Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.3, p. 782

Section Standardization rules for short symbols

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail: Standardization rules for short symbols

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The symbols of Bravais lattices and glide planes depend on the choice of basis vectors. As shown in the preceding section, additional translation vectors in centred unit cells produce new symmetry operations with the same rotation but different glide/screw parts. Moreover, it was shown that for diagonal orientations symmetry operations may be represented by different symbols. Thus, different short symbols for the same space group can be derived even if the rules for the selection of the generators and indicators are obeyed.

For the unique designation of a space-group type, a standardization of the short symbol is necessary. Rules for standardization were given first by Hermann (1931[link]) and later in a slightly modified form in IT (1952[link]).

These rules, which are generally followed in the present tables, are given below. Because of the historical development of the symbols (cf. Section 3.3.4[link]), some of the present symbols do not obey the rules, whereas others depending on the crystal class need additional rules for them to be uniquely determined. These exceptions and additions are not explicitly mentioned, but may be discovered from Table[link] in which the short symbols are listed for all space groups. A table for all settings may be found in Section 1.5.4[link] .

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Standard space-group symbols

No.Schoenflies symbolShubnikov symbolSymbols of International TablesComments
1935 EditionPresent Edition
1 [C_{1}^{1}] [(a/b/c)\cdot 1] P1 P1 P1 P1  
2 [C_{i}^{1}] [(a/b/c)\cdot \overline{2}] [P\overline{1}] [P\overline{1}] [P\overline{1}] [P\overline{1}] [(a/b/c)\cdot \overline{1}] (Sh–K)
3 [C_{2}^{1}] [(b\!:\!(c/a))\!:\!2] P2 P2 P2 P121  
    [(c\!:\!(a/b))\!:\!2]       P112  
4 [C_{2}^{2}] [(b\!:\!(c/a))\!:\!2_{1}] [P2_{1}] [P2_{1}] [P2_{1}] [P12_{1}1]  
    [(c\!:\!(a/b))\!:\!2_{1}]       [P112_{1}]  
5 [C_{2}^{3}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\!:\!2] C2 C2 C2 C121 B2, B112 (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\!:\!2]       A112 [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\!:\!2] (Sh–K)
6 [C_{2}^{1}] [(b\!:\!(c/a))\cdot m] Pm Pm Pm P1m1  
    [(c\!:\!(a/b))\cdot m]       P11m  
7 [C_{s}^{2}] [(b\!:\!(c/a))\cdot \tilde{c}] Pc Pc Pc P1c1 Pb, P11b (IT, 1952[link])
    [(c\!:\!(b/a))\cdot \tilde{a}]       P11a [(c\!:\!(a/b))\cdot \tilde{b}] (Sh–K)
8 [C_{s}^{3}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot m] Cm Cm Cm C1m1 Bm, B11m (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot m]       A11m [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot m] (Sh–K)
9 [C_{s}^{4}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot \tilde{c}] Cc Cc Cc C1c1 Bb, B11b (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot \tilde{a}]       A11a [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot \tilde{b}] (Sh–K)
10 [C_{2h}^{1}] [(b\!:\!(c/a))\cdot m\!:\!2] [P2/m] [P2/m] [P2/m] [P1\ {2/m}1]  
    [(c\!:\!(a/b))\cdot m\!:\!2]       [P11\ 2/m]  
11 [C_{2h}^{2}] [(b\!:\!(c/a))\cdot m\!:\!2_{1}] [P2_{1}/m] [P2_{1}/m] [P2_{1}/m] [P1\ 2_{1}/m\ 1]  
    [(c\!:\!(a/b))\cdot m\!:\!2_{1}]       [P11\ 2_{1}/m]  
12 [C_{2h}^{3}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot m\!:\!2] [C2/m] [C2/m] [C2/m] [C1\ 2/m\ 1] [B2/m, B11\ 2/m] (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot m\!:\!2]       [A11\ 2/m] [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot m\!:\!2] (Sh–K)
13 [C_{2h}^{4}] [(b\!:\!(c/a))\cdot \tilde{c}\!:\!2] [P2/c] [P2/c] [P2/c] [P1\ 2/c\ 1] [P2/b, P11\ 2/b] (IT, 1952[link])
    [(c\!:\!(a/b))\cdot \tilde{a}\!:\!2]       [P11\ 2/a] [(c\!:\!(a/b))\cdot \tilde{b}\!:\!2] (Sh–K)
14 [C_{2h}^{5}] [(b\!:\!(c/a))\cdot \tilde{c}\!:\!2_{1}] [P2_{1}/c] [P2_{1}/c] [P2_{1}/c] [P1\ 2_{1}/c\ 1] [P2_{1}/b,P112_{1}/b] (IT, 1952[link])
    [(c\!:\!(a/b))\cdot \tilde{a}\!:\!2_{1}]       [P11\ 2_{1}/a] [(c\!:\!(a/b))\cdot b\!:\!2_{1}] (Sh–K)
15 [C_{2h}^{6}] [\left(\displaystyle{a + b \over 2}\bigg/b\!:\!(c/a)\right)\cdot \tilde{c}\!:\!2] [C2/c] [C2/c] [C2/c] [C1\ 2/c\ 1] [B2/b, B11\ 2/b] (IT, 1952[link])
    [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(b/a)\right)\cdot \tilde{a}\!:\!2]       A11 2a [\left(\displaystyle{a + c \over 2}\bigg/c\!:\!(a/b)\right)\cdot \tilde{b}\!:\!2] (Sh–K)
