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List of space groups

None

There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point group of the unit cell.

Symbols

In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions:

P primitive
I body-centered (from the German Innenzentriert)
F face-centered (from the German Flächenzentriert)
S base-centered (from the German Seitenflächenzentriert), or specifically:
- A centered on A faces only
- B centered on B faces only
- C centered on C faces only
R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

a, b, or c: glide translation along half the lattice vector of this face
n: glide translation along half the diagonal of this face
d: glide planes with translation along a quarter of a face diagonal
e: two glides with the same glide plane and translation along two (different) half-lattice vectors.

A gyration point can be replaced by a screw axis denoted by a number, n, where the angle of rotation is \color{Black}\tfrac{360^\circ}{n}. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (twofold) rotation followed by a translation of of the lattice vector. 31 is a 120° (threefold) rotation followed by a translation of of the lattice vector. The possible screw axes are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.

Wherever there is both a rotation or screw axis n and a mirror or glide plane m along the same crystallographic direction, they are represented as a fraction \frac{n}{m} or n/m. For example, 41/a means that the crystallographic axis in question contains both a 41 screw axis as well as a glide plane along a.

In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is sometimes supplemented with a symbol of the form \Gamma_x^y which specifies the Bravais lattice. Here x \in {t, m, o, q, rh, h, c} is the lattice system, and y \in {\empty, b, v, f} is the centering type.

In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups.

Symmorphic

The 73 symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Example for point group 4/mmm (\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}): the symmorphic space groups are P4/mmm (P\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}, 36s) and I4/mmm (I\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}, 37s).

Hemisymmorphic

The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Example for point group 4/mmm (\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}): hemisymmorphic space groups contain the axial combination 422, but at least one mirror plane m will be substituted with glide plane, for example P4/mcc (P\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{c}, 35h), P4/nbm (P\tfrac{4}{n}\tfrac{2}{b}\tfrac{2}{m}, 36h), P4/nnc (P\tfrac{4}{n}\tfrac{2}{n}\tfrac{2}{c}, 37h), and I4/mcm (I\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{m}, 38h).

Asymmorphic

The remaining 103 space groups are asymmorphic. Example for point group 4/mmm (\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}): P4/mbm (P\tfrac{4}{m}\tfrac{2_1}{b}\tfrac{2}{m}, 54a), P42/mmc (P\tfrac{4_2}{m}\tfrac{2}{m}\tfrac{2}{c}, 60a), I41/acd (I\tfrac{4_1}{a}\tfrac{2}{c}\tfrac{2}{d}, 58a) - none of these groups contains the axial combination 422.

List of triclinic

[[File:Triclinic.svg	80px]]

Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
1	1	1	P1	P 1	\Gamma_tC_1^1	1s	(a/b/c)\cdot 1	(\circ)
2		\times	P	P	\Gamma_tC_i^1	2s	(a/b/c)\cdot \tilde 2	(2222)

List of monoclinic

Simple (P)	Base (S)
[[File:Monoclinic.svg	80px]]	[[File:Base-centered monoclinic.svg	80px]]

Number	Point group	Orbifold	Short name	Full name(s)	Schoenflies	Fedorov	Shubnikov	Fibrifold (primary)	Fibrifold (secondary)
3-53	2	22	P2	P 1 2 1	P 1 1 2	\Gamma_mC_2^1	3s	(b:(c/a)):2	(2_02_02_02_0)
4	P21	P 1 21 1	P 1 1 21	\Gamma_mC_2^2	1a	(b:(c/a)):2_1	(2_12_12_12_1)	(\bar{\times}\bar{\times})
5	C2	C 1 2 1	B 1 1 2	\Gamma_m^bC_2^3	4s	\left ( \tfrac{a+b}{2}/b:(c/a)\right ) :2	(2_02_02_12_1)	({}_1{}_1), ({*}\bar{\times})
6-96	m	*	Pm	P 1 m 1	P 1 1 m	\Gamma_mC_s^1	5s	(b:(c/a))\cdot m	[\circ_0]
7	Pc	P 1 c 1	P 1 1 b	\Gamma_mC_s^2	1h	(b:(c/a))\cdot \tilde c	(\bar\circ_0)	({}{:}{}{:}), ({\times}{\times}_0)
8	Cm	C 1 m 1	B 1 1 m	\Gamma_m^bC_s^3	6s	\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m	[\circ_1]	({}{\cdot}{}{:}), ({*}{\cdot}{\times})
9	Cc	C 1 c 1	B 1 1 b	\Gamma_m^bC_s^4	2h	\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c	(\bar\circ_1)	({*}{:}{\times}), ({\times}{\times}_1)
10-1510	2/m	2*	P2/m	P 1 2/m 1	P 1 1 2/m	\Gamma_mC_{2h}^1	7s	(b:(c/a))\cdot m:2	[2_02_02_02_0]
11	P21/m	P 1 21/m 1	P 1 1 21/m	\Gamma_mC_{2h}^2	2a	(b:(c/a))\cdot m:2_1	[2_12_12_12_1]	(22{*}{\cdot})
12	C2/m	C 1 2/m 1	B 1 1 2/m	\Gamma_m^bC_{2h}^3	8s	\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m:2	[2_02_02_12_1]	(2{\cdot}22{:}2), (2\bar{}2{\cdot}2)
13	P2/c	P 1 2/c 1	P 1 1 2/b	\Gamma_mC_{2h}^4	3h	(b:(c/a))\cdot \tilde c:2	(2_02_022)	(2{:}22{:}2), (22{}_0)
14	P21/c	P 1 21/c 1	P 1 1 21/b	\Gamma_mC_{2h}^5	3a	(b:(c/a))\cdot \tilde c:2_1	(2_12_122)	(22{*}{:}), (22{\times})
15	C2/c	C 1 2/c 1	B 1 1 2/b	\Gamma_m^bC_{2h}^6	4h	\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c:2	(2_02_122)	(2\bar{}2{:}2), (22{}_1)

List of orthorhombic

Simple (P)	Body (I)	Face (F)	Base (S)
[[File:Orthorhombic.svg	80px]]	[[File:Orthorhombic-body-centered.svg	80px]]	[[File:Orthorhombic-face-centered.svg	80px]]	[[File:Orthorhombic-base-centered.svg	80px]]

Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold (primary)	Fibrifold (secondary)
16-2416	222	222	P222	P 2 2 2	\Gamma_oD_2^1	9s	(c:a:b):2:2	(*2_02_02_02_0)
17	P2221	P 2 2 21	\Gamma_oD_2^2	4a	(c:a:b):2_1:2	(*2_12_12_12_1)	(2_02_0{*})
18	P21212	P 21 21 2	\Gamma_oD_2^3	7a	(c:a:b):2 [[File:Circled_colon.png	16px]] 2_1	(2_02_0\bar{\times})	(2_12_1{*})
19	P212121	P 21 21 21	\Gamma_oD_2^4	8a	(c:a:b):2_1 [[File:Circled_colon.png	16px]] 2_1	(2_12_1\bar{\times})
20	C2221	C 2 2 21	\Gamma_o^bD_2^5	5a	\left ( \tfrac{a+b}{2}:c:a:b\right ) :2_1:2	(2_1{*}2_12_1)	(2_02_1{*})
21	C222	C 2 2 2	\Gamma_o^bD_2^6	10s	\left ( \tfrac{a+b}{2}:c:a:b\right ) :2:2	(2_0{*}2_02_0)	(*2_02_02_12_1)
22	F222	F 2 2 2	\Gamma_o^fD_2^7	12s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :2:2	(*2_02_12_02_1)
23	I222	I 2 2 2	\Gamma_o^vD_2^8	11s	\left ( \tfrac{a+b+c}{2}/c:a:b\right ) :2:2	(2_1{*}2_02_0)
24	I212121	I 21 21 21	\Gamma_o^vD_2^9	6a	\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :2:2_1	(2_0{*}2_12_1)
25-3525	mm2	*22	Pmm2	P m m 2	\Gamma_oC_{2v}^1	13s	(c:a:b):m \cdot 2	(*{\cdot}2{\cdot}2{\cdot}2{\cdot}2)	[{}_0{\cdot}{}_0{\cdot}]
26	Pmc21	P m c 21	\Gamma_oC_{2v}^2	9a	(c:a:b): \tilde c \cdot 2_1	(*{\cdot}2{:}2{\cdot}2{:}2)	(\bar{}{\cdot}\bar{}{\cdot}), [{\times_0}{\times_0}]
27	Pcc2	P c c 2	\Gamma_oC_{2v}^3	5h	(c:a:b): \tilde c \cdot 2	(*{:}2{:}2{:}2{:}2)	(\bar{}_0\bar{}_0)
28	Pma2	P m a 2	\Gamma_oC_{2v}^4	6h	(c:a:b): \tilde a \cdot 2	(2_02_0{*}{\cdot})	[{}_0{:}{}_0{:}], ({\cdot}{}_0)
29	Pca21	P c a 21	\Gamma_oC_{2v}^5	11a	(c:a:b): \tilde a \cdot 2_1	(2_12_1{*}{:})	(\bar{}{:}\bar{}{:})
30	Pnc2	P n c 2	\Gamma_oC_{2v}^6	7h	(c:a:b): \tilde c \odot 2	(2_02_0{*}{:})	(\bar{}_1\bar{}_1), ({*}_0{\times}_0)
31	Pmn21	P m n 21	\Gamma_oC_{2v}^7	10a	(c:a:b): \widetilde{ac} \cdot 2_1	(2_12_1{*}{\cdot})	(*{\cdot}\bar{\times}), [{\times}_0{\times}_1]
32	Pba2	P b a 2	\Gamma_oC_{2v}^8	9h	(c:a:b): \tilde a \odot 2	(2_02_0{\times}_0)	({:}{}_0)
33	Pna21	P n a 21	\Gamma_oC_{2v}^9	12a	(c:a:b): \tilde a \odot 2_1	(2_12_1{\times})	(*{:}{\times}), ({\times}{\times}_1)
34	Pnn2	P n n 2	\Gamma_oC_{2v}^{10}	8h	(c:a:b): \widetilde{ac} \odot 2	(2_02_0{\times}_1)	(*_0{\times}_1)
35	Cmm2	C m m 2	\Gamma_o^bC_{2v}^{11}	14s	\left ( \tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2	(2_0{*}{\cdot}2{\cdot}2)	[_0{\cdot}{}_0{:}]
36-4636	Cmc21	C m c 21	\Gamma_o^bC_{2v}^{12}	13a	\left ( \tfrac{a+b}{2}:c:a:b\right ) :\tilde c \cdot 2_1	(2_1{*}{\cdot}2{:}2)	(\bar{}{\cdot}\bar{}{:}), [{\times}_1{\times}_1]
37	Ccc2	C c c 2	\Gamma_o^bC_{2v}^{13}	10h	\left ( \tfrac{a+b}{2}:c:a:b\right ) : \tilde c \cdot 2	(2_0{*}{:}2{:}2)	(\bar{}_0\bar{}_1)
38	Amm2	A m m 2	\Gamma_o^bC_{2v}^{14}	15s	\left ( \tfrac{b+c}{2}/c:a:b\right ):m \cdot 2	(*{\cdot}2{\cdot}2{\cdot}2{:}2)	[{}_1{\cdot}{}_1{\cdot}], [*{\cdot}{\times}_0]
39	Aem2	A b m 2	\Gamma_o^bC_{2v}^{15}	11h	\left ( \tfrac{b+c}{2}/c:a:b\right ) :m \cdot 2_1	(*{\cdot}2{:}2{:}2{:}2)	[{}_1{:}{}_1{:}], (\bar{}{\cdot}\bar{}_0)
40	Ama2	A m a 2	\Gamma_o^bC_{2v}^{16}	12h	\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2	(2_02_1{*}{\cdot})	({\cdot}{}_1), [*{:}{\times}_1]
41	Aea2	A b a 2	\Gamma_o^bC_{2v}^{17}	13h	\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2_1	(2_02_1{*}{:})	({:}{}_1), (\bar{}{:}\bar{}_1)
42	Fmm2	F m m 2	\Gamma_o^fC_{2v}^{18}	17s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2	(*{\cdot}2{\cdot}2{:}2{:}2)	[{}_1{\cdot}{}_1{:}]
43	Fdd2	F d d 2	\Gamma_o^fC_{2v}^{19}	16h	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b \right ) : \tfrac{1}{2} \widetilde{ac} \odot 2	(2_02_1{\times})	({*}_1{\times})
44	Imm2	I m m 2	\Gamma_o^vC_{2v}^{20}	16s	\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :m \cdot 2	(2_1{*}{\cdot}2{\cdot}2)	[*{\cdot}{\times}_1]
45	Iba2	I b a 2	\Gamma_o^vC_{2v}^{21}	15h	\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde c \cdot 2	(2_1{*}{:}2{:}2)	(\bar{}{:}\bar{}_0)
46	Ima2	I m a 2	\Gamma_o^vC_{2v}^{22}	14h	\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde a \cdot 2	(2_0{*}{\cdot}2{:}2)	(\bar{}{\cdot}\bar{}_1), [*{:}{\times}_0]
47-5547	\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}	*222	Pmmm	P 2/m 2/m 2/m	\Gamma_oD_{2h}^1	18s	\left ( c:a:b \right ) \cdot m:2 \cdot m	[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]
