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

International Tables for Crystallography (2006). Vol. A, ch. 4.3, pp. 62-76
doi: 10.1107/97809553602060000509

Chapter 4.3. Symbols for space groups

E. F. Bertauta

aLaboratoire de Cristallographie, CNRS, Grenoble, France

Chapters 4.1[link] , 4.2[link] and 4.3 contain extensive explanations and tabulations of the various types of space-group symbols. Chapter 4.3 contains for each crystal system extensive tables of the short, full and extended space-group symbols for various settings and cells. The explanatory text treats, system by system, the `additional symmetry elements' of the space groups. In addition, many examples of those group–subgroup relations are presented which can be read directly from the extended space-group symbols.

4.3.1. Triclinic system

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There are only two triclinic space groups, [P1\;(1)] and [P\bar{1}\;(2)]. P1 is quite outstanding because all its subgroups are also P1. They are listed in Table 13.2.2.1[link] for indices up to [7]. [P\bar{1}] has subgroups [P\bar{1}], isomorphic, and P1, non-isomorphic.

In the triclinic system, a primitive unit cell can always be selected. In some cases, however, it may be advantageous to select a larger cell, with A, B, C, I or F centring.

The two types of reduced bases (reduced cells) are discussed in Section 9.2.2.[link]

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).

4.3.3. Orthorhombic system

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4.3.3.1. Historical note and arrangement of the tables

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The synoptic table of IT (1935)[link] contained space-group symbols for the six orthorhombic `settings', corresponding to the six permutations of the basis vectors a, b, c. In IT (1952),[link] left-handed systems like [\bar{\bf c}\bf{ba}] were changed to right-handed systems by reversing the orientation of the c axis, as in [{\bf cba}]. Note that reversal of two axes does not change the handedness of a coordinate system, so that the settings [{\bar{\bf c}}{\bf ba}], [{\bf c}{\bar{\bf b}}{\bf a}], [{\bf cb}{\bar{\bf a}}] and [{\bar{\bf c}}{\bar{\bf b}}{\bar{\bf a}}] are equivalent in this respect. The tabulation thus deals with the [6 \times 4 = 24] possible right-handed settings. For further details see Section 2.2.6.4.[link]

An important innovation of IT (1952)[link] was the introduction of extended symbols for the centred groups A, B, C, I, F. These symbols are systematically developed in Table 4.3.2.1.[link] Settings which permute the two axes a and b are listed side by side so that the two C settings appear together, followed by the two A and the two B settings.

In crystal classes mm2 and 222, the last symmetry element is the product of the first two and thus is not independent. It was omitted in the short Hermann–Mauguin symbols of IT (1935)[link] for all space groups of class mm2, but was restored in IT (1952)[link]. In space groups of class 222, the last symmetry element cannot be omitted (see examples below).

For the new `double' glide plane symbol `e', see the Foreword to the Fourth Edition (IT 1995)[link] and Section 1.3.2, Note (x)[link] .

4.3.3.2. Group–subgroup relations

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The present section emphasizes the use of the extended and full symbols for the derivation of maximal subgroups of types I and IIa; maximal orthorhombic subgroups of types IIb and IIc cannot be recognized by inspection of the synoptic Table 4.3.2.1[link].

4.3.3.2.1. Maximal non-isomorphic k subgroups of type IIa (decentred)

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  • (i) Extended symbols of centred groups A, B, C, I

    By convention, the second line of the extended space-group symbol is the result of the multiplication of the first line by the centring translation (cf. Table 4.1.2.3[link] ). As a consequence, the product of any two terms in one line is equal to the product of the corresponding two terms in the other line.

    • (a) Class 222

      The extended symbol of I222 (23) is [\matrix{\noalign{\vskip 13pt}I 2\ 2\ 2{\hbox{;}}\hfill\cr 2_{1} 2_{1} 2_{1}\hfill\cr}] the twofold axes intersect and one obtains [2_{x}\;{\times\; 2_{y} = 2_{z} = 2_{1x}}\; {\times\; 2_{1y}}].

      Maximal k subgroups are P222 and [P2_{1}2_{1}2] (plus permutations) but not [P2_{1}2_{1}2_{1}].

      The extended symbol of [I2_{1}2_{1}2_{1}\;(24)] is [\matrix{\noalign{\vskip 13pt}I 2_{1} 2_{1} 2_{1}\hfill\cr 222\hfill\cr}], where one obtains [2_{1x} \times 2_{1y} = 2_{1z} = 2_{x} \times 2_{y}]; the twofold axes do not intersect. Thus, maximal non-isomorphic k subgroups are [P2_{1}2_{1}2_{1}] and [P222_{1}] (plus permutations), but not P222.

    • (b) Class mm2

      The extended symbol of Aea2 (41) is [\matrix{\noalign{\vskip 12pt}Aba2{\hbox{;}}\hfill\cr cn2_{1}\hfill\cr}] the following relations hold: [b \times a = 2 = c \times n] and [b \times n = 2_{1} = c \times a].

      Maximal k subgroups are Pba2; Pcn2 (Pnc2); [Pbn2_{1}] ([Pna2_{1}]); [Pca2_{1}].

    • (c) Class mmm

      By convention, the first line of the extended symbol contains those symmetry elements for which the coordinate triplets are explicitly printed under Positions. From the two-line symbols, as defined in the example below, one reads not only the eight maximal k subgroups P of class mmm but also the location of their centres of symmetry, by applying the following rules:

      If in the symbol of the P subgroup the number of symmetry planes, chosen from the first line of the extended symbol, is odd (three or one), the symmetry centre is at 0, 0, 0; if it is even (two or zero), the symmetry centre is at [{1 \over 4},{1 \over 4},0] for the subgroups of C groups and at [{1 \over 4},{1 \over 4},{1 \over 4}] for the subgroups of I groups (Bertaut, 1976[link]).

    Examples

    • (1) According to these rules, the extended symbol of Cmce (64) is [\!\matrix{\noalign{\vskip 13pt}C m c b\hfill\cr b n a\hfill\cr}] (see above). The four k subgroups with symmetry centres at 0, 0, 0 are Pmcb (Pbam); Pmna; Pbca; Pbnb (Pccn); those with symmetry centres at [{1 \over 4},{1 \over 4},0] are Pbna (Pbcn); Pmca (Pbcm); Pmnb (Pnma); Pbcb (Pcca). These rules can easily be transposed to other settings.

    • (2) The extended symbol of Ibam (72) is [\matrix{\noalign{\vskip 12pt}I b a m\hfill\cr c c n\hfill\cr}]. The four subgroups with symmetry centre at 0, 0, 0 are Pbam; Pbcn; Pcan (Pbcn); Pccm;

      those with symmetry centre at [{1 \over 4},{1 \over 4},{1 \over 4}] are Pccn; Pcam (Pbcm); Pbcm; Pban.