16 [D_{2}^{1}] [(c\!:\!(a\!:\!b))\!:\!2\!:\!2] P222 P222 P222 P222  
17 [D_{2}^{2}] [(c\!:\!(a\!:\!b))\!:\!2_{1}\!:\!2] [P222_{1}] [P222_{1}] [P222_{1}] [P222_{1}]  
18 [D_{2}^{3}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!b))\!:\!2\circcol\,2_{1}] [P2_{1}2_{1}2] [P2_{1}2_{1}2] [P2_{1}2_{1}2] [P2_{1}2_{1}2]  
19 [D_{2}^{4}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!b))\!:\!2_{1}\circcol\, 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}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!2_{1}\!:\!2] [C222_{1}] [C222_{1}] [C222_{1}] [C222_{1}]  
21 [D_{2}^{6}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!2\!:\!2] C222 C222 C222 C222  
22 [D_{2}^{7}] [\displaylines{\left(\displaystyle{a + c \over 2}\bigg/{b + c \over 2}\bigg/{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr \quad :\!2\!:\!2\hfill}] F222 F222 F222 F222  
23 [D_{2}^{8}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!2\!:\!2] I222 I222 I222 I222  
24 [D_{2}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!2\!:\!2_{1}] [I2_{1}2_{1}2_{1}] [I2_{1}2_{1}2_{1}] [I2_{1}2_{1}2_{1}] [I2_{1}2_{1}2_{1}]  
25 [C_{2v}^{1}] [(c\!:\!(a\!:\!b))\!:\!m\cdot 2] Pmm Pmm2 Pmm2 Pmm2  
26 [C_{2v}^{2}] [(c\!:\!(a\!:\!b))\!:\!\tilde{c}\cdot 2_{1}] Pmc [Pmc2_{1}] [Pmc2_{1}] [Pmc2_{1}]  
27 [C_{2v}^{3}] [(c\!:\!(a\!:\!b))\!:\!\tilde{c}\cdot 2] Pcc Pcc2 Pcc2 Pcc2  
28 [C_{2v}^{4}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\cdot 2] Pma Pma2 Pma2 Pma2  
29 [C_{2v}^{5}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\cdot 2_{1}] Pca [Pca2_{1}] [Pca2_{1}] [Pca2_{1}]  
30 [C_{2v}^{6}] [(c\!:\!(a\!:\!b))\!:\!\tilde{c} \bigodot 2] Pnc Pnc2 Pnc2 Pnc2 [(c\!:\!(a\!:\!b))\!:\!\widetilde{ac}\cdot 2] (Sh–K)
31 [C_{2v}^{7}] [(c\!:\!(a\!:\!b))\!:\!\widetilde{ac}\cdot 2_{1}] Pmn [Pmn2_{1}] [Pmn2_{1}] [Pmn2_{1}]  
32 [C_{2v}^{8}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\bigodot 2] Pba Pba2 Pba2 Pba2  
33 [C_{2v}^{9}] [(c\!:\!(a\!:\!b))\!:\!\tilde{a}\bigodot 2_{1}] Pna [Pna2_{1}] [Pna2_{1}] [Pna2_{1}]  
34 [C_{2v}^{10}] [(c\!:\!(a\!:\!b))\!:\!\widetilde{ac}\bigodot 2] Pnn Pnn2 Pnn2 Pnn2  
35 [C_{2v}^{11}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2] Cmm Cmm2 Cmm2 Cmm2  
36 [C_{2v}^{12}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2_{1}] Cmc [Cmc2_{1}] [Cmc2_{1}] [Cmc2_{1}]  
37 [C_{2v}^{13}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2] Ccc Ccc2 Ccc2 Ccc2  
38 [C_{2v}^{14}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2] Amm Amm2 Amm2 Amm2  
39 [C_{2v}^{15}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2_{1}] Abm Abm2 Aem2 Aem2 [\cases{\!\!\left(\displaystyle{b + c \over 2}\!\big/\!c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2\cr\quad (\rm{Sh\!-\!K})\cr \hbox{Use former symbol}\cr Abm2\ \hbox{for generation}\cr}]
40 [C_{2v}^{16}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2] Ama Ama2 Ama2 Ama2  
41 [C_{2v}^{17}] [\left(\displaystyle{b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2_{1}] Aba Aba2 Aea2 Aea2 [\cases{\!\!\left(\displaystyle{b + c \over 2}\!\big/c\!:\!(a\!:\!b)\right)\!:\!\widetilde{ac}\cdot 2 \cr\quad (\rm{Sh\!-\!K})\cr \hbox{Use former symbol}\cr Aba2\ \hbox{for generation}\cr}]
42 [C_{2v}^{18}] [\displaylines{\left(\displaystyle{a + c \over 2}\bigg/{b + c \over 2}\bigg/{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr\quad:m\cdot 2\hfill}] Fmm Fmm2 Fmm2 Fmm2  
43 [C_{2v}^{19}] [\left(\displaystyle{a + c \over 2}\bigg/{b + c \over 2}\bigg/{a + b \over 2}\!:\!\tilde{c}\!:\!(a\!:\!b)\right)][:{1\over 2}\widetilde{ac}\bigodot 2] Fdd Fdd2 Fdd2 Fdd2  
44 [C_{2v}^{20}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!m\cdot 2] Imm Imm2 Imm2 Imm2  
45 [C_{2v}^{21}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{c}\cdot 2] Iba Iba2 Iba2 Iba2 [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2_{1}](Sh–K)
46 [C_{2v}^{22}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\!:\!\tilde{a}\cdot 2] Ima Ima2 Ima2 Ima2  
47 [D_{2h}^{1}] [(c\!:\!(a\!:\!b))\cdot m\!:\!2\cdot m] Pmmm [P2/m\ 2/m\ 2/m] Pmmm [P2/m\ 2/m\ 2/m]  
48 [D_{2h}^{2}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\circdot \widetilde{ac}] Pnnn [P2/n\ 2/n\ 2/n] Pnnn [P2/n\ 2/n\ 2/n]  
49 [D_{2h}^{3}] [(c\!:\!(a\!:\!b))\cdot m\!:\!2\cdot \tilde{c}] Pccm [P2/c\ 2/c\ 2/m] Pccm [P2/c\ 2/c\ 2/m]  
50 [D_{2h}^{4}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\circdot \tilde{a}] Pban [P2/b\ 2/a\ 2/n] Pban [P2/b\ 2/a\ 2/n]  