48	Pnnn	P 2/n 2/n 2/n	\Gamma_oD_{2h}^2	19h	\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \widetilde{ac}	(2\bar{*}_12_02_0)
49	Pccm	P 2/c 2/c 2/m	\Gamma_oD_{2h}^3	17h	\left ( c:a:b \right ) \cdot m:2 \cdot \tilde c	[*{:}2{:}2{:}2{:}2]	(*2_02_02{\cdot}2)
50	Pban	P 2/b 2/a 2/n	\Gamma_oD_{2h}^4	18h	\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \tilde a	(2\bar{*}_02_02_0)	(*2_02_02{:}2)
51	Pmma	P 21/m 2/m 2/a	\Gamma_oD_{2h}^5	14a	\left ( c:a:b \right ) \cdot \tilde a :2 \cdot m	[2_02_0{*}{\cdot}]	[{\cdot}2{:}2{\cdot}2{:}2], [2{\cdot}2{\cdot}2{\cdot}2]
52	Pnna	P 2/n 21/n 2/a	\Gamma_oD_{2h}^6	17a	\left ( c:a:b \right ) \cdot \tilde a:2 \odot \widetilde{ac}	(2_02\bar{*}_1)	(2_0{}2{:}2), (2\bar{}2_12_1)
53	Pmna	P 2/m 2/n 21/a	\Gamma_oD_{2h}^7	15a	\left ( c:a:b \right ) \cdot \tilde a:2_1 \cdot \widetilde{ac}	[2_02_0{*}{:}]	(2_12_12{\cdot}2), (2_0{}2{\cdot}2)
54	Pcca	P 21/c 2/c 2/a	\Gamma_oD_{2h}^8	16a	\left ( c:a:b \right ) \cdot \tilde a:2 \cdot \tilde c	(2_02\bar{*}_0)	(2{:}2{:}2{:}2), (2_12_12{:}2)
55	Pbam	P 21/b 21/a 2/m	\Gamma_oD_{2h}^9	22a	\left ( c:a:b \right ) \cdot m:2 \odot \tilde a	[2_02_0{\times}_0]	(*2{\cdot}2{:}2{\cdot}2)
56-6456	Pccn	P 21/c 21/c 2/n	\Gamma_oD_{2h}^{10}	27a	\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot \tilde c	(2\bar{*}{:}2{:}2)	(2_12\bar{*}_0)
57	Pbcm	P 2/b 21/c 21/m	\Gamma_oD_{2h}^{11}	23a	\left ( c:a:b \right ) \cdot m:2_1 \odot \tilde c	(2_02\bar{*}{\cdot})	(2{:}2{\cdot}2{:}2), [2_12_1{}{:}]
58	Pnnm	P 21/n 21/n 2/m	\Gamma_oD_{2h}^{12}	25a	\left ( c:a:b \right ) \cdot m:2 \odot \widetilde{ac}	[2_02_0{\times}_1]	(2_1{*}2{\cdot}2)
59	Pmmn	P 21/m 21/m 2/n	\Gamma_oD_{2h}^{13}	24a	\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot m	(2\bar{*}{\cdot}2{\cdot}2)	[2_12_1{*}{\cdot}]
60	Pbcn	P 21/b 2/c 21/n	\Gamma_oD_{2h}^{14}	26a	\left ( c:a:b \right ) \cdot \widetilde{ab}:2_1 \odot \tilde c	(2_02\bar{*}{:})	(2_1{}2{:}2), (2_12\bar{}_1)
61	Pbca	P 21/b 21/c 21/a	\Gamma_oD_{2h}^{15}	29a	\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot \tilde c	(2_12\bar{*}{:})
62	Pnma	P 21/n 21/m 21/a	\Gamma_oD_{2h}^{16}	28a	\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot m	(2_12\bar{*}{\cdot})	(2\bar{*}{\cdot}2{:}2), [2_12_1{\times}]
63	Cmcm	C 2/m 2/c 21/m	\Gamma_o^bD_{2h}^{17}	18a	\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2_1 \cdot \tilde c	[2_02_1{*}{\cdot}]	(2{\cdot}2{\cdot}2{:}2), [2_1{}{\cdot}2{:}2]
64	Cmce	C 2/m 2/c 21/a	\Gamma_o^bD_{2h}^{18}	19a	\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2_1 \cdot \tilde c	[2_02_1{*}{:}]	(2{\cdot}2{:}2{:}2), (2_12{\cdot}2{:}2)
65-7465	Cmmm	C 2/m 2/m 2/m	\Gamma_o^bD_{2h}^{19}	19s	\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m	[2_0{*}{\cdot}2{\cdot}2]	[*{\cdot}2{\cdot}2{\cdot}2{:}2]
66	Cccm	C 2/c 2/c 2/m	\Gamma_o^bD_{2h}^{20}	20h	\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot \tilde c	[2_0{*}{:}2{:}2]	(*2_02_12{\cdot}2)
67	Cmme	C 2/m 2/m 2/e	\Gamma_o^bD_{2h}^{21}	21h	\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot m	(*2_02{\cdot}2{\cdot}2)	[*{\cdot}2{:}2{:}2{:}2]
68	Ccce	C 2/c 2/c 2/e	\Gamma_o^bD_{2h}^{22}	22h	\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c	(*2_02{:}2{:}2)	(*2_02_12{:}2)
69	Fmmm	F 2/m 2/m 2/m	\Gamma_o^fD_{2h}^{23}	21s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m	[*{\cdot}2{\cdot}2{:}2{:}2]
70	Fddd	F 2/d 2/d 2/d	\Gamma_o^fD_{2h}^{24}	24h	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot \tfrac{1}{2}\widetilde{ab}:2 \odot \tfrac{1}{2}\widetilde{ac}	(2\bar{*}2_02_1)
71	Immm	I 2/m 2/m 2/m	\Gamma_o^vD_{2h}^{25}	20s	\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot m	[2_1{*}{\cdot}2{\cdot}2]
72	Ibam	I 2/b 2/a 2/m	\Gamma_o^vD_{2h}^{26}	23h	\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot \tilde c	[2_1{*}{:}2{:}2]	(*2_02{\cdot}2{:}2)
73	Ibca	I 2/b 2/c 2/a	\Gamma_o^vD_{2h}^{27}	21a	\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c	(*2_12{:}2{:}2)
74	Imma	I 2/m 2/m 2/a	\Gamma_o^vD_{2h}^{28}	20a	\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot m	(*2_12{\cdot}2{\cdot}2)	[2_0{*}{\cdot}2{:}2]