  • (ii) Extended symbols of F-centred space groups

    Maximal k subgroups of the groups F222, Fmm2 and Fmmm are C, A and B groups. The corresponding centring translations are [w = t({1 \over 2},{1 \over 2},0), u = t(0,{1 \over 2},{1 \over 2})] and [v = w \times u = t({1 \over 2},0,{1 \over 2})].

    The (four-line) extended symbols of these groups can be obtained from the following scheme: [\halign to \hsize{\hskip0pc# &$#$\hfill\hskip2.7pc &$#$\hfill\hskip2.7pc &$#$\hfill\hskip2.7pc &$#$\hfill\cr &&F222\ (22) &Fmm2\ (42) &Fmmm\ (69)\cr \noalign{\vskip3pt}\noalign{\hrule}\cr &1 &222 &mm2 &mmm\cr &w &2_{1}2_{1}2^{w} &ba2^{w} &ban\cr &u &2^{u}2_{1}^{v}2_{1} &nc2_{1} &ncb\cr &v &2_{1}^{u}2^{v}2_{1}^{w} &cn2_{1}^{w} &cna\cr\noalign{\vskip 3pt}\noalign{\hrule}\cr}] The second, third, and fourth lines are the result of the multiplication of the first line by the centring translations w, u and v, respectively.

    The following abbreviations are used: [2_{z}^{w} = w \times 2_{z}{\hbox{;}} \quad 2_{1z}^{w} = w \times 2_{1z}{\hbox{;}}\ etc.] For the location of the symmetry elements in the above scheme, see Table 4.1.2.3[link] . In Table 4.3.2.1[link], the centring translations and the superscripts u, v, w have been omitted. The first two lines of the scheme represent the extended symbols of C222, Cmm2 and Cmmm. An interchange of the symmetry elements in the first two lines does not change the group. To obtain further maximal C subgroups, one has to replace symmetry elements of the first line by corresponding elements of the third or fourth line. Note that the symbol `e' is not used in the four-line symbols for Fmm2 and Fmmm in order to keep the above scheme transparent.

    Examples

    • (1) F222 (22). In the first line replace [2_{x}] by [2_{x}^{u}] (third line, same column) and keep [2_{y}]. Complete the first line by the product [2_{x}^{u} \times 2_{y} = 2_{1z}] and obtain the maximal C subgroup [C2^{u}22_{1}].

      Similarly, in the first line keep [2_{x}] and replace [2_{y}] with [2_{y}^{v}] (fourth line, same column). Complete the first line by the product [2_{x} \times 2_{y}^{v} = 2_{1z}] and obtain the maximal C subgroup [C22^{v}2_{1}].

      Finally, replace [2_{x}] and [2_{y}] by [2_{x}^{u}] and [2_{y}^{v}] and form the product [2_{x}^{u} \times 2_{y}^{v} = 2_{z}^{w}], to obtain the maximal C subgroup [C2^{u}2^{v}2^{w}] (where [2^{w}] can be replaced by 2). Note that C222 and [C2^{u}2^{v}2] are two different subgroups, as are [C2^{u}22_{1}] and [C22^{v}2_{1}].

    • (2) Fmm2 (42). A similar procedure leads to the four maximal k subgroups [Cmm2{\hbox{;}}\; Cmc2_{1}{\hbox{;}}\; Ccm2_{1}^{w}\ \hbox{(}Cmc2_{1}\hbox{)}]; and Ccc2.

    • (3) Fmmm (69). One finds successively the eight maximal k subgroups Cmmm; Cmma; Cmcm; Ccmm (Cmcm); Cmca; Ccma (Cmca); Cccm; and Ccca.

    Maximal A- and B-centred subgroups can be obtained from the C subgroups by simple symmetry arguments.

    In space groups Fdd2 (43) and Fddd (70), the nature of the d planes is not altered by the translations of the F lattice; for this reason, a two-line symbol for Fdd2 and a one-line symbol for Fddd are sufficient. There exist no maximal non-isomorphic k subgroups for these two groups.

4.3.3.2.2. Maximal t subgroups of type I

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  • (i) Orthorhombic subgroups

    The standard full symbol of a P group of class mmm indicates all the symmetry elements, so that maximal t subgroups can be read at once.

    Example

    [P\;2_{1}/m2/m2/a] (51) has the following four t subgroups: [P2_{1}22] [(P222_{1})]; [Pmm2]; [P2_{1}ma] [(Pmc2_{1})]; Pm2a (Pma2).

    From the standard full symbol of an I group of class mmm, the t subgroup of class 222 is read directly. It is either I222 [for Immm (71) and Ibam (72)] or [I2_{1}2_{1}2_{1}] [for Ibca (73) and Imma (74)]. Use of the two-line symbols results in three maximal t subgroups of class mm2.

    Example

    [\matrix{\noalign{\vskip 13pt}I b a m\hfill\cr c c n\hfill\cr}] (72) has the following three maximal t subgroups of class mm2: [Iba2]; [Ib2_{1}m] [(Ima2)]; [I2_{1}am] [(Ima2)].

    From the standard full symbol of a C group of class mmm, one immediately reads the maximal t subgroup of class 222, which is either [C222_{1}] [for Cmcm (63) and Cmce (64)] or C222 (for all other cases). For the three maximal t subgroups of class mm2, the two-line symbols are used.

    Example

    [\matrix{\noalign{\vskip 12pt}C m c e \hfill\cr b n a\hfill\cr}] (64) has the following three maximal t subgroups of class mm2: [Cmc2_{1}]; [Cm2e] ([Aem2]); [C2ce] ([Aea2]).

    Finally, Fmmm (69) has maximal t subgroups F222 and Fmm2 (plus permutations), whereas Fddd (70) has F222 and Fdd2 (plus permutations).

  • (ii) Monoclinic subgroups

    These subgroups are obtained by substituting the symbol `l' in two of the three positions. Non-standard centred cells are reduced to primitive cells.

    Examples

    • (1) [C222_{1}] (20) has the maximal t subgroups C211 (C2), C121 (C2) and [C112_{1}]. The last one reduces to [P112_{1}] ([P2_{1}]).

    • (2) Ama2 (40) has the maximal t subgroups Am11, reducible to Pm, A1a1 (Cc) and A112 (C2).

    • (3) Pnma (62) has the standard full symbol [P2_{1}/n2_{1}/m2_{1}/a], from which the maximal t subgroups [P2_{1}/n11] [(P2_{1}/c)], [P12_{1}/m1] [(P2_{1}/m)] and [P112_{1}/a] [(P2_{1}/c)] are obtained.

    • (4) Fddd (70) has the maximal t subgroups [F2/d11], [F12/d1] and [F112/d], each one reducible to [C2/c].