51 [D_{2h}^{5}] [(c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2\cdot m] Pmma [P2_{1}/m\ 2/m\ 2/a] Pmma [P2_{1}/m\ 2/m\ 2/a]  
52 [D_{2h}^{6}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2\circdot \widetilde{ac}] Pnna [P2/n\ 2_{1}/n\ 2/a] Pnna [P2/n\ 2_{1}/n\ 2/a]  
53 [D_{2h}^{7}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2_{1}\cdot \widetilde{ac}] Pmna [P2/m\ 2/n\ 2_{1}/a] Pmna [P2/m\ 2/n\ 2_{1}/a]  
54 [D_{2h}^{8}] [(c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2\cdot \tilde{c}] Pcca [P2_{1}/c\ 2/c\ 2/a] Pcca [P2_{1}/c\ 2/c\ 2/a]  
55 [D_{2h}^{9}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot m\!:\!2\circdot \tilde{a}] Pbam [P2_{1}/b\ 2_{1}/a\ 2/m] Pbam [P2_{1}/b\ 2_{1}/a\ 2/m]  
56 [D_{2h}^{10}] [(c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\cdot \tilde{c}] Pccn [P2_{1}/c\ 2_{1}/c\ 2/n] Pccn [P2_{1}/c\ 2_{1}/c\ 2/n]  
57 [D_{2h}^{11}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot m\!:\!2_{1}\circdot \tilde{c}] Pbcm [P2/b\ 2_{1}/c\ 2_{1}/m] Pbcm [P2/b\ 2_{1}/c\ 2_{1}/m]  
58 [D_{2h}^{12}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot m\!:\!2\circdot \widetilde{ac}] Pnnm [P2_{1}/n\ 2_{1}/n\ 2/m] Pnnm [P2_{1}/n\ 2_{1}/n\ 2/m]  
59 [D_{2h}^{13}] [(c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2\cdot m] Pmmn [P2_{1}/m\ 2_{1}/m\ 2/n] Pmmn [P2_{1}/m\ 2_{1}/m\ 2/n]  
60 [D_{2h}^{14}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \widetilde{ab}\!:\!2_{1}\circdot \tilde{c}] Pbcn [P2_{1}/b\ 2/c\ 2_{1}/n] Pbcn [P2_{1}/b\ 2/c\ 2_{1}/n]  
61 [D_{2h}^{15}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2_{1}\circdot \tilde{c}] Pbca [P2_{1}/b\ 2_{1}/c\ 2_{1}/a] Pbca [P2_{1}/b\ 2_{1}/c\ 2_{1}/a]  
62 [D_{2h}^{16}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!(a\!:\!b))\cdot \tilde{a}\!:\!2_{1}\circdot m] Pnma [P2_{1}/n\ 2_{1}/m\ 2_{1}/a] Pnma [P2_{1}/n\ 2_{1}/m\ 2_{1}/a]  
63 [D_{2h}^{17}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot m\!:\!2_{1}\cdot \tilde{c}] Cmcm [C2/m\ 2/c\ 2_{1}/m] Cmcm [C2/m\ 2/c\ 2_{1}/m]  
64 [D_{2h}^{18}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2_{1}\cdot \tilde{c}] Cmca [C2/m\ 2/c\ 2_{1}/a] Cmce [C2/m\ 2/c\ 2_{1}/e] Use former symbol Cmca for generation
65 [D_{2h}^{19}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot m] Cmmm [C2/m\ 2/m\ 2/m] Cmmm [C2/m\ 2/m\ 2/m]  
66 [D_{2h}^{20}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot \tilde{c}] Cccm [C2/c\ 2/c\ 2/m] Cccm [C2/c\ 2/c\ 2/m]  
67 [D_{2h}^{21}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot m] Cmma [C2/m\ 2/m\ 2/a] Cmme [C2/m\ 2/m\ 2/e] Use former symbol Cmma for generation
68 [D_{2h}^{22}] [\left(\displaystyle{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot \tilde{c}] Ccca [C2/c\ 2/c\ 2/a] Ccce [C2/c\ 2/c\ 2/e] Use former symbol Ccca for generation
69 [D_{2h}^{23}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr \quad\cdot \,m\!:\!2\cdot m\hfill\cr}] Fmmm [F2/m\ 2/m\ 2/m] Fmmm [F2/m\ 2/m\ 2/m]  
70 [D_{2h}^{24}] [\def\circdot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} \displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!c\!:\!(a\!:\!b)\right)\hfill\cr \quad\cdot\, {\textstyle{1 \over 2}}\widetilde{ab}\!:\!2\circdot {\textstyle{1 \over 2}}\widetilde{ac}\hfill\cr}] Fddd [F2/d\ 2/d\ 2/d] Fddd [F2/d\ 2/d\ 2/d]  
71 [D_{2h}^{25}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot m] Immm [I2/m\ 2/m\ 2/m] Immm I2/m 2/m 2/m  
72 [D_{2h}^{26}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot m\!:\!2\cdot \tilde{c}] Ibam [I2/b\ 2/a\ 2/m] Ibam [I2/b\ 2/a\ 2/m]  
73 [D_{2h}^{27}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot \tilde{c}] Ibca [I2_{1}/b\ 2_{1}/c\ 2_{1}/a] Ibca [I2_{1}/b\ 2_{1}/c\ 2_{1}/a] [I2/b\ 2/c\ 2/a] (IT, 1952[link])
74 [D_{2h}^{28}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!b)\right)\cdot \tilde{a}\!:\!2\cdot m] Imma [I2_{1}/m\ 2_{1}/m\ 2_{1}/a] Imma [I2_{1}/m\ 2_{1}/m\ 2_{1}/a] [I2/m\ 2/m\ 2/a] (IT, 1952[link])
75 [C_{4}^{1}] [(c\!:\!(a\!:\!a))\!:\!4] P4 P4 P4 P4  
76 [C_{4}^{2}] [(c\!:\!(a\!:\!a))\!:\!4_{1}] [P4_{1}] [P4_{1}] [P4_{1}] [P4_{1}]  
77 [C_{4}^{3}] [(c\!:\!(a\!:\!a))\!:\!4_{2}] [P4_{2}] [P4_{2}] [P4_{2}] [P4_{2}]  
78 [C_{4}^{4}] [(c\!:\!(a\!:\!a))\!:\!4_{3}] [P4_{3}] [P4_{3}] [P4_{3}] [P4_{3}]  
79 [C_{4}^{5}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4] I4 I4 I4 I4  
80 [C_{4}^{6}] [\left(\displaystyle{a - b - c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}] [I4_{1}] [I4_{1}] [I4_{1}] [I4_{1}]  