List of tetragonal

Simple (P)	Body (I)
[[File:Tetragonal.svg	80px]]	[[File:Tetragonal-body-centered.svg	80px]]

Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
75-8075	4	44	P4	P 4	\Gamma_qC_4^1	22s	(c:a:a):4	(4_04_02_0)
76	P41	P 41	\Gamma_qC_4^2	30a	(c:a:a) :4_1	(4_14_12_1)
77	P42	P 42	\Gamma_qC_4^3	33a	(c:a:a) :4_2	(4_24_22_0)
78	P43	P 43	\Gamma_qC_4^4	31a	(c:a:a) :4_3	(4_14_12_1)
79	I4	I 4	\Gamma_q^vC_4^5	23s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4	(4_24_02_1)
80	I41	I 41	\Gamma_q^vC_4^6	32a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1	(4_34_12_0)
81-8281		2\times	P	P	\Gamma_qS_4^1	26s	(c:a:a):\tilde 4	(442_0)
82	I	I	\Gamma_q^vS_4^2	27s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4	(442_1)
83-8883	4/m	4*	P4/m	P 4/m	\Gamma_qC_{4h}^1	28s	(c:a:a)\cdot m:4	[4_04_02_0]
84	P42/m	P 42/m	\Gamma_qC_{4h}^2	41a	(c:a:a)\cdot m:4_2	[4_24_22_0]
85	P4/n	P 4/n	\Gamma_qC_{4h}^3	29h	(c:a:a)\cdot \widetilde{ab}:4	(44_02)
86	P42/n	P 42/n	\Gamma_qC_{4h}^4	42a	(c:a:a)\cdot \widetilde{ab}:4_2	(44_22)
87	I4/m	I 4/m	\Gamma_q^vC_{4h}^5	29s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4	[4_24_02_1]
88	I41/a	I 41/a	\Gamma_q^vC_{4h}^6	40a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1	(44_12)
89-9889	422	224	P422	P 4 2 2	\Gamma_qD_4^1	30s	(c:a:a):4:2	(*4_04_02_0)
90	P4212	P4212	\Gamma_qD_4^2	43a	(c:a:a):4 [[File:circled_colon.png	16px]] 2_1	(4_0{*}2_0)
91	P4122	P 41 2 2	\Gamma_qD_4^3	44a	(c:a:a):4_1:2	(*4_14_12_1)
92	P41212	P 41 21 2	\Gamma_qD_4^4	48a	(c:a:a):4_1 [[File:circled_colon.png	16px]] 2_1	(4_1{*}2_1)
93	P4222	P 42 2 2	\Gamma_qD_4^5	47a	(c:a:a):4_2:2	(*4_24_22_0)
94	P42212	P 42 21 2	\Gamma_qD_4^6	50a	(c:a:a):4_2 [[File:circled_colon.png	16px]] 2_1	(4_2{*}2_0)
95	P4322	P 43 2 2	\Gamma_qD_4^7	45a	(c:a:a):4_3:2	(*4_14_12_1)
96	P43212	P 43 21 2	\Gamma_qD_4^8	49a	(c:a:a):4_3 [[File:circled_colon.png	16px]] 2_1	(4_1{*}2_1)
97	I422	I 4 2 2	\Gamma_q^vD_4^9	31s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2	(*4_24_02_1)
98	I4122	I 41 2 2	\Gamma_q^vD_4^{10}	46a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2_1	(*4_34_12_0)
99-11099	4mm	*44	P4mm	P 4 m m	\Gamma_qC_{4v}^1	24s	(c:a:a):4\cdot m	(*{\cdot}4{\cdot}4{\cdot}2)
100	P4bm	P 4 b m	\Gamma_qC_{4v}^2	26h	(c:a:a):4\odot \tilde a	(4_0{*}{\cdot}2)
101	P42cm	P 42 c m	\Gamma_qC_{4v}^3	37a	(c:a:a):4_2\cdot \tilde c	(*{:}4{\cdot}4{:}2)
102	P42nm	P 42 n m	\Gamma_qC_{4v}^4	38a	(c:a:a):4_2\odot \widetilde{ac}	(4_2{*}{\cdot}2)
103	P4cc	P 4 c c	\Gamma_qC_{4v}^5	25h	(c:a:a):4\cdot \tilde c	(*{:}4{:}4{:}2)
104	P4nc	P 4 n c	\Gamma_qC_{4v}^6	27h	(c:a:a):4\odot \widetilde{ac}	(4_0{*}{:}2)
105	P42mc	P 42 m c	\Gamma_qC_{4v}^7	36a	(c:a:a):4_2\cdot m	(*{\cdot}4{:}4{\cdot}2)
106	P42bc	P 42 b c	\Gamma_qC_{4v}^8	39a	(c:a:a):4\odot \tilde a	(4_2{*}{:}2)
107	I4mm	I 4 m m	\Gamma_q^vC_{4v}^9	25s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot m	(*{\cdot}4{\cdot}4{:}2)
108	I4cm	I 4 c m	\Gamma_q^vC_{4v}^{10}	28h	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot \tilde c	(*{\cdot}4{:}4{:}2)
109	I41md	I 41 m d	\Gamma_q^vC_{4v}^{11}	34a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot m	(4_1{*}{\cdot}2)
110	I41cd	I 41 c d	\Gamma_q^vC_{4v}^{12}	35a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot \tilde c	(4_1{*}{:}2)
111-122111	2m	2{*}2	P2m	P 2 m	\Gamma_qD_{2d}^1	32s	(c:a:a):\tilde 4 :2	(*4{\cdot}42_0)