4.3.4. Tetragonal system

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4.3.4.1. Historical note and arrangement of the tables

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In the 1935 edition of International Tables[link], for each tetragonal P and I space group an additional C-cell and F-cell description was given. In the corresponding space-group symbols, secondary and tertiary symmetry elements were simply interchanged. Coordinate triplets for these larger cells were not printed, except for the space groups of class [\bar{4}m2]. In IT (1952)[link], the C and F cells were dropped from the space-group tables but kept in the comparative tables.

In the present edition, the C and F cells reappear in the sub- and supergroup tabulations of Part 7[link] , as well as in the synoptic Table 4.3.2.1,[link] where short and extended (two-line) symbols are given for P and C cells, as well as for I and F cells.

4.3.4.2. Relations between symmetry elements

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In the crystal classes 42(2), 4m(m), [\bar{4}2\hbox{(}m\hbox{)}] or [\bar{4}m\hbox{(}2\hbox{)}], [4/m\;2/m\;\hbox{(}2/m\hbox{)}], where the tertiary symmetry elements are between parentheses, one finds [4 \times m = (m) = \bar{4} \times 2; \ 4 \times 2 = (2) = \bar{4} \times m.] Analogous relations hold for the space groups. In order to have the symmetry direction of the tertiary symmetry elements along [[1\bar{1}0]] (cf. Table 2.2.4.1[link] ), one has to choose the primary and secondary symmetry elements in the product rule along [001] and [010].

Example

In [P4_{1}2(2)\ (91)], one has [4_{1} \times 2 = (2)] so that [P4_{1}2] would be the short symbol. In fact, in IT (1935)[link], the tertiary symmetry element was suppressed for all groups of class 422, but re-established in IT (1952)[link], the main reason being the generation of the fourfold rotation as the product of the secondary and tertiary symmetry operations: [4 = (m) \times m] etc.

4.3.4.3. Additional symmetry elements

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As a result of periodicity, in all space groups of classes 422, [\bar{4}m2] and [4/m\; 2/m\; 2/m], the two tertiary diagonal axes 2, along [[1\bar{1}0]] and [110], alternate with axes [2_{1}], the screw component being [{1 \over 2},\mp] [{1 \over 2}], 0 (cf. Table 4.1.2.2[link] ).

Likewise, tertiary diagonal mirrors m in x, x, z and [x,\bar{x},z] in space groups of classes 4mm, [\bar{4}2m] and [4/m\; 2/m\; 2/m] alternate with glide planes called g,1 the glide components being [{1 \over 2}], [\pm {1 \over 2}], 0. The same glide components produce also an alternation of diagonal glide planes c and n (cf. Table 4.1.2.2[link] ).

4.3.4.4. Multiple cells

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The transformations from the P to the two C cells, or from the I to the two F cells, are [\eqalign{C_1 \hbox{ or } F_1\hbox{: }(\rm i)\quad {\bf a}' &= {\bf a} - {\bf b},\quad{\bf b}' = \phantom{-} {\bf a} + {\bf b},\quad{\bf c}' = {\bf c}\cr C_2 \hbox{ or } F_2\hbox{: }(\rm ii)\quad {\bf a}' &= {\bf a} + {\bf b},\quad{\bf b}' = - {\bf a} + {\bf b},\quad{\bf c}' = {\bf c}}] (cf. Fig. 5.1.3.5[link] ). The secondary and tertiary symmetry directions are interchanged in the double cells. It is important to know how primary, secondary and tertiary symmetry elements change in the new cells [{\bf a}',{\bf b}',{\bf c}'].

  • (i) Primary symmetry elements

    In P groups, only two kinds of planes, m and n, occur perpendicular to the fourfold axis: a and b planes are forbidden. A plane m in the P cell corresponds to a plane in the C cell which has the character of both a mirror plane m and a glide plane n. This is due to the centring translation [{1 \over 2},{1 \over 2},0] (cf. Chapter 4.1[link] ). Thus, the C-cell description shows2 that [P4/m..] (cell a, b, c) has two maximal k subgroups of index [2], [P4/m..] and [P4/n..] (cells [{\bf a}',{\bf b}',{\bf c}']), originating from the decentring of the C cell. The same reasoning is valid for [P4_{2}/m..] .

    A glide plane n in the P cell is associated with glide planes a and b in the C cell. Since such planes do not exist in tetragonal P groups, the C cell cannot be decentred, i.e. [P4/n..] and [P4_{2}/n..] have no k subgroups of index [2] and cells [{\bf a}',{\bf b}',{\bf c}'].

    Glide planes a perpendicular to c only occur in [I4_{1}/a\;(88)] and groups containing [I4_{1}/a] [[I4_{1}/amd\ (141)] and [I4_{1}/acd\ (142)]]; they are associated with d planes in the F cell. These groups cannot be decentred, i.e. they have no P subgroups at all.

  • (ii) Secondary symmetry elements

    In the tetragonal space-group symbols, one finds two kinds of secondary symmetry elements:

    • (1) 2, m, c without glide components in the ab plane occur in P and I groups. They transform to tertiary symmetry elements 2, m, c in the C or F cells, from which k subgroups can be obtained by decentring.

    • (2) [2_{1},b,n] with glide components [{1 \over 2},0,0{\hbox{;}}\; 0,{1 \over 2},0{\hbox{;}}\; {1 \over 2},{1 \over 2},0] in the ab plane occur only in P groups. In the C cell, they become tertiary symmetry elements with glide components [{1 \over 4},\! - {1 \over 4}, 0]; [{1 \over 4},{1 \over 4},0]; [{1 \over 4}], [{1 \over 4}], [{1 \over 2}]. One has the following correspondence between P- and C-cell symbols: [\eqalign{&P . 2_{1} . = C .. 2_{1}\cr &P . b . = C .. g_{1} \hbox{ with } g({\textstyle{1 \over 4}}, {\textstyle{1 \over 4}}, 0)\hbox{ in } \quad x, x - {\textstyle{1 \over 4}}, z\cr &P . n . = C .. g_{2} \hbox{ with } g({\textstyle{1 \over 4}}, {\textstyle{1 \over 4}}, {\textstyle{1 \over 2}})\hbox{ in } \quad x, x - {\textstyle{1 \over 4}}, z,}] where [(g_{1})^{2}] and [(g_{2})^{2}] are the centring translations [{1 \over 2},{1 \over 2},0] and [{1 \over 2},{1 \over 2},1]. Thus, the C cell cannot be decentred, i.e. tetragonal P groups having secondary symmetry elements [2_{1}], b or n cannot have klassengleiche P subgroups of index [2] and cells [{\bf a}',{\bf b}',{\bf c}'].