81 [S_{4}^{1}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}] [P\overline{4}] [P\overline{4}] [P\overline{4}] [P\overline{4}]  
82 [S_{4}^{2}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}] [I\overline{4}] [I\overline{4}] [I\overline{4}] [I\overline{4}]  
83 [C_{4h}^{1}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4] [P4/m] [P4/m] [P4/m] [P4/m]  
84 [C_{4h}^{2}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}] [P4_{2}/m] [P4_{2}/m] [P4_{2}/m] [P4_{2}/m]  
85 [C_{4h}^{3}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4] [P4/n] [P4/n] [P4/n] [P4/n]  
86 [C_{4h}^{4}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}] [P4_{2}/n] [P4_{2}/n] [P4_{2}/n] [P4_{2}/n]  
87 [C_{4h}^{5}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\cdot m\!:\!4] [I4/m] [I4/m] [I4/m] [I4/m]  
88 [C_{4h}^{6}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\cdot \tilde{a}\!:\!4_{1}] [I4_{1}/a] [I4_{1}/a] [I4_{1}/a] [I4_{1}/a]  
89 [D_{4}^{1}] (c:(a:a)):4:2 P42 P422 P422 P422  
90 [D_{4}^{2}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4 \circcol\, 2_{1}] [P42_{1}] [P42_{1}2] [P42_{1}2] [P42_{1}2]  
91 [D_{4}^{3}] [(c\!:\!(a\!:\!a))\!:\!4_{1}\!:\!2] [P4_{1}2] [P4_{1}22] [P4_{1}22] [P4_{1}22]  
92 [D_{4}^{4}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4_{1}\circcol\, 2_{1}] [P4_{1}2_{1}] [P4_{1}2_{1}2] [P4_{1}2_{1}2] [P4_{1}2_{1}2]  
93 [D_{4}^{5}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\!:\!2] [P4_{2}2] [P4_{2}22] [P4_{2}22] [P4_{2}22]  
94 [D_{4}^{6}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4_{2}\circcol\, 2_{1}] [P4_{2}2_{1}] [P4_{2}2_{1}2] [P4_{2}2_{1}2] [P4_{2}2_{1}2]  
95 [D_{4}^{7}] [(c\!:\!(a\!:\!a))\!:\!4_{3}\!:\!2] [P4_{3}2] [P4_{3}22] [P4_{3}22] [P4_{3}22]  
96 [D_{4}^{8}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!4_{3}\circcol\, 2_{1}] [P4_{3}2_{1}] [P4_{3}2_{1}2] [P4_{3}2_{1}2] [P4_{3}2_{1}2]  
97 [D_{4}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4\!:\!2] I42 I422 I422 I422  
98 [D_{4}^{10}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}\!:\!2] [I4_{1}2] [I4_{1}22] [I4_{1}22] [I4_{1}22]  
99 [C_{4v}^{1}] [(c\!:\!(a\!:\!a))\!:\!4\cdot m] P4mm P4mm P4mm P4mm  
100 [C_{4v}^{2}] [ (c\!:\!(a\!:\!a))\!:\!4\bigodot \tilde{a}] P4bm P4bm P4bm P4bm  
101 [C_{4v}^{3}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\cdot \tilde{c}] P4cm [P4_{2}cm] [P4_{2}cm] [P4_{2}cm]  
102 [C_{4v}^{4}] [ (c\!:\!(a\!:\!a))\!:\!4_{2}\bigodot \widetilde{ac}] P4nm [P4_{2}nm] [P4_{2}nm] [P4_{2}nm]  
103 [C_{4v}^{5}] [(c\!:\!(a\!:\!a))\!:\!4\cdot \tilde{c}] P4cc P4cc P4cc P4cc  
104 [C_{4v}^{6}] [ (c\!:\!(a\!:\!a))\!:\!4\bigodot \widetilde{ac}] P4nc P4nc P4nc P4nc  
105 [C_{4v}^{7}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\cdot m] P4mc [P4_{2}mc] [P4_{2}mc] [P4_{2}mc]  
106 [C_{4v}^{8}] [(c\!:\!(a\!:\!a))\!:\!4_{2}\bigodot \tilde{a}] P4bc [P4_{2}bc] [P4_{2}bc] [P4_{2}bc]  
107 [C_{4v}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4\cdot m] I4mm I4mm I4mm I4mm  
108 [C_{4v}^{10}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4\cdot \tilde{c}] I4cm I4cm I4cm I4cm  
109 [C_{4v}^{11}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}\bigodot m] I4md [I4_{1}md] [I4_{1}md] [I4_{1}md]  
110 [C_{4v}^{12}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!4_{1}\bigodot \tilde{c}] I4cd [I4_{1}cd] [I4_{1}cd] [I4_{1}cd] [\left(\displaystyle{a + b + c \over 2}\!\!\bigg/\!\!c\!:\!a\!:\!a\right)\!:\!4_{1}\cdot \tilde{a}](Sh–K)
111 [D_{2d}^{1}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\!:\!2] [P\overline{4}2m] [P\overline{4}2m] [P\overline{4}2m] [P\overline{4}2m]  
112 [D_{2d}^{2}] [\def\circcol{\mathop{\bigcirc\hskip -5pt{\raise.05pt\hbox{$\!:\!$}} \ }} (c\!:\!(a\!:\!a))\!:\!\tilde{4}\circcol\, 2] [P\overline{4}2c] [P\overline{4}2c] [P\overline{4}2c] [P\overline{4}2c]  
113 [D_{2d}^{3}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \widetilde{ab}] [P\overline{4}2_{1}m] [P\overline{4}2_{1}m] [P\overline{4}2_{1}m] [P\overline{4}2_{1}m]  
114 [D_{2d}^{4}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \widetilde{abc}] [P\overline{4}2_{1}c] [P\overline{4}2_{1}c] [P\overline{4}2_{1}c] [P\overline{4}2_{1}c]  
115 [D_{2d}^{5}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot m] [C\overline{4}2m] [C\overline{4}2m] [P\overline{4}m2] [P\overline{4}m2]  
116 [D_{2d}^{6}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \tilde{c}] [C\overline{4}2c] [C\overline{4}2c] [P\overline{4}c2] [P\overline{4}c2]  
117 [D_{2d}^{7}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\bigodot \tilde{a}] [C\overline{4}2b] [C\overline{4}2b] [P\overline{4}b2] [P\overline{4}b2]  
