12)], or due to both [[Cc] (9), [C2/c] (15)]. For these groups, three different choices of the two oblique axes are possible which are called `cell choices' 1, 2 and 3 (see Section 2.2.16[link] ). If this is combined with the three choices of the unique axis, [3 \times 3 = 9] symbols result. If we add the effect of the permutation of the two oblique axes (and simultaneously reversing the sense of the unique axis to keep the system right-handed, as in abc and [{\bf c}{\bar{\bf b}}{\bf a}]), we arrive at the [9 \times 2 = 18] symbols listed in Table 4.3.2.1[link] for each of the eight space groups mentioned above.

The space-group symbols [P2] (3), [P2_{1}] (4), [Pm] (6), [P2/m] (10) and [P2_{1}/m] (11) do not depend on the cell choice: in these cases, one line of six space-group symbols is sufficient.

For space groups with centred lattices (A, B, C, I), extended symbols are given; the `additional symmetry elements' (due to the centring) are printed in the half line below the space-group symbol.

The use of the present tabulation is illustrated by two examples, Pm, which does not depend on the cell choice, and [C2/c], which does.

Examples

  • (1) Pm (6)

    • (i) Unique axis b

      In the first column, headed by abc, one finds the full symbol P1m1. Interchanging the labels of the oblique axes a and c does not change this symbol, which is found again in the second column headed by [{\bf c}\bar{\underline{\bf b}}{\bf a}].

    • (ii) Unique axis c

      In the third column, headed by abc, one finds the symbol P11m. Again, this symbol is conserved in the interchange of the oblique axes a and b, as seen in the fourth column headed by [{\bf ba}\bar{\underline{\bf c}}].

      The same applies to the setting with unique axis a, columns five and six.

  • (2) [C2/c\ (15)]

    The short symbol [C2/c] is followed by three lines, corresponding to the cell choices 1, 2, 3. Each line contains six full space-group symbols.

    • (i) Unique axis b

      The column headed by abc contains the three symbols [C\;1\; 2/c\; 1], [A1\; 2/n\; 1] and [I1\; 2/a\; 1], equivalent to the short symbol [C2/c] and corresponding to the cell choices 1, 2, 3. In the half line below each symbol, the additional symmetry elements are indicated (extended symbol). If the oblique axes a and c are interchanged, the column under cba lists the symbols [A1\; 2/a\; 1], [C1\; 2/n\; 1] and [I1\; 2/c\; 1] for the three cell choices.

    • (ii) Unique axis c

      The column under abc contains the symbols [A112/a], [B112/n] and [I112/b], corresponding to the cell choices 1, 2 and 3. If the oblique axes a and b are interchanged, the column under [{\bf ba}\underline{\bar{{\bf c}}}] applies.

      Similar considerations apply to the a-axis setting.

4.3.2.2. Transformation of space-group symbols

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How does a monoclinic space-group symbol transform for the various settings of the same unit cell? This can be easily recognized with the help of the headline of Table 4.3.2.1,[link] completed to the following scheme: [\matrix{{\bf a}{\underline{\bf b}}{\bf c} &{\bf c}\bar{\underline{\bf b}}{\bf a} &{\bf cab} &{\bf ac}\bar{\bf b} &{\bf bca} &{\bar{\bf b}\bf{ac}} &\hbox{Unique axis}\ b\cr {\bf bca} &{\bf a}{\bar{\bf c}}{\bf b} &{\bf ab}{\underline{\bf c}} &{\bf ba}\bar{\underline{\bf c}} &{\bf cab} &{\bar{{\bf c}}\bf ba} &\hbox{Unique axis}\ c\cr {\bf cab} &{\bf b}{\bar{\bf a}}{\bf c} &{\bf bca} &{\bf cb}\bar{\bf a} &{\underline{\bf a}}{\bf bc} &\bar{\underline{\bf a}}{\bf cb} &\hbox{ Unique axis}\ a.\cr}] The use of this three-line scheme is illustrated by the following examples.

Examples

  • (1) [C2/c] (15, unique axis b, cell choice 1)

    Extended symbol: [\matrix{\noalign{\vskip 12pt}C1\ 2/c\ 1.\hfill\cr 2_{1}/n\ \hfill\cr}]

    Consider the setting cab, first line, third column. Compared to the initial setting abc, it contains the `unique axis b' in the third place and, consequently, must be identified with the setting abc, unique axis c, in the third column, for which in Table 4.3.2.1[link] the new symbol for cell choice 1 is listed as [\matrix{\noalign{\vskip 12pt}A11\ 2/a\hfill\cr 2_{1}/n.\hfill\cr}]

  • (2) [C2/c] (15, unique axis b, cell choice 3)

    Extended symbol: [\matrix{\noalign{\vskip 12pt}I1\ 2/a\ \ 1.\hfill\cr 2_{1}/c \hfill\cr}]

    Consider the setting [{\bar{\bf b}}{\bf ac}] in the first line, sixth column. It contains the `unique axis b' in the first place and thus must be identified with the setting [\bar{\underline{\bf a}}{\bf cb}], unique axis a, in the sixth column. From Table 4.3.2.1,[link] the appropriate space-group symbol for cell choice 3 is found as [\matrix{\noalign{\vskip 12pt}I\ 2/b\ 11.\hfill\cr 2_{1}/ c\ \hfill\cr}]

4.3.2.3. Group–subgroup relations

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It is easy to read all monoclinic maximal t and k subgroups of types I and IIa directly from the extended full symbols of a space group. Maximal subgroups of types IIb and IIc cannot be recognized by simple inspection of the synoptic Table 4.3.2.1[link]

Example: [\matrix{\noalign{\vskip 12pt}B\ 2/b\ 11\hfill\cr 2_{1}/n\ \hfill\cr}] (15, unique axis a)

The t subgroups of index [2] (type I) are B211(C2); Bb11(Cc); [B\bar{1}(P\bar{1})].

The k subgroups of index [2] (type IIa) are P2[/]b11(P2[/]c): [P2_{1}/b 11]([P2_{1}/c]); P2[/]n11(P2[/]c); [P2_{1}/n 11]([P2_{1}/c]).

Some subgroups of index [4] (not maximal) are P211(P2); [P2_{1} 11]([P2_{1}]); Pb11(Pc); Pn11(Pc); [P\bar{1}]; B1(P1).

References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]








































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