  • (iii) Tertiary symmetry elements

    Tertiary symmetry elements 2, m, c in P groups transform to secondary symmetry elements in the C cell, from which k subgroups can easily be deduced [(\rightarrow)]: [\matrix{P..m\ =&C.m.\longrightarrow& P.m.{\hbox to 3pt{}}\cr g&b{\hbox to 12pt{}}& P.b.{\hbox to 3pt{}}\cr P..c\ =&C.c.\longrightarrow &P.c.{\hbox to 4pt{}}\cr n&n\qquad&P.n.{\hbox to 4pt{}}\cr P..2\ =&C.2.\longrightarrow & P.2.{\hbox to 4pt{}}\cr {\hbox to 6pt{}}2_{1}&2_{1}{\hbox to 3pt{}}&P.2_{1}.}] Decentring leads in each case to two P subgroups (cell [{\bf a}',{\bf b}',{\bf c}']), when allowed by (i) and (ii).

    In I groups, 2, m and d occur as tertiary symmetry elements. They are transformed to secondary symmetry elements in the F cells. I groups with tertiary d glides cannot be decentred to P groups, whereas I groups with diagonal symmetry elements 2 and m have maximal P subgroups, due to decentring.

4.3.4.5. Group–subgroup relations

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Examples are given for maximal k subgroups of P groups (i), of I groups (ii), and for maximal tetragonal, orthorhombic and monoclinic t subgroups.

4.3.4.5.1. Maximal k subgroups

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  • (i) Subgroups of P groups

    The discussion is limited to maximal P subgroups, obtained by decentring the larger C cell (cf. Section 4.3.4.4[link] Multiple cells).

    • Classes [\bar{4}], 4 and 422

      Examples

      • (1) Space groups [P\bar{4}] (81) and [P4_{p}\ (p = 0, 1, 2, 3)] (75–78) have isomorphic k subgroups of index [2], cell [{\bf a}',{\bf b}',{\bf c}'].

      • (2) Space groups [P4_{p}22\ (p = 0, 1, 2, 3)] (89, 91, 93, 95) have the extended C-cell symbol [\matrix{\noalign{\vskip 13pt}C4_{p} 2\ 2\hfill\cr }2_{1}\hfill\cr}], from which one deduces two k subgroups, [P4_{p}22] (isomorphic, type [{\bf IIc}]) and [P4_{p}2_{1}2] (non-isomorphic, type IIb), cell a′, b′, c′.

      • (3) Space groups [P4_{p}2_{1}2] (90, 92, 94, 96) have no k subgroups of index [2], cell [{\bf a}',{\bf b}',{\bf c}'].

    • Classes [\bar{4}m2], 4mm, 4/m, and 4/mmm

      Examples

      • (1) [P\bar{4}c2\ (116)] has the C-cell symbol [\matrix{\noalign{\vskip 13pt}C\bar{4}2\ c\hfill\cr }2_{1}\hfill\cr}], wherefrom one deduces two k subgroups, [P\bar{4}2c] and [P\bar{4}2_{1}c], cell a′, b′, c′.

      • (2) [P4_{2}mc\ (105)] has the C-cell symbol [\matrix{\noalign{\vskip 13pt}C4_{2}cm\hfill\cr }n\hfill\cr}], from which the k subgroups [P4_{2}cm] (101) and [P4_{2}nm\ (102)], cell [{\bf a}', {\bf b}', {\bf c}'], are obtained.

      • (3) [P4_{2}/mcm\ (132)] has the extended C symbol [\matrix{\noalign{\vskip 13pt}C4_{2}/mmc\hfill\cr / }n\; b\hfill\cr}], wherefrom one reads the following k subgroups of index [2], cell [{\bf a}', {\bf b}', {\bf c}']: [P4_{2}/mmc], [P4_{2}/mbc], [P4_{2}/nmc], [P4_{2}/nbc].

      • (4) [P4/nbm\ (125)] has the extended C symbol [\matrix{\noalign{\vskip 13pt}C4/amg_{1}\hfill\cr  bb\hfill\cr}] and has no k subgroups of index [2], as explained above in Section 4.3.4.4[link] .

  • (ii) Subgroups of I groups

    Note that I groups with a glides perpendicular to [001] or with diagonal d planes cannot be decentred (cf. above). The discussion is limited to P subgroups of index [2], obtained by decentring the I cell. These subgroups are easily read from the two-line symbols of the I groups in Table 4.3.2.1[link].

    Examples

    • (1) I4cm (108) has the extended symbol [\matrix{\noalign{\vskip 13pt}I 4 c e\hfill\cr4_{2} b m\hfill\cr}]. The multiplication rules [4 \times b = m = 4_{2} \times c] give rise to the maximal k subgroups: P4cc, [P4_{2}bc], P4bm, [P4_{2}cm].

      Similarly, I4mm (107) has the P subgroups P4mm, [P4_{2}nm], P4nc, [P4_{2}mc], i.e. I4mm and I4cm have all P groups of class 4mm as maximal k subgroups.

    • (2) [I4/mcm\ (140)] has the extended symbol [\matrix{\noalign{\vskip 13pt}I 4/m \;ce\hfill\cr4_{2}/n b m\hfill\cr}]. One obtains the subgroups of example (1) with an additional m or n plane perpendicular to c.

      As in example (1), [I4/mcm] (140) and [I4/mmm] (139) have all P groups of class [4/mmm] as maximal k subgroups.

4.3.4.5.2. Maximal t subgroups

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  • (i) Tetragonal subgroups

    The class [4/mmm] contains the classes [4/m, 422, 4mm] and [\bar{4}2m]. Maximal t subgroups belonging to these classes are read directly from the standard full symbol.

    Examples

    • (1) [P4_{2}/m\;bc\ (135)] has the full symbol [P4_{2}/m\;2_{1}/b\;2/c] and the tetragonal maximal t subgroups: [P4_{2}/m], [P4_{2}2_{1}2], [P4_{2}bc], [P\bar{4}2_{1}c], [P\bar{4}b2].

    • (2) [I4/m\;cm\ (140)] has the extended full symbol [{\matrix{\noalign{\vskip 13pt}I 4/m{\hbox to 1pt{}}2/c{\hbox to 5pt{}}2/e\hfill\cr4_{2}/n2_{1}/b 2_{1}/m\hfill\cr}}] and the tetragonal maximal t subgroups [I4/m], I422, I4cm, [I\bar{4}2m], [I\bar{4}c2]. Note that the t subgroups of class [\bar{4}m2] always exist in pairs.

  • (ii) Orthorhombic subgroups

    In the orthorhombic subgroups, the symmetry elements belonging to directions [100] and [010] are the same, except that a glide plane b perpendicular to [100] is accompanied by a glide plane a perpendicular to [010].

    Examples

    • (1) [P4_{2}/m bc\ (135)]. From the full symbol, the first maximal t subgroup is found to be [P2_{1}/b\;2_{1}/a \;2/m] (Pbam). The C-cell symbol is [C4_{2}/m\;cg_{1}] and gives rise to the second maximal orthorhombic t subgroup Cccm, cell [{\bf a}',{\bf b}',{\bf c}'].