118 [D_{2d}^{8}] [(c\!:\!(a\!:\!a))\!:\!\tilde{4}\cdot \widetilde{ac}] [C\overline{4}2n] [C\overline{4}2n] [P\overline{4}n2] [P\overline{4}n2]  
119 [D_{2d}^{9}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\cdot m] [F\overline{4}2m] [F\overline{4}2m] [I\overline{4}m2] [I\overline{4}m2]  
120 [D_{2d}^{10}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\cdot \tilde{c}] [F\overline{4}2c] [F\overline{4}2c] [I\overline{4}c2] [I\overline{4}c2]  
121 [D_{2d}^{11}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\!:\!2] [I\overline{4}2m] [I\overline{4}2m] [I\overline{4}2m] [I\overline{4}2m]  
122 [D_{2d}^{12}] [\left(\displaystyle{a + b + c \over 2}\bigg/c\!:\!(a\!:\!a)\right)\!:\!\tilde{4}\bigodot {1 \over 2}\widetilde{abc}] [I\overline{4}2d] [I\overline{4}2d] [I\overline{4}2d] [I\overline{4}2d]  
123 [D_{4h}^{1}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\cdot m] [P4/mmm] [P4/m\ 2/m\ 2/m] [P4/mmm] [P4/m\ 2/m\ 2/m]  
124 [D_{4h}^{2}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\cdot \tilde{c}] [P4/mcc] [P4/m\ 2/c\ 2/c] [P4/mcc] [P4/m\ 2/c\ 2/c]  
125 [D_{4h}^{3}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4\bigodot \tilde{a}] [P4/nbm] [P4/n\ 2/b\ 2/m] [P4/nbm] [P4/n\ 2/b\ 2/m] [\def\bigodot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!a\!:\!a)\cdot \widetilde{ab}\!:\!4\bigodot \tilde{b}] (Sh–K)
126 [D_{4h}^{4}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4\bigodot \widetilde{ac}] [P4/nnc] [P4/n\ 2/n\ 2/c] [P4/nnc] [P4/n\ 2/n\ 2/c]  
127 [D_{4h}^{5}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\bigodot \tilde{a}] [P4/mbm] [P4/m\ 2_{1}/b\ 2/m] [P4/mbm] [P4/m\ 2_{1}/b\ 2/m] [\def\bigodot{\mathop{\bigcirc\hskip -6.5pt{\raise.05pt\hbox{$\cdot$}} \ }} (c\!:\!a\!:\!a)\cdot m\!:\!4\bigodot \tilde{b}] (Sh–K)
128 [D_{4h}^{6}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4\bigodot \widetilde{ac}] [P4/mnc] [P4/m\ 2_{1}/n\ 2/c] [P4/mnc] [P4/m\ 2_{1}/n\ 2/c]  
129 [D_{4h}^{7}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4\cdot m] [P4/nmm] [P4/n\ 2_{1}/m\ 2/m] [P4/nmm] [P4/n\ 2_{1}/m\ 2/m]  
130 [D_{4h}^{8}] [(c\!:\!(a\!:\!a)\cdot \widetilde{ab}\!:\!4\cdot \tilde{c}] [P4/ncc] [P4/n\ 2/c\ 2/c] [P4/ncc] [P4/n\ 2/c\ 2/c]  
131 [D_{4h}^{9}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}\cdot m] [P4/mmc] [P4_{2}/m\ 2/m\ 2/c] [P4_{2}/mmc] [P4_{2}/m\ 2/m\ 2/c]  
132 [D_{4h}^{10}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}\cdot \tilde{c}] [P4/mcm] [P4_{2}/m\ 2/c\ 2/m] [P4_{2}/mcm] [P4_{2}/m\ 2/c\ 2/m]  
133 [D_{4h}^{11}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}\bigodot \tilde{a}] [P4/nbc] [P4_{2}/n\ 2/b\ 2/c] [P4_{2}/nbc] [P4_{2}/n\ 2/b\ 2/c] [(c\!:\!a\!:\!a)\cdot \widetilde{ab}\!:\!4_{2}\bigodot \tilde{b}] (Sh–K)
134 [D_{4h}^{12}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}\bigodot \widetilde{ac}] [P4/nnm] [P4_{2}/n\ 2/n\ 2/m] [P4_{2}/nnm] [P4_{2}/n\ 2/n\ 2/m]  
135 [D_{4h}^{13}] [(c\!:\!(a\!:\!a))\cdot n\!:\!4_{2}\bigodot \tilde{a}] [P4/mbc] [P4_{2}/m\ 2_{1}/b\ 2/c] [P4_{2}/mbc] [P4_{2}/m\ 2_{1}/b\ 2/c] [(c\!:\!a\!:\!a)\cdot m\!:\!4_{2}\bigodot \tilde{b}] (Sh–K)
136 [D_{4h}^{14}] [(c\!:\!(a\!:\!a))\cdot m\!:\!4_{2}\bigodot \widetilde{ac}] [P4/mnm] [P4_{2}/m\ 2_{1}/n\ 2/m] [P4_{2}/mnm] [P4_{2}/m\ 2_{1}/n\ 2/m]  
137 [D_{4h}^{15}] [(c\!:\!(a\!:\!a))\cdot \widetilde{ab}\!:\!4_{2}\cdot m] [P4/nmc] [P4_{2}/n\ 2_{1}/m\ 2/c] [P4_{2}/nmc] [P4_{2}/n\ 2_{1}/m\ 2/c]  
138 [D_{4h}^{16}] [(c\!:\!(a\!:\!a))\cdot ab\!:\!4_{2}\cdot \tilde{c}] [P4/ncm] [P4_{2}/n\ 2_{1}/c\ 2/m] [P4_{2}/ncm] [P4_{2}/n\ 2_{1}/c\ 2/m]  
139 [D_{4h}^{17}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot m\!:\!4\cdot m] [I4/mmm] [I4/m\ 2/m\ 2/m] [I4/mmm] [I4/m\ 2/m\ 2/m]  
140 [D_{4h}^{18}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot m\!:\!4\cdot \tilde{c}] [I4/mcm] [I4/m\ 2/c\ 2/m] [I4/mcm] [I4/m\ 2/c\ 2/m]  
141 [D_{4h}^{19}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot \tilde{a}\!:\!4_{1}\bigodot m] [I4/amd] [I4_{1}/a\ 2/m\ 2/d] [I4_{1}/amd] [I4_{1}/a\ 2/m\ 2/d]  
142 [D_{4h}^{20}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!c\!:\!(a\!:\!a)\right)\cdot \tilde{a}\!:\!4_{1}\bigodot \tilde{c}] [I4/acd] [I4_{1}/a\ 2/c\ 2/d] [I4_{1}/acd] [I4_{1}/a\ 2/c\ 2/d]  
143 [C_{3}^{1}] [(c\!:\!(a/a))\!:\!3] C3 C3 P3 P3  
144 [C_{3}^{2}] [(c\!:\!(a/a))\!:\!3_{1}] [C3_{1}] [C3_{1}] [P3_{1}] [P3_{1}]  
145 [C_{3}^{3}] [(c\!:\!(a/a))\!:\!3_{2}] [C3_{2}] [C3_{2}] [P3_{2}] [P3_{2}]  