112	P2c	P 2 c	\Gamma_qD_{2d}^2	30h	(c:a:a):\tilde 4 [[File:circled_colon.png	16px]] 2	(*4{:}42_0)
113	P21m	P 21 m	\Gamma_qD_{2d}^3	52a	(c:a:a):\tilde 4 \cdot \widetilde{ab}	(4\bar{*}{\cdot}2)
114	P21c	P 21 c	\Gamma_qD_{2d}^4	53a	(c:a:a):\tilde 4 \cdot \widetilde{abc}	(4\bar{*}{:}2)
115	Pm2	P m 2	\Gamma_qD_{2d}^5	33s	(c:a:a):\tilde 4 \cdot m	(*{\cdot}44{\cdot}2)
116	Pc2	P c 2	\Gamma_qD_{2d}^6	31h	(c:a:a):\tilde 4 \cdot \tilde c	(*{:}44{:}2)
117	Pb2	P b 2	\Gamma_qD_{2d}^7	32h	(c:a:a):\tilde 4 \odot \tilde a	(4\bar{*}_02_0)
118	Pn2	P n 2	\Gamma_qD_{2d}^8	33h	(c:a:a):\tilde 4 \cdot \widetilde{ac}	(4\bar{*}_12_0)
119	Im2	I m 2	\Gamma_q^vD_{2d}^9	35s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot m	(*4{\cdot}42_1)
120	Ic2	I c 2	\Gamma_q^vD_{2d}^{10}	34h	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot \tilde c	(*4{:}42_1)
121	I2m	I 2 m	\Gamma_q^vD_{2d}^{11}	34s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 :2	(*{\cdot}44{:}2)
122	I2d	I 2 d	\Gamma_q^vD_{2d}^{12}	51a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \odot \tfrac{1}{2}\widetilde{abc}	(4\bar{*}2_1)
123-132123	4/m 2/m 2/m	*224	P4/mmm	P 4/m 2/m 2/m	\Gamma_qD_{4h}^1	36s	(c:a:a)\cdot m:4\cdot m	[*{\cdot}4{\cdot}4{\cdot}2]
124	P4/mcc	P 4/m 2/c 2/c	\Gamma_qD_{4h}^2	35h	(c:a:a)\cdot m:4\cdot \tilde c	[*{:}4{:}4{:}2]
125	P4/nbm	P 4/n 2/b 2/m	\Gamma_qD_{4h}^3	36h	(c:a:a)\cdot \widetilde{ab}:4\odot \tilde a	(*4_04{\cdot}2)
126	P4/nnc	P 4/n 2/n 2/c	\Gamma_qD_{4h}^4	37h	(c:a:a)\cdot \widetilde{ab}:4\odot \widetilde{ac}	(*4_04{:}2)
127	P4/mbm	P 4/m 21/b 2/m	\Gamma_qD_{4h}^5	54a	(c:a:a)\cdot m:4\odot \tilde a	[4_0{*}{\cdot}2]
128	P4/mnc	P 4/m 21/n 2/c	\Gamma_qD_{4h}^6	56a	(c:a:a)\cdot m:4\odot \widetilde{ac}	[4_0{*}{:}2]
129	P4/nmm	P 4/n 21/m 2/m	\Gamma_qD_{4h}^7	55a	(c:a:a)\cdot \widetilde{ab}:4\cdot m	(*4{\cdot}4{\cdot}2)
130	P4/ncc	P 4/n 21/c 2/c	\Gamma_qD_{4h}^8	57a	(c:a:a)\cdot \widetilde{ab}:4\cdot \tilde c	(*4{:}4{:}2)
131	P42/mmc	P 42/m 2/m 2/c	\Gamma_qD_{4h}^9	60a	(c:a:a)\cdot m:4_2\cdot m	[*{\cdot}4{:}4{\cdot}2]
132	P42/mcm	P 42/m 2/c 2/m	\Gamma_qD_{4h}^{10}	61a	(c:a:a)\cdot m:4_2\cdot \tilde c	[*{:}4{\cdot}4{:}2]
133-142133	P42/nbc	P 42/n 2/b 2/c	\Gamma_qD_{4h}^{11}	63a	(c:a:a)\cdot \widetilde{ab}:4_2\odot \tilde a	(*4_24{:}2)
134	P42/nnm	P 42/n 2/n 2/m	\Gamma_qD_{4h}^{12}	62a	(c:a:a)\cdot \widetilde{ab}:4_2\odot \widetilde{ac}	(*4_24{\cdot}2)
135	P42/mbc	P 42/m 21/b 2/c	\Gamma_qD_{4h}^{13}	66a	(c:a:a)\cdot m:4_2\odot \tilde a	[4_2{*}{:}2]
136	P42/mnm	P 42/m 21/n 2/m	\Gamma_qD_{4h}^{14}	65a	(c:a:a)\cdot m:4_2\odot \widetilde{ac}	[4_2{*}{\cdot}2]
137	P42/nmc	P 42/n 21/m 2/c	\Gamma_qD_{4h}^{15}	67a	(c:a:a)\cdot \widetilde{ab}:4_2\cdot m	(*4{\cdot}4{:}2)
138	P42/ncm	P 42/n 21/c 2/m	\Gamma_qD_{4h}^{16}	65a	(c:a:a)\cdot \widetilde{ab}:4_2\cdot \tilde c	(*4{:}4{\cdot}2)
139	I4/mmm	I 4/m 2/m 2/m	\Gamma_q^vD_{4h}^{17}	37s	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot m	[*{\cdot}4{\cdot}4{:}2]
140	I4/mcm	I 4/m 2/c 2/m	\Gamma_q^vD_{4h}^{18}	38h	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot \tilde c	[*{\cdot}4{:}4{:}2]
141	I41/amd	I 41/a 2/m 2/d	\Gamma_q^vD_{4h}^{19}	59a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot m	(*4_14{\cdot}2)
142	I41/acd	I 41/a 2/c 2/d	\Gamma_q^vD_{4h}^{20}	58a	\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot \tilde c	(*4_14{:}2)