    • (2) [I4/m\;cm\ (140)]. Similarly, the first orthorhombic maximal t subgroup is [\matrix{\noalign{\vskip 13pt}I c c m\hfill\crb a n\hfill\cr}] (Ibam); the second maximal orthorhombic t subgroup is obtained from the F-cell symbol as [\matrix{\noalign{\vskip 13pt}F c{\hbox to 3pt{}} c{\hbox to 3pt{}} m\hfill\crm m n\hfill\cr}] (Fmmm), cell a′, b′, c′.

    These examples show that P- and C-cell, as well as I- and F-cell descriptions of tetragonal groups have to be considered together.

  • (iii) Monoclinic subgroups

    Only space groups of classes 4, [\bar{4}] and [4/m] have maximal monoclinic t subgroups.

    Examples

    • (1) [P4_{1}\;(76)] has the subgroup [P112_{1}\ (P2_{1})]. The C-cell description does not add new features: [C112_{1}] is reducible to [P2_{1}].

    • (2) [I4_{1}/a\ (88)] has the subgroup [I112_{1}/a], equivalent to [I11 2/a\;(C2/c)]. The F-cell description yields the same subgroup [F11\;2/d], again reducible to [C2/c].

4.3.5. Trigonal and hexagonal systems

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The trigonal and hexagonal crystal systems are considered together, because they form the hexagonal `crystal family', as explained in Chapter 2.1[link] . Hexagonal lattices occur in both systems, whereas rhombohedral lattices occur only in the trigonal system.

4.3.5.1. Historical note

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The 1935[link] edition of International Tables contains the symbols C and H for the hexagonal lattice and R for the rhombohedral lattice. C recalls that the hexagonal lattice can be described by a double rectangular C-centred cell (orthohexagonal axes); H was used for a hexagonal triple cell (see below); R designates the rhombohedral lattice and is used for both the rhombohedral description (primitive cell) and the hexagonal description (triple cell).

In the 1952[link] edition the following changes took place (cf. pages x, 51 and 544 of IT 1952)[link]: The lattice symbol C was replaced by P for reasons of consistency; the H description was dropped. The symbol R was kept for both descriptions, rhombohedral and hexagonal. The tertiary symmetry element in the short Hermann–Mauguin symbols of class 622, which was omitted in IT (1935)[link], was re-established.

In the present volume, the use of P and R is the same as in IT (1952)[link]. The H cell, however, reappears in the sub- and supergroup data of Part 7[link] and in Table 4.3.2.1[link] of this section, where short symbols for the H description of trigonal and hexagonal space groups are given. The C cell reappears in the subgroup data for all trigonal and hexagonal space groups having symmetry elements orthogonal to the main axis.

4.3.5.2. Primitive cells

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The primitive cells of the hexagonal and the rhombohedral lattice, hP and hR, are defined in Table 2.1.2.1[link] In Part 7[link] , the `rhombohedral' description of the hR lattice is designated by `rhombohedral axes'; cf. Chapter 1.2[link] .

4.3.5.3. Multiple cells

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Multiple cells are frequently used to describe both the hexagonal and the rhombohedral lattice.

  • (i) The triple hexagonal R cell; cf. Chapters 1.2[link] and 2.1[link]

    When the lattice is rhombohedral hR (primitive cell a, b, c), the triple R cell [{\bf a}',{\bf b}',{\bf c}'] corresponds to the `hexagonal description' of the rhombohedral lattice. There are three right-handed obverse R cells: [\matrix{R_{1}: \quad {\bf a}' = {\bf a} - {\bf b}; &{\bf b}' = {\bf b} - {\bf c}; &{\bf c}' = {\bf a} + {\bf b} + {\bf c};\cr R_{2}: \quad {\bf a}' = {\bf b} - {\bf c}; &{\bf b}' = {\bf c} - {\bf a}; &{\bf c}' = {\bf a} + {\bf b} + {\bf c};\cr R_{3}: \quad {\bf a}' = {\bf c} - {\bf a}; &{\bf b}' = {\bf a} - {\bf b}; &{\bf c}' = {\bf a} + {\bf b} + {\bf c}.\cr}] Three further right-handed R cells are obtained by changing [{\bf a}'] and [{\bf b}'] to [- {\bf a}'] and [- {\bf b}'], i.e. by a [180^{\circ}] rotation around [{\bf c}']. These cells are reverse. The transformations between the triple R cells and the primitive rhombohedral cell are given in Table 5.1.3.1[link] and Fig. 5.1.3.6[link] .

    The obverse triple R cell has `centring points' at [0,0,0; \quad {\textstyle{2 \over 3}, {1 \over 3}, {1 \over 3}; \quad {1 \over 3}, {2 \over 3}, {2 \over 3}},] whereas the reverse R cell has `centring points' at [0,0,0; \quad {\textstyle{1 \over 3}, {2 \over 3}, {1 \over 3}; \quad {2 \over 3}, {1 \over 3}, {2 \over 3}}.] In the space-group tables of Part 7[link] , the obverse [R_{1}] cell is used, as illustrated in Fig. 2.2.6.9[link] . This `hexagonal description' is designated by `hexagonal axes'.

  • (ii) The triple rhombohedral D cell

    Parallel to the `hexagonal description of the rhombohedral lattice' there exists a `rhombohedral description of the hexagonal lattice'. Six right-handed rhombohedral cells (here denoted by D) with cell vectors [{\bf a}',{\bf b}',{\bf c}'] of equal lengths are obtained from the hexagonal P cell a, b, c by the following transformations and by cyclic permutations of [{\bf a}',{\bf b}',{\bf c}']: [\matrix{D_{1}: &{\bf a}' = \phantom{-}{\bf a} + {\bf c};\hfill &{\bf b}' = \phantom{-}{\bf b} + {\bf c};\hfill &{\bf c}' = - ({\bf a} + {\bf b}) + {\bf c}\hfill\cr D_{2}: &{\bf a}' = - {\bf a} + {\bf c}; &{\bf b}' = - {\bf b} + {\bf c}; &{\bf c}' = \phantom{-(}{\bf a} + {\bf b}\phantom{)} + {\bf c}.\hfill\cr}] The transformation matrices are listed in Table 5.1.3.1[link] . [D_{2}] follows from [D_{1}] by a [180^{\circ}] rotation around [111]. The D cells are triple rhombohedral cells with `centring' points at [0,0,0; \quad {\textstyle{1 \over 3}, {1 \over 3}, {1 \over 3}; \quad {2 \over 3}, {2 \over 3}, {2 \over 3}}.] The D cell, not used in practice and not considered explicitly in the present volume, is useful for a deeper understanding of the relations between hexagonal and rhombohedral lattices.

  • (iii) The triple hexagonal H cell; cf. Chapter 1.2[link]

    Generally, a hexagonal lattice hP is described by means of the smallest hexagonal P cell. An alternative description employs a larger hexagonal H-centred cell of three times the volume of the P cell; this cell was extensively used in IT (1935)[link], see Historical note [link] above.