146 [C_{3}^{4}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\!\bigg/\!\!{a + 2b + 2c \over 3}\!\!\bigg/\!\!c\!:\!(a/a)\right)\hfill\cr\quad:3\hfill}] R3 R3 R3 R3 Hexagonal setting (Sh–K)
    [(a/a/a)/3]         Rhombohedral setting (Sh–K)
147 [C_{3i}^{1}] [(c\!:\!(a/a))\!:\!\tilde{6}] [C\overline{3}] [C\overline{3}] [P\overline{3}] [P\overline{3}]  
148 [C_{3i}^{2}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr\quad :\tilde{6}\hfill}] [R\overline{3}] [R\overline{3}] [R\overline{3}] [R\overline{3}] Hexagonal setting (Sh–K)
    [(a/a/a)/\tilde{6}]         Rhombohedral setting (Sh–K)
149 [D_{3}^{1}] [(c\!:\!(a/a))\!:\!2\!:\!3] H32 H321 P312 P312  
150 [D_{3}^{2}] [(c\!:\!(a/a))\!:\!2\!:\!3] C32 C321 P321 P321  
151 [D_{3}^{3}] [(c\!:\!(a/a))\!:\!2\!:\!3_{1}] [H3_{1}2] [H3_{1}21] [P3_{1}12] [P3_{1}12]  
152 [D_{3}^{4}] [(c\!:\!(a/a))\!:\!2\!:\!3_{1}] [C3_{1}2] [C3_{1}21] [P3_{1}21] [P3_{1}21]  
153 [D_{3}^{5}] [(c\!:\!(a/a))\!:\!2\!:\!3_{2}] [H3_{2}2] [H3_{2}21] [P3_{2}12] [P3_{2}12]  
154 [D_{3}^{6}] [(c\!:\!(a/a))\!:\!2\!:\!3_{2}] [C3_{2}2] [C3_{2}21] [P3_{2}21] [P3_{2}21]  
155 [D_{3}^{7}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr \quad\cdot 2\!:\!3\hfill\cr}] R32 R32 R32 R32 Hexagonal setting (Sh–K)
    [(a/a/a)/3\!:\!2]         Rhombohedral setting (Sh–K)
156 [C_{3v}^{1}] [(c\!:\!(a/a))\!:\!m\!:\!3] C3m C3m1 P3m1 P3m1  
157 [C_{3v}^{2}] [(a\!:\!c\!:\!a)\!:\!m\!:\!3] H3m H3m1 P31m P31m [(c\!:\!(a/a))\cdot m\cdot 3] (Sh–K) with special comment
158 [C_{3v}^{3}] [(c\!:\!(a/a))\!:\!\tilde{c}\!:\!3] C3c C3c1 P3c1 P3c1  
159 [C_{3v}^{4}] [(a\!:\!c\!:\!a)\!:\!\tilde{c}\!:\!3] H3c H3c1 P31c P31c [(c\!:\!(a/a))\cdot \tilde{c}\cdot 3] (Sh–K) with special comment
160 [C_{3v}^{5}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr \quad\cdot\, m\cdot 3\hfill\cr}] R3m R3m R3m R3m Hexagonal setting (Sh–K)
    [(a/a/a)/3\cdot m]         Rhombohedral setting (Sh–K)
161 [C_{3v}^{6}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right) \hfill\cr\quad \cdot \,\tilde{c}\cdot 3\hfill\cr}] R3c R3c R3c R3c Hexagonal setting (Sh–K)
    [(a/a/a)/3\cdot \widetilde{abc}]         Rhombohedral setting (Sh–K)
162 [D_{3d}^{1}] [(a\!:\!c\!:\!a)\cdot m\cdot \tilde{6}] [H\overline{3}m] [H\overline{3}\ 2/m\ 1] [P\overline{3}1m] [P\overline{3}1\ 2/m] [(c\!:\!(a/a))\cdot m\cdot \tilde{6}] (Sh–K) with special comment
163 [D_{3d}^{2}] [(a\!:\!c\!:\!a)\cdot \tilde{c}\cdot \tilde{6}] [H\overline{3}c] [H\overline{3}\ 2/c\ 1] [P\overline{3}1c] [P\overline{3}1\ 2/c] [(c\!:\!(a/a)\cdot \tilde{c}\cdot \tilde{6}] (Sh–K) with special comment
164 [D_{3d}^{3}] [(c\!:\!(a/a))\!:\!m\cdot \tilde{6}] [C\overline{3}m] [C\overline{3}\ 2/m\ 1] [P\overline{3}m1] [P\overline{3}\ 2/m\ 1]  
165 [D_{3d}^{4}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot \tilde{6}] [C\overline{3}c] [C\overline{3}\ 2/c\ 1] [P\overline{3}c1] [P\overline{3}\ 2/c\ 1]  
166 [D_{3d}^{5}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right) \hfill\cr \quad:m\cdot \tilde{6}\hfill\cr}] [R\overline{3}m] [R\overline{3}\ 2/m] [R\overline{3}m] [R\overline{3}\ 2/m] Hexagonal setting (Sh–K)
    [(a/a/a)/\tilde{6}\cdot m]         Rhombohedral setting (Sh–K)
167 [D_{3d}^{6}] [\displaylines{\left(\displaystyle{2a + b + c \over 3}\!\bigg/\!{a + 2b + 2c \over 3}\!\bigg/\!c\!:\!(a/a)\right)\hfill\cr \quad:\tilde{c}\cdot \tilde{6}\hfill\cr}] [R\overline{3}c] [R\overline{3}\ 2/c] [R\overline{3}c] [R\overline{3}\ 2/c] Hexagonal setting (Sh–K)
    [(a/a/a)/\tilde{6}\cdot \widetilde{abc}]         Rhombohedral setting (Sh–K)
168 [C_{6}^{1}] [(c\!:\!(a/a))\!:\!6] C6 C6 P6 P6  
169 [C_{6}^{2}] [(c\!:\!(a/a))\!:\!6_{1}] [C6_{1}] [C6_{1}] [P6_{1}] [P6_{1}]  
170 [C_{6}^{3}] [(c\!:\!(a/a))\!:\!6_{5}] [C6_{5}] [C6_{5}] [P6_{5}] [P6_{5}]  
171 [C_{6}^{4}] [(c\!:\!(a/a))\!:\!6_{2}] [C6_{2}] [C6_{2}] [P6_{2}] [P6_{2}]  
172 [C_{6}^{5}] [(c\!:\!(a/a))\!:\!6_{4}] [C6_{4}] [C6_{4}] [P6_{4}] [P6_{4}]  
173 [C_{6}^{6}] [(c\!:\!(a/a))\!:\!6_{3}] [C6_{3}] [C6_{3}] [P6_{3}] [P6_{3}]  
174 [C_{3h}^{1}] [(c\!:\!(a/a))\!:\!3\!:\!m] [C\overline{6}] [C\overline{6}] [P\overline{6}] [P\overline{6}]  
175 [C_{6h}^{1}] [(c\!:\!(a/a))\cdot m\!:\!6] [C6/m] [C6/m] [P6/m] [P6/m]  
176 [C_{6h}^{2}] [(c\!:\!(a/a))\cdot m\!:\!6_{3}] [C6_{3}/m] [C6_{3}/m] [P6_{3}/m] [P6_{3}/m]  
177 [D_{6}^{1}] [(c\!:\!(a/a))\cdot 2\!:\!6] C62 C622 P622 P622  
178 [D_{6}^{2}] [(c\!:\!(a/a))\cdot 2\!:\!6_{1}] [C6_{1}2] [C6_{1}22] [P6_{1}22] [P6_{1}22]  
179 [D_{6}^{3}] [(c\!:\!(a/a))\cdot 2\!:\!6_{5}] [C6_{5}2] [C6_{5}22] [P6_{5}22] [P6_{5}22]  