    There are three right-handed orientations of the H cell (basis vectors [{\bf a}',{\bf b}',{\bf c}']) with respect to the basis vectors a, b, c of the P cell: [\matrix{H_{1}: &{\bf a}' = \phantom{2}{\bf a} - \phantom{2}{\bf b}{\hbox{;}} &{\bf b}' = \phantom{-2}{\bf a} + 2{\bf b}{\hbox{;}} &{\bf c}' = {\bf c}\hfill\cr H_{2}: \hfill&{\bf a}' = 2 {\bf a} + \phantom{2}{\bf b}{\hbox{;}} &{\bf b}' = - \phantom{2}{\bf a} + {\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}\hfill\cr H_{3}: &{\bf a}' = \phantom{2}{\bf a} + 2{\bf b}{\hbox{;}} &{\bf b}' = - 2 {\bf a} - {\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}.\hfill\cr}] The transformations are given in Table 5.1.3.1[link] and Fig. 5.1.3.8[link] . The new vectors [{\bf a}'] and [{\bf b}'] are rotated in the ab plane by [- 30^{\circ} (H_{1})], [+ 30^{\circ} (H_{2})], [+ 90^{\circ} (H_{3})] with respect to the old vectors a and b. Three further right-handed H cells are obtained by changing [{\bf a}'] and [{\bf b}'] to [- {\bf a}'] and [- {\bf b}'], i.e. by a rotation of [180^{\circ}] around [{\bf c}'].

    The H cell has `centring' points at [0,0,0; \quad {\textstyle{2 \over 3}, {1 \over 3}, 0; \quad {1 \over 3}, {2 \over 3}}, 0.] Secondary and tertiary symmetry elements of the P cell are interchanged in the H cell, and the general position in the H cell is easily obtained, as illustrated by the following example.

    Example

    The space-group symbol P3m1 in the P cell a, b, c becomes H31m in the H cell [{\bf a}',{\bf b}',{\bf c}']. To obtain the general position of H31m, consider the coordinate triplets of P31m and add the centring translations [0,0,0{\hbox{;}}\ {2 \over 3},{1 \over 3},0{\hbox{;}}\ {1 \over 3},{2 \over 3},0].

  • (iv) The double orthohexagonal C cell

    The C-centred cell which is defined by the so-called `orthohexagonal' vectors [{\bf a}'], [{\bf b}'], [{\bf c}'] has twice the volume of the P cell. There are six right-handed orientations of the C cell, which are [C_{1}], [C_{2}] and [C_{3}] plus three further ones obtained by changing [{\bf a}'] and [{\bf b}'] to [-{\bf a}'] and [-{\bf b}']: [\matrix{C_{1}: {\bf a}' = {\bf a}\hfill{\hbox{;}} &{\bf b}' = \phantom{-2}{\bf a} + 2\hfill{\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}\cr C_{2}: {\bf a}' = {\bf a} + {\bf b}{\hbox{;}} &{\bf b}' = -\phantom{2}{\bf a} + \phantom{2}{\bf b}{\hbox{;}}\hfill &{\bf c}' = {\bf c}\cr C_{3}: {\bf a}' = \hfill{\bf b}{\hbox{;}} &{\bf b}' = -2{\bf a} - \phantom{2}\hfill{\bf b}{\hbox{;}}\hfill &{\hbox to 2pt{}}{\bf c}' = {\bf c}.\cr}] Transformation matrices are given in Table 5.1.3.1[link] and illustrations in Fig. 5.1.3.7[link] . Here [{\bf b}'] is the long axis.

4.3.5.4. Relations between symmetry elements

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In the hexagonal crystal classes 62(2), 6m(m) and [\bar{6}2](m) or [\bar{6}m](2), where the tertiary symmetry element is between parentheses, the following products hold: [6 \times 2 = (2) = \bar{6} \times m;\quad 6 \times m = (m) = \bar{6} \times 2] or [6 \times 2 \times (2) = 6 \times m \times (m) = \bar{6} \times 2 \times (m) = \bar{6} \times m \times (2) = 1.] The same relations hold for the corresponding Hermann–Mauguin space-group symbols.

4.3.5.5. Additional symmetry elements

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Parallel axes 2 and [2_{1}] occur perpendicular to the principal symmetry axis. Examples are space groups R32 (155), P321 (150) and P312 (149), where the screw components are [{1 \over 2},{1 \over 2},0] (rhombohedral axes) or [{1 \over 2},0,0] (hexagonal axes) for R32; [{1 \over 2},0,0] for P321; and [{1 \over 2},1,0] for P312. Hexagonal examples are P622 (177) and [P\bar{6}2c] (190).

Likewise, mirror planes m parallel to the main symmetry axis alternate with glide planes, the glide components being perpendicular to the principal axis. Examples are P3m1 (156), P31m (157), R3m (160) and P6mm (183).

Glide planes c parallel to the main axis are interleaved by glide planes n. Examples are P3c1 (158), P31c (159), R3c (161, hexagonal axes), [P\bar{6}c2] (188). In R3c and [R\bar{3}c], the glide component [0,0,{1 \over 2}] for hexagonal axes becomes [{1 \over 2},{1 \over 2},{1 \over 2}] for rhombohedral axes, i.e. the c glide changes to an n glide. Thus, if the space group is referred to rhombohedral axes, diagonal n planes alternate with diagonal a, b or c planes (cf. Section 1.4.4[link] ).

In R space groups, all additional symmetry elements with glide and screw components have their origin in the action of an integral lattice translation. This is also true for the axes [3_{1}] and [3_{2}] which appear in all R space groups (cf. Table 4.1.2.2[link] ). For this reason, the `rhombohedral centring' R is not included in Table 4.1.2.3[link] , which contains only the centrings A, B, C, I, F.

4.3.5.6. Group–subgroup relations

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4.3.5.6.1. Maximal k subgroups

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Maximal k subgroups of index [3] are obtained by `decentring' the triple cells R (hexagonal description), D and H in the trigonal system, H in the hexagonal system. Any one of the three centring points may be taken as origin of the subgroup.

  • (i) Trigonal system

    Examples

    • (1) P3m1 (156) (cell a, b, c) is equivalent to H31m (a′, b′, c). Decentring of the H cell yields maximal non-isomorphic k subgroups of type P31m. Similarly, P31m (157) has maximal subgroups of type P3m1; thus, one can construct infinite chains of subgroup relations of index [3], tripling the cell at each step: [P3m1 \rightarrow P31m \rightarrow P3m1 \ldots]

    • (2) R3 (146), by decentring the triple hexagonal R cell [{\bf a}',{\bf b}',{\bf c}'], yields the subgroups P3, [P3_{1}] and [P3_{2}] of index [3].