180 [D_{6}^{4}] [(c\!:\!(a/a))\cdot 2\!:\!6_{2}] [C6_{2}2] [C6_{2}22] [P6_{2}22] [P6_{2}22]  
181 [D_{6}^{5}] [(c\!:\!(a/a))\cdot 2\!:\!6_{4}] [C6_{4}2] [C6_{4}22] [P6_{4}22] [P6_{4}22]  
182 [D_{6}^{6}] [(c\!:\!(a/a))\cdot 2\!:\!6_{3}] [C6_{3}2] [C6_{3}22] [P6_{3}22] [P6_{3}22]  
183 [C_{6v}^{1}] [(c\!:\!(a/a))\!:\!m\cdot 6] C6mm C6mm P6mm P6mm  
184 [C_{6v}^{2}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot 6] C6cc C6cc P6cc P6cc  
185 [C_{6v}^{3}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot 6_{3}] C6cm [C6_{3}cm] [P6_{3}cm] [P6_{3}cm]  
186 [C_{6v}^{4}] [(c\!:\!(a/a))\!:\!m\cdot 6_{3}] C6mc [C6_{3}mc] [P6_{3}mc] [P6_{3}mc]  
187 [D_{3h}^{1}] [(c\!:\!(a/a))\!:\!m\cdot 3\!:\!m] [C\overline{6}m2] [C\overline{6}m2] [P\overline{6}m2] [P\overline{6}m2]  
188 [D_{3h}^{2}] [(c\!:\!(a/a))\!:\!\tilde{c}\cdot 3\!:\!m] [C\overline{6}c2] [C\overline{6}c2] [P\overline{6}c2] [P\overline{6}c2]  
189 [D_{3h}^{3}] [(c\!:\!(a/a))\cdot m\!:\!3\cdot m] [H\overline{6}m2] [H\overline{6}m2] [P\overline{6}2m] [P\overline{6}2m]  
190 [D_{3h}^{4}] [(c\!:\!(a/a))\cdot m\!:\!3\cdot \tilde{c}] [H\overline{6}c2] [H\overline{6}c2] [P\overline{6}2c] [P\overline{6}2c]  
191 [D_{6h}^{1}] [(c\!:\!(a/a))\cdot m\!:\!6\cdot m] [C6/mmm] [C6/m\ 2/m\ 2/m] [P6/mmm] [P6/m\ 2/m\ 2/m]  
192 [D_{6h}^{2}] [(c\!:\!(a/a))\cdot m\!:\!6\cdot \tilde{c}] [C6/mcc] [C6/m\ 2/c\ 2/c] [P6/mcc] [P6/m\ 2/c\ 2/c]  
193 [D_{6h}^{3}] [(c\!:\!(a/a))\cdot m\!:\!6_{3}\cdot \tilde{c}] [C6/mcm] [C6_{3}/m\ 2/c\ 2/m] [P6_{3}/mcm] [P6_{3}/m\ 2/c\ 2/m]  
194 [D_{6h}^{4}] [(c\!:\!(a/a))\cdot m\!:\!6_{3}\cdot m] [C6/mmc] [C6_{3}/m\ 2/m\ 2/c] [P6_{3}/mmc] [P6_{3}/m\ 2/m\ 2/c]  
195 [T^{1}] [(a\!:\!(a/a))\!:\!2/3] P23 P23 P23 P23  
196 [T^{2}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad:2/3\hfill}] F23 F23 F23 F23  
197 [T^{3}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!2/3] I23 I23 I23 I23  
198 [T^{4}] [(a\!:\!(a\!:\!a))\!:\!2_{1}//3] [P2_{1}3] [P2_{1}3] [P2_{1}3] [P2_{1}3]  
199 [T^{5}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!2_{1}//3] [I2_{1}3] [I2_{1}3] [I2_{1}3] [I2_{1}3]  
200 [T_{h}^{1}] [(a\!:\!(a\!:\!a))\cdot m/\tilde{6}] Pm3 [P2/m\ \overline{3}] [Pm\overline{3}] [P2/m\ \overline{3}] Pm3 (IT, 1952[link])
201 [T_{h}^{2}] [(a\!:\!(a\!:\!a))\cdot \widetilde{ab}/\tilde{6}] Pn3 [P2/n\ \overline{3}] [Pn\overline{3}] [P2/n\ \overline{3}] Pn3 (IT, 1952[link])
202 [T_{h}^{3}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad\cdot\, m/\tilde{6}\hfill}] Fm3 [F2/m\ \overline{3}] [Fm\overline{3}] [F2/m\ \overline{3}] Fm3 (IT, 1952[link])
203 [T_{h}^{4}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right) \hfill\cr\quad\cdot \,{\textstyle{1 \over 2}}ab/\tilde{6}\hfill}] Fd3 [F2/d\ \overline{3}] [Fd\overline{3}] [F2/d\ \overline{3}] Fd3 (IT, 1952[link])
204 [T_{h}^{5}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\cdot m/\tilde{6}] Im3 [I2/m\ \overline{3}] [Im\overline{3}] [I2/m\ \overline{3}] Im3 (IT, 1952[link])
205 [T_{h}^{6}] [(a\!:\!(a\!:\!a))\cdot \tilde{a}/\tilde{6}] Pa3 [P2_{1}/a\ \overline{3}] [Pa\overline{3}] [P2_{1}/a\ \overline{3}] Pa3 (IT, 1952[link])
206 [T_{h}^{7}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\cdot \tilde{a}/\tilde{6}] Ia3 [I2_{1}/a\ \overline{3}] [Ia\overline{3}] [I2_{1}/a\ \overline{3}] Ia3 (IT, 1952[link])
207 [O^{1}] [(a\!:\!(a\!:\!a))\!:\!4/3] P43 P432 P432 P432  
208 [O^{2}] [(a\!:\!(a\!:\!a))\!:\!4_{2}//3] [P4_{2}3] [P4_{2}32] [P4_{2}32] [P4_{2}32]  
209 [O^{3}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :4/3\hfill}] F43 F432 F432 F432  
210 [O^{4}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :4_{1}//3\hfill}] [F4_{1}3] [F4_{1}32] [F4_{1}32] [F4_{1}32]  
211 [O^{5}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4/3] I43 I432 I432 I432  
212 [O^{6}] [(a\!:\!(a\!:\!a))\!:\!4_{3}//3] [P4_{3}3] [P4_{3}32] [P4_{3}32] [P4_{3}32]  
213 [O^{7}] [(a\!:\!(a\!:\!a))\!:\!4_{1}//3] [P4_{1}3] [P4_{1}32] [P4_{1}32] [P4_{1}32]  
214 [O^{8}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4_{1}//3] [I4_{1}3] [I4_{1}32] [I4_{1}32] [I4_{1}32]  
215 [T_{d}^{1}] [(a\!:\!(a\!:\!a))\!:\!\tilde{4}/3] [P\overline{4}3m] [P\overline{4}3m] [P\overline{4}3m] [P\overline{4}3m]  
216 [T_{d}^{2}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :\tilde{4}/3\hfill}] [F\overline{4}3m] [F\overline{4}3m] [F\overline{4}3m] [F\overline{4}3m]  
217 [T_{d}^{3}] [\displaylines{\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :\tilde{4}/3\hfill}] [I\overline{4}3m] [I\overline{4}3m] [I\overline{4}3m] [I\overline{4}3m]  