    • (3) Likewise, decentring of the triple rhombohedral cells [D_{1}] and [D_{2}] gives rise, for each cell, to the rhombohedral subgroups of a trigonal P group, again of index [3].

      Combining (2) and (3), one may construct infinite chains of subgroup relations, tripling the cell at each step: [P3 \rightarrow R3 \rightarrow P3 \rightarrow R3 \ldots] These chains illustrate best the connections between rhombohedral and hexagonal lattices.

    • (4) Special care must be applied when secondary or tertiary symmetry elements are present. Combining (1), (2) and (3), one has for instance: [P31c \rightarrow R3c \rightarrow P3c1 \rightarrow P31c \rightarrow R3c \ldots]

    • (5) Rhombohedral subgroups, found by decentring the triple cells [D_{1}] and [D_{2}], are given under block IIb and are referred there to hexagonal axes, [{\bf a}',{\bf b}',{\bf c}] as listed below. Examples are space groups P3 (143) and [P\bar{3}1c] (163) [\matrix{{\bf a}' = \phantom{2}{\bf a} - {\bf b}, &{\bf b}' = \phantom{-}{\bf a} + 2{\bf b}, &{\bf c}' = 3{\bf c};\cr {\bf a}' = 2{\bf a} + {\bf b}, &{\bf b}' = -{\bf a} + \phantom{2}{\bf b}, &{\bf c}' = 3{\bf c}.\cr}]

  • (ii) Hexagonal system

    Examples

    • (1) [P\bar{6}2c] (190) is described as [H\bar{6}c2] in the triple cell [{\bf a}',{\bf b}',{\bf c}']; decentring yields the non-isomorphic subgroup [P\bar{6}c2].

    • (2) [P6/mcc] (192) (cell a, b, c) keeps the same symbol in the H cell and, consequently, gives rise to the maximal isomorphic subgroup [P6/mcc] with cell [{\bf a}',{\bf b}',{\bf c}']. An analogous result applies whenever secondary and tertiary symmetry elements in the Hermann–Mauguin symbol are the same and also to space groups of classes 6, [\bar{6}] and [6/m].

4.3.5.6.2. Maximal t subgroups

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Maximal t subgroups of index [2] are read directly from the full symbol of the space groups of classes 32, 3m, [\bar{3}m], 622, 6mm, [\bar{6}2m], [6/mmm].

Maximal t subgroups of index [3] follow from the third power of the main-axis operation. Here the C-cell description is valuable.

  • (i) Trigonal system

    • (a) Trigonal subgroups

      Examples

      • (1) [R\bar{3}2/c] (167) has R3c, R32 and [R\bar{3}] as maximal t subgroups of index [2].

      • (2) [P\bar{3}c1] (165) has P3c1, P321 and [P\bar{3}] as maximal t subgroups of index [2].

    • (b) Orthorhombic subgroups

      No orthorhombic subgroups of trigonal space groups exist, in spite of the existence of an orthohexagonal C cell.

    • (c) Monoclinic subgroups

      All trigonal space groups with secondary or tertiary symmetry elements have monoclinic C-centred maximal t subgroups of index [3].

      Example

      [P\bar{3}1c] (163), [P\bar{3}c1] (165) and [R\bar{3}c] (167) have subgroups of type [C2/c].

    • (d) Triclinic subgroups

      All trigonal space groups without secondary or tertiary symmetry elements have triclinic maximal t subgroups of index [3].

      Example

      [P\bar{3}] (147) and [R\bar{3}] (148) have subgroups [P\bar{1}].

  • (ii) Hexagonal system

    • (a) Hexagonal subgroups

      Example

      [P6_{3}/m\;2/c\;2/m] (193) has maximal t subgroups [P6_{3}/m], [P6_{3}22], [P6_{3}cm], [P\bar{6}2m] and [P\bar{6}c2] of index [2].

    • (b) Trigonal subgroups

      The second and fourth powers of sixfold operations are threefold operations; thus, all hexagonal space groups have maximal trigonal t subgroups of index [2]. In space groups of classes 622, 6mm, [\bar{6}2m], [6/mmm] with secondary and tertiary symmetry elements, trigonal t subgroups always occur in pairs.

      Examples

      • (1) [P6_{1}] (169) contains [P3_{1}] of index [2].

      • (2) [P\bar{6}2c] (190) has maximal t subgroups P321 and P31c; [P6_{1} 22] (178) has subgroups [P3_{1}21] and [P3_{1}12], all of index [2].

      • (3) [P6_{3}/mcm] (193) contains the operation [\bar{3}\;[= (6_{3})^{2} \times \bar{1}]] and thus has maximal t subgroups [P\bar{3}c1] and [P\bar{3}1m] of index [2].

    • (c) Orthorhombic and monoclinic subgroups

      The third power of the sixfold operation is a twofold operation: accordingly, maximal orthorhombic t subgroups of index [3] are derived from the C-cell description of space groups of classes 622, 6mm, [\bar{6}2m] and [6/mmm]. Monoclinic P subgroups of index [3] occur in crystal classes 6, [\bar{6}] and [6/m].

      Examples

      • (1) [P\bar{6}2c] (190) becomes [C\bar{6}2c] in the C cell; with [(\bar{6})^{3} = m], one obtains C2cm (sequence a, b, c) as a maximal t subgroup of index [3]. The standard symbol is Ama2.

      • (2) [P6_{3}/mcm] (193) has maximal orthorhombic t subgroups of type Cmcm of index [3]. With the examples under (a) and (b), this exhausts all maximal t subgroups of [P6_{3}/mcm].

      • (3) [P6_{1}] (169) has a maximal t subgroup [P2_{1}]; [P6_{3}/m] (176) has [P2_{1}/m] as a maximal t subgroup.

4.3.6. Cubic system

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4.3.6.1. Historical note and arrangement of tables

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In the synoptic tables of IT (1935)[link] and IT (1952)[link], for cubic space groups short and full Hermann–Mauguin symbols were listed. They agree, except that in IT (1935)[link] the tertiary symmetry element of the space groups of class 432 was omitted; it was re-established in IT (1952)[link].

In the present edition, the symbols of IT (1952)[link] are retained, with one exception. In the space groups of crystal classes [m\bar{3}] and [m\bar{3}m], the short symbols contain [\bar{3}] instead of 3 (cf. Section 2.2.4[link] ). In Table 4.3.2.1,[link] short and full symbols for all cubic space groups are given. In addition, for centred groups F and I and for P groups with tertiary symmetry elements, extended space-group symbols are listed. In space groups of classes 432 and [\bar{4}3m], the product rule (as defined below) is applied in the first line of the extended symbol.