218 [T_{d}^{4}] [(a\!:\!(a\!:\!a))\!:\!\tilde{4}//3] [P\overline{4}3n] [P\overline{4}3n] [P\overline{4}3n] [P\overline{4}3n]  
219 [T_{d}^{5}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :\tilde{4}//3\hfill}] [F\overline{4}3c] [F\overline{4}3c] [F\overline{4}3c] [F\overline{4}3c]  
220 [T_{d}^{6}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!\tilde{4}//3] [I\overline{4}3d] [I\overline{4}3d] [I\overline{4}3d] [I\overline{4}3d]  
221 [O_{h}^{1}] [(a\!:\!(a\!:\!a))\!:\!4/\tilde{6}\cdot m] Pm3m [P4/m\ \overline{3}\ 2/m] [Pm\overline{3}m] [P4/m\ \overline{3}\ 2/m] Pm3m (IT, 1952[link])
222 [O_{h}^{2}] [(a\!:\!(a\!:\!a))\!:\!4/\tilde{6}\cdot \widetilde{abc}] Pn3n [P4/n\ \overline{3}\ 2/n] [Pn\overline{3}n] [P4/n\ \overline{3}\ 2/n] Pn3n (IT, 1952[link])
223 [O_{h}^{3}] [(a\!:\!(a\!:\!a))\!:\!4_{2}//\tilde{6}\cdot \widetilde{abc}] Pm3n [P4_{2}/m\ \overline{3}\ 2/n] [Pm\overline{3}n] [P4_{2}/m\ \overline{3}\ 2/n] Pm3n (IT, 1952[link])
224 [O_{h}^{4}] [(a\!:\!(a\!:\!a))\!:\!4_{2}//\tilde{6}\cdot m] Pn3m [P4_{2}/n\ \overline{3}\ 2/m] [Pn\overline{3}m] [P4_{2}/n\ \overline{3}\ 2/m] Pn3m (IT ,1952[link])
225 [O_{h}^{5}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right) \hfill\cr\quad :4/\tilde{6}\cdot m\hfill}] Fm3m [F4/m\ \overline{3}\ 2/m] [Fm\overline{3}m] [F4/m\ \overline{3}\ 2/m] Fm3m (IT, 1952[link])
226 [O_{h}^{6}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr\quad :4/\tilde{6}\cdot \tilde{c}\hfill}] Fm3c [F4/m\ \overline{3}\ 2/c] [Fm\overline{3}c] [F4/m\ \overline{3}\ 2/c] Fm3c (IT, 1952[link])
227 [O_{h}^{7}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a(a\!:\!a)\right)\hfill\cr\quad :\!4_{1}//\tilde{6}\cdot m\hfill\cr}] Fd3m [F4_{1}/d\ \overline{3}\ 2/m] [Fd\overline{3}m] [F4_{1}/d\ \overline{3}\ 2/m] Fd3m (IT, 1952[link])
228 [O_{h}^{8}] [\displaylines{\left(\displaystyle{a + c \over 2}\!\bigg/\!{b + c \over 2}\!\bigg/\!{a + b \over 2}\!:\!a\!:\!(a\!:\!a)\right)\hfill\cr \quad:\!4_{1}//\tilde{6}\cdot \tilde{c}\hfill\cr}] Fd3c [F4_{1}/d\ \overline{3}\ 2/c] [Fd\overline{3}c] [F4_{1}/d\ \overline{3}\ 2/c] Fd3c (IT, 1952[link])
229 [O_{h}^{9}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4/\tilde{6}\cdot m] Im3m [I4/m\ \overline{3}\ 2/m] [Im\overline{3}m] [I4/m\ \overline{3}\ 2/m] Im3m (IT, 1952[link])
230 [O_{h}^{10}] [\left(\displaystyle{a + b + c \over 2}\!\bigg/\!a\!:\!(a\!:\!a)\right)\!:\!4_{1}//\tilde{6}\cdot {1 \over 2}\widetilde{abc}] Ia3d [I4_{1}/a\ \overline{3}\ 2/d] [Ia\overline{3}d] [I4_{1}/a\ \overline{3}\ 2/d] Ia3d (IT, 1952[link])
Abbreviations used in the column Comments: IT, 1952[link]: International Tables for X-ray Crystallography, Vol. I (1952[link]); Sh–K; Shubnikov & Koptsik (1972[link]). Note that this table contains only one notation for the b-unique setting and one notation for the c-unique setting in the monoclinic case, always referring to cell choice 1 of the space-group tables.

Triclinic symbols are unique if the unit cell is primitive. For the standard setting of monoclinic space groups, the lattice symmetry direction is labelled b. From the three possible centrings A, I and C, the latter one is favoured. If glide components occur in the plane perpendicular to [010], the glide direction c is preferred. In the space groups corresponding to the orthorhombic group mm2, the unique direction of the twofold axis is chosen along c. Accordingly, the face centring C is employed for centrings perpendicular to the privileged direction. For space groups with possible A or B centring, the first one is preferred. For groups 222 and mmm, no privileged symmetry direction exists, so the different possibilities of one-face centring can be reduced to C centring by change of the setting. The choices of unit cell and centring type are fixed by the conventional basis in systems with higher symmetry.

When more than one kind of symmetry elements exist in one representative direction, in most cases the choice for the space-group symbol is made in order of decreasing priority: for reflections and glide reflections m, a, b, c, n, d; for proper rotations and screw rotations [6,\ 6_{1},\ 6_{2},\ 6_{3},\ 6_{4},\ 6_{5}]; [4,\ 4_{1},\ 4_{2},\ 4_{3}]; [3,\ 3_{1},\ 3_{2}]; [2,\ 2_{1}] [cf. IT (1952[link]), p. 55, and Section 1.4.1[link] ].


International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Revised editions: 1965, 1969 and 1977. Abbreviated as IT (1952).]
Hermann, C. (1931). Bemerkungen zu der vorstehenden Arbeit von Ch. Mauguin. Z. Kristallogr. 76, 559–561.

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