4.3.6.2. Relations between symmetry elements

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Conventionally, the representative directions of the primary, secondary and tertiary symmetry elements are chosen as [001], [111], and [[1\bar{1}0]] (cf. Table 2.2.4.1[link] for the equivalent directions). As in tetragonal and hexagonal space groups, tertiary symmetry elements are not independent. In classes 432, [\bar{4}3m] and [m\bar{3}m], there are product rules [4 \times 3 = (2){\hbox{;}}\quad \bar{4} \times 3 = (m) = 4 \times \bar{3},] where the tertiary symmetry element is in parentheses; analogous rules hold for the space groups belonging to these classes. When the symmetry directions of the primary and secondary symmetry elements are chosen along [001] and [111], respectively, the tertiary symmetry direction is [011], according to the product rule. In order to have the tertiary symmetry direction along [[1\bar{1}0]], one has to choose the somewhat awkward primary and secondary symmetry directions [010] and [[\bar{1}1\bar{1}]].

Examples

  • (1) In [P\bar{4}3n] (218), with the choice of the 3 axis along [[\bar{1}1\bar{1}]] and of the [\bar{4}] axis parallel to [010], one finds [\bar{4} \times 3 = n], the n glide plane being in x, x, z, as shown in the space-group diagram.

  • (2) In [F\bar{4}3c] (219), one has the same product rule as above; the centring translation [t({1 \over 2},{1 \over 2},0)], however, associates with the n glide plane a c glide plane, also located in x, x, z (cf. Table 4.1.2.3[link] ). In the space-group diagram and symbol, c was preferred to n.

4.3.6.3. Additional symmetry elements

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Owing to periodicity, the tertiary symmetry elements alternate; diagonal axes 2 alternate with parallel screw axes [2_{1}]; diagonal planes m alternate with parallel glide planes g; diagonal n planes, i.e. planes with glide components [{1 \over 2}, {1 \over 2}, {1 \over 2}], alternate with glide planes a, b or c (cf. Chapter 4.1[link] and Tables 4.1.2.2[link] and 4.1.2.3[link] ). For the meaning of the various glide planes g, see Section 11.1.2[link] and the entries Symmetry operations in Part 7[link] .

4.3.6.4. Group–subgroup relations

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4.3.6.4.1. Maximal k subgroups

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The extended symbol of [Fm\bar{3}] (202) shows clearly that [Pm\bar{3}], [Pn\bar{3}], [Pb\bar{3}\; (Pa\bar{3})] and [Pa\bar{3}] are maximal subgroups. [Pm\bar{3}m], [Pn\bar{3}n], [Pm\bar{3}n] and [Pn\bar{3}m] are maximal subgroups of [Im\bar{3}m] (229). Space groups with d glide planes have no k subgroup of lattice P.

4.3.6.4.2. Maximal t subgroups

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  • (a) Cubic subgroups

    The cubic space groups of classes [m\bar{3}], 432 and [\bar{4}3m] have maximal cubic subgroups of class 23 which are found by simple inspection of the full symbol.

    Examples

    [Ia\bar{3}] (206), full symbol [I2_{1}/a\bar{3}], contains [I2_{1}3]. [P2_{1}3] is a maximal subgroup of [P4_{1}32] (213) and its enantiomorph [P4_{3}32] (212). A more difficult example is [I\bar{4}3d] (220) which contains [I2_{1}3].3

    The cubic space groups of class [m\bar{3}m] have maximal subgroups which belong to classes 432 and [\bar{4}3m].

    Examples

    [F4/m\bar{3}2/c] (226) contains F432 and [F\bar{4}3c]; [I4_{1}/a\bar{3}2/d] (230) contains [I4_{1}32] and [I\bar{4}3d].

  • (b) Tetragonal subgroups

    In the cubic space groups of classes 432 and [\bar{4}3m], the primary and tertiary symmetry elements are relevant for deriving maximal tetragonal subgroups.

    Examples

    The groups P432 (207), [P4_{2}32] (208), [P4_{3}32] (212) and [P4_{1}32] (213) have maximal tetragonal t subgroups of index [3]: P422, [P4_{2}22, P4_{3}2_{1}2] and [P4_{1}2_{1}2]. I432 (211) gives rise to I422 with the same cell. F432 (209) also gives rise to I422, but via F422, so that the final unit cell is [a\sqrt{2}/2, a\sqrt{2}/2, a].

    In complete analogy, the groups [P\bar{4}3m] (215) and [P\bar{4}3n] (218) have maximal subgroups [P\bar{4}2m] and [P\bar{4}2c].4

    For the space groups of class [m\bar{3}m], the full symbols are needed to recognize their tetragonal maximal subgroups of class [4/mmm]. The primary symmetry planes of the cubic space group are conserved in the primary and secondary symmetry elements of the tetragonal subgroup: m, n and d remain in the tetragonal symbol; a remains a in the primary and becomes c in the secondary symmetry element of the tetragonal symbol.

    Example

    [P4_{2}/n\;\bar{3}\;2/m] (224) and [I4_{1}/a\;\bar{3}\;2/d] (230) have maximal subgroups [P4_{2}/n\;2/n\;2/m] and [I4_{1}/a\;2/c\;2/d], respectively, [F4_{1}/d\;\bar{3}\;2/c] (228) gives rise to [F4_{1}/d\;2/d\;2/c], which is equivalent to [I4_{1}/a\;2/c\;2/d], all of index [3].

  • (c) Rhombohedral subgroups5

    Here the secondary and tertiary symmetry elements of the cubic space-group symbols are relevant. For space groups of classes 23, [m\bar{3}], 432, the maximal R subgroups are R3, [R\bar{3}] and R32, respectively. For space groups of class [\bar{4}3m], the maximal R subgroup is [R3m] when the tertiary symmetry element is m and R3c otherwise. Finally, for space groups of class [m\bar{3}m], the maximal R subgroup is [R\bar{3}m] when the tertiary symmetry element is m and [R\bar{3}c] otherwise. All subgroups are of index [4].

  • (d) Orthorhombic subgroups

    Maximal orthorhombic space groups of index [3] are easily derived from the cubic space-group symbols of classes 23 and [m\bar{3}].5 Thus, P23, F23, I23, [P2_{1}3], [I2_{1}3] (195–199) have maximal subgroups P222, F222, I222, [P2_{1}2_{1}2_{1}], [I2_{1}2_{1}2_{1}], respectively. Likewise, maximal subgroups of [Pm\bar{3}], [Pn\bar{3}], [Fm\bar{3}], [Fd\bar{3}], [Im\bar{3}], [Pa\bar{3}], [Ia\bar{3}] (200–206) are Pmmm, Pnnn, Fmmm, Fddd, Immm, Pbca, Ibca, respectively. The lattice type (P, F, I) is conserved and only the primary symmetry element has to be considered.

References

Bertaut, E. F. (1976). Study of principal subgroups and of their general positions in C and I groups of class mmm–Dzh. Acta Cryst. A32, 380–387.
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 Crystallography (1995). Vol. A, fourth, revised ed., edited by Th. Hahn. Dordrecht: Kluwer Academic Publishers. [Abbreviated as IT (1995).]
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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