List of trigonal

Rhombohedral (R)	Hexagonal (P)
[[File:Hexagonal latticeR.svg	100px]]	[[File:Hexagonal latticeFRONT.svg	100px]]

Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
143-146143	3	33	P3	P 3	\Gamma_hC_3^1	38s	(c:(a/a)):3	(3_03_03_0)
144	P31	P 31	\Gamma_hC_3^2	68a	(c:(a/a)):3_1	(3_13_13_1)
145	P32	P 32	\Gamma_hC_3^3	69a	(c:(a/a)):3_2	(3_13_13_1)
146	R3	R 3	\Gamma_{rh}C_3^4	39s	(a/a/a)/3	(3_03_13_2)
147-148147		3\times	P	P	\Gamma_hC_{3i}^1	51s	(c:(a/a)):\tilde 6	(63_02)
148	R	R	\Gamma_{rh}C_{3i}^2	52s	(a/a/a)/\tilde 6	(63_12)
149-155149	32	223	P312	P 3 1 2	\Gamma_hD_3^1	45s	(c:(a/a)):2:3	(*3_03_03_0)
150	P321	P 3 2 1	\Gamma_hD_3^2	44s	(c:(a/a))\cdot 2:3	(3_0{*}3_0)
151	P3112	P 31 1 2	\Gamma_hD_3^3	72a	(c:(a/a)):2:3_1	(*3_13_13_1)
152	P3121	P 31 2 1	\Gamma_hD_3^4	70a	(c:(a/a))\cdot 2:3_1	(3_1{*}3_1)
153	P3212	P 32 1 2	\Gamma_hD_3^5	73a	(c:(a/a)):2:3_2	(*3_13_13_1)
154	P3221	P 32 2 1	\Gamma_hD_3^6	71a	(c:(a/a))\cdot 2:3_2	(3_1{*}3_1)
155	R32	R 3 2	\Gamma_{rh}D_3^7	46s	(a/a/a)/3:2	(*3_03_13_2)
156-161156	3m	*33	P3m1	P 3 m 1	\Gamma_hC_{3v}^1	40s	(c:(a/a)):m\cdot 3	(*{\cdot}3{\cdot}3{\cdot}3)
157	P31m	P 3 1 m	\Gamma_hC_{3v}^2	41s	(c:(a/a))\cdot m\cdot 3	(3_0{*}{\cdot}3)
158	P3c1	P 3 c 1	\Gamma_hC_{3v}^3	39h	(c:(a/a)):\tilde c:3	(*{:}3{:}3{:}3)
159	P31c	P 3 1 c	\Gamma_hC_{3v}^4	40h	(c:(a/a))\cdot\tilde c :3	(3_0{*}{:}3)
160	R3m	R 3 m	\Gamma_{rh}C_{3v}^5	42s	(a/a/a)/3\cdot m	(3_1{*}{\cdot}3)
161	R3c	R 3 c	\Gamma_{rh}C_{3v}^6	41h	(a/a/a)/3\cdot\tilde c	(3_1{*}{:}3)
162-167162	2/m	2{*}3	P1m	P 1 2/m	\Gamma_hD_{3d}^1	56s	(c:(a/a))\cdot m\cdot\tilde 6	(*{\cdot}63_02)
163	P1c	P 1 2/c	\Gamma_hD_{3d}^2	46h	(c:(a/a))\cdot\tilde c \cdot\tilde 6	(*{:}63_02)
164	Pm1	P 2/m 1	\Gamma_hD_{3d}^3	55s	(c:(a/a)):m\cdot\tilde 6	(*6{\cdot}3{\cdot}2)
165	Pc1	P 2/c 1	\Gamma_hD_{3d}^4	45h	(c:(a/a)):\tilde c \cdot\tilde 6	(*6{:}3{:}2)
166	Rm	R 2/m	\Gamma_{rh}D_{3d}^5	57s	(a/a/a)/\tilde 6 \cdot m	(*{\cdot}63_12)
167	Rc	R 2/c	\Gamma_{rh}D_{3d}^6	47h	(a/a/a)/\tilde 6 \cdot\tilde c	(*{:}63_12)

List of hexagonal

[[File:Hexagonal latticeFRONT.svg	80px]]

Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
168-173168	6	66	P6	P 6	\Gamma_hC_6^1	49s	(c:(a/a)):6	(6_03_02_0)
169	P61	P 61	\Gamma_hC_6^2	74a	(c:(a/a)):6_1	(6_13_12_1)
170	P65	P 65	\Gamma_hC_6^3	75a	(c:(a/a)):6_5	(6_13_12_1)
171	P62	P 62	\Gamma_hC_6^4	76a	(c:(a/a)):6_2	(6_23_22_0)
172	P64	P 64	\Gamma_hC_6^5	77a	(c:(a/a)):6_4	(6_23_22_0)
173	P63	P 63	\Gamma_hC_6^6	78a	(c:(a/a)):6_3	(6_33_02_1)
174-176174		3*	P	P	\Gamma_hC_{3h}^1	43s	(c:(a/a)):3:m	[3_03_03_0]
175	6/m	6*	P6/m	P 6/m	\Gamma_hC_{6h}^1	53s	(c:(a/a))\cdot m :6	[6_03_02_0]
176	P63/m	P 63/m	\Gamma_hC_{6h}^2	81a	(c:(a/a))\cdot m :6_3	[6_33_02_1]
177-182177	622	226	P622	P 6 2 2	\Gamma_hD_6^1	54s	(c:(a/a))\cdot 2 :6	(*6_03_02_0)
178	P6122	P 61 2 2	\Gamma_hD_6^2	82a	(c:(a/a))\cdot 2 :6_1	(*6_13_12_1)
179	P6522	P 65 2 2	\Gamma_hD_6^3	83a	(c:(a/a))\cdot 2 :6_5	(*6_13_12_1)
180	P6222	P 62 2 2	\Gamma_hD_6^4	84a	(c:(a/a))\cdot 2 :6_2	(*6_23_22_0)
181	P6422	P 64 2 2	\Gamma_hD_6^5	85a	(c:(a/a))\cdot 2 :6_4	(*6_23_22_0)
182	P6322	P 63 2 2	\Gamma_hD_6^6	86a	(c:(a/a))\cdot 2 :6_3	(*6_33_02_1)
183-186183	6mm	*66	P6mm	P 6 m m	\Gamma_hC_{6v}^1	50s	(c:(a/a)):m\cdot 6	(*{\cdot}6{\cdot}3{\cdot}2)
184	P6cc	P 6 c c	\Gamma_hC_{6v}^2	44h	(c:(a/a)):\tilde c \cdot 6	(*{:}6{:}3{:}2)
185	P63cm	P 63 c m	\Gamma_hC_{6v}^3	80a	(c:(a/a)):\tilde c \cdot 6_3	(*{\cdot}6{:}3{:}2)
186	P63mc	P 63 m c	\Gamma_hC_{6v}^4	79a	(c:(a/a)):m\cdot 6_3	(*{:}6{\cdot}3{\cdot}2)
187-190187	m2	*223	Pm2	P m 2	\Gamma_hD_{3h}^1	48s	(c:(a/a)):m\cdot 3:m	[*{\cdot}3{\cdot}3{\cdot}3]
188	Pc2	P c 2	\Gamma_hD_{3h}^2	43h	(c:(a/a)):\tilde c \cdot 3:m	[*{:}3{:}3{:}3]
189	P2m	P 2 m	\Gamma_hD_{3h}^3	47s	(c:(a/a))\cdot m:3\cdot m	[3_0{*}{\cdot}3]
190	P2c	P 2 c	\Gamma_hD_{3h}^4	42h	(c:(a/a))\cdot m:3\cdot \tilde c	[3_0{*}{:}3]
191-194191	6/m 2/m 2/m	*226	P6/mmm	P 6/m 2/m 2/m	\Gamma_hD_{6h}^1	58s	(c:(a/a))\cdot m:6\cdot m	[*{\cdot}6{\cdot}3{\cdot}2]
192	P6/mcc	P 6/m 2/c 2/c	\Gamma_hD_{6h}^2	48h	(c:(a/a))\cdot m:6\cdot\tilde c	[*{:}6{:}3{:}2]
193	P63/mcm	P 63/m 2/c 2/m	\Gamma_hD_{6h}^3	87a	(c:(a/a))\cdot m:6_3\cdot\tilde c	[*{\cdot}6{:}3{:}2]
194	P63/mmc	P 63/m 2/m 2/c	\Gamma_hD_{6h}^4	88a	(c:(a/a))\cdot m:6_3\cdot m	[*{:}6{\cdot}3{\cdot}2]

List of cubic

Simple (P)	Body centered (I)	Face centered (F)
[[File:Cubic.svg	100px]]	[[File:Cubic-body-centered.svg	100px]]	[[File:Cubic-face-centered.svg	100px]]

Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Conway	Fibrifold (preserving z)	Fibrifold (preserving x, y, z)
195-199195	23	332	P23	P 2 3	\Gamma_cT^1	59s	\left ( a:a:a\right ) :2/3	2^\circ	(*2_02_02_02_0){:}3	(*2_02_02_02_0){:}3
196	F23	F 2 3	\Gamma_c^fT^2	61s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :2/3	1^\circ	(*2_02_12_02_1){:}3	(*2_02_12_02_1){:}3
197	I23	I 2 3	\Gamma_c^vT^3	60s	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2/3	4^{\circ\circ}	(2_1{*}2_02_0){:}3	(2_1{*}2_02_0){:}3
198	P213	P 21 3	\Gamma_cT^4	89a	\left ( a:a:a\right ) :2_1/3	1^\circ/4	(2_12_1\bar{\times}){:}3	(2_12_1\bar{\times}){:}3
199	I213	I 21 3	\Gamma_c^vT^5	90a	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2_1/3	2^\circ/4	(2_0{*}2_12_1){:}3	(2_0{*}2_12_1){:}3
200-206200	2/m	3{*}2	Pm	P 2/m	\Gamma_cT_h^1	62s	\left ( a:a:a\right ) \cdot m/ \tilde 6	4^-	[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}3	[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}3
201	Pn	P 2/n	\Gamma_cT_h^2	49h	\left ( a:a:a\right ) \cdot \widetilde{ab} / \tilde 6	4^{\circ+}	(2\bar{*}_12_02_0){:}3	(2\bar{*}_12_02_0){:}3
202	Fm	F 2/m	\Gamma_c^fT_h^3	64s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot m/ \tilde 6	2^-	[*{\cdot}2{\cdot}2{:}2{:}2]{:}3	[*{\cdot}2{\cdot}2{:}2{:}2]{:}3
203	Fd	F 2/d	\Gamma_c^fT_h^4	50h	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot \tfrac{1}{2}\widetilde{ab} / \tilde 6	2^{\circ+}	(2\bar{*}2_02_1){:}3	(2\bar{*}2_02_1){:}3
204	Im	I 2/m	\Gamma_c^vT_h^5	63s	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot m/\tilde 6	8^{-\circ}	[2_1{*}{\cdot}2{\cdot}2]{:}3	[2_1{*}{\cdot}2{\cdot}2]{:}3
205	Pa	P 21/a	\Gamma_cT_h^6	91a	\left ( a:a:a\right ) \cdot \tilde a /\tilde 6	2^-/4	(2_12\bar{*}{:}){:}3	(2_12\bar{*}{:}){:}3
206	Ia	I 21/a	\Gamma_c^vT_h^7	92a	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot \tilde a /\tilde 6	4^-/4	(*2_12{:}2{:}2){:}3	(*2_12{:}2{:}2){:}3
207-214207	432	432	P432	P 4 3 2	\Gamma_cO^1	68s	\left ( a:a:a\right ) :4/3	4^{\circ-}	(*4_04_02_0){:}3	(*2_02_02_02_0){:}6
208	P4232	P 42 3 2	\Gamma_cO^2	98a	\left ( a:a:a\right ) :4_2//3	4^+	(*4_24_22_0){:}3	(*2_02_02_02_0){:}6
209	F432	F 4 3 2	\Gamma_c^fO^3	70s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/3	2^{\circ-}	(*4_24_02_1){:}3	(*2_02_12_02_1){:}6
210	F4132	F 41 3 2	\Gamma_c^fO^4	97a	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//3	2^+	(*4_34_12_0){:}3	(*2_02_12_02_1){:}6
211	I432	I 4 3 2	\Gamma_c^vO^5	69s	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/3	8^{+\circ}	(4_24_02_1){:}3	(2_1{*}2_02_0){:}6
212	P4332	P 43 3 2	\Gamma_cO^6	94a	\left ( a:a:a\right ) :4_3//3	2^+/4	(4_1{*}2_1){:}3	(2_12_1\bar{\times}){:}6
213	P4132	P 41 3 2	\Gamma_cO^7	95a	\left ( a:a:a\right ) :4_1//3	2^+/4	(4_1{*}2_1){:}3	(2_12_1\bar{\times}){:}6
214	I4132	I 41 3 2	\Gamma_c^vO^8	96a	\left ( \tfrac{a+b+c}{2}/:a:a:a\right ) :4_1//3	4^+/4	(*4_34_12_0){:}3	(2_0{*}2_12_1){:}6
215-220215	3m	*332	P3m	P 3 m	\Gamma_cT_d^1	65s	\left ( a:a:a\right ) :\tilde 4 /3	2^\circ{:}2	(*4{\cdot}42_0){:}3	(*2_02_02_02_0){:}6
216	F3m	F 3 m	\Gamma_c^fT_d^2	67s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 /3	1^\circ{:}2	(*4{\cdot}42_1){:}3	(*2_02_12_02_1){:}6
217	I3m	I 3 m	\Gamma_c^vT_d^3	66s	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 /3	4^\circ{:}2	(*{\cdot}44{:}2){:}3	(2_1{*}2_02_0){:}6
218	P3n	P 3 n	\Gamma_cT_d^4	51h	\left ( a:a:a\right ) :\tilde 4 //3	4^\circ	(*4{:}42_0){:}3	(*2_02_02_02_0){:}6
219	F3c	F 3 c	\Gamma_c^fT_d^5	52h	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 //3	2^{\circ\circ}	(*4{:}42_1){:}3	(*2_02_12_02_1){:}6
220	I3d	I 3 d	\Gamma_c^vT_d^6	93a	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 //3	4^\circ/4	(4\bar{*}2_1){:}3	(2_0{*}2_12_1){:}6
221-230221	4/m 2/m	*432	Pmm	P 4/m 2/m	\Gamma_cO_h^1	71s	\left ( a:a:a\right ) :4/\tilde 6 \cdot m	4^-{:}2	[*{\cdot}4{\cdot}4{\cdot}2]{:}3	[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}6
222	Pnn	P 4/n 2/n	\Gamma_cO_h^2	53h	\left ( a:a:a\right ) :4/\tilde 6 \cdot \widetilde{abc}	8^{\circ\circ}	(*4_04{:}2){:}3	(2\bar{*}_12_02_0){:}6
223	Pmn	P 42/m 2/n	\Gamma_cO_h^3	102a	\left ( a:a:a\right ) :4_2//\tilde 6 \cdot \widetilde{abc}	8^\circ	[*{\cdot}4{:}4{\cdot}2]{:}3	[*{\cdot}2{\cdot}2{\cdot}2{\cdot}2]{:}6
224	Pnm	P 42/n 2/m	\Gamma_cO_h^4	103a	\left ( a:a:a\right ) :4_2//\tilde 6 \cdot m	4^+{:}2	(*4_24{\cdot}2){:}3	(2\bar{*}_12_02_0){:}6
225	Fmm	F 4/m 2/m	\Gamma_c^fO_h^5	73s	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot m	2^-{:}2	[*{\cdot}4{\cdot}4{:}2]{:}3	[*{\cdot}2{\cdot}2{:}2{:}2]{:}6
226	Fmc	F 4/m 2/c	\Gamma_c^fO_h^6	54h	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot \tilde c	4^{--}	[*{\cdot}4{:}4{:}2]{:}3	[*{\cdot}2{\cdot}2{:}2{:}2]{:}6
227	Fdm	F 41/d 2/m	\Gamma_c^fO_h^7	100a	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot m	2^+{:}2	(*4_14{\cdot}2){:}3	(2\bar{*}2_02_1){:}6
228	Fdc	F 41/d 2/c	\Gamma_c^fO_h^8	101a	\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot \tilde c	4^{++}	(*4_14{:}2){:}3	(2\bar{*}2_02_1){:}6
229	Imm	I 4/m 2/m	\Gamma_c^vO_h^9	72s	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/\tilde 6 \cdot m	8^\circ{:}2	[*{\cdot}4{\cdot}4{:}2]{:}3	[2_1{*}{\cdot}2{\cdot}2]{:}6
230	Iad	I 41/a 2/d	\Gamma_c^vO_h^{10}	99a	\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4_1//\tilde 6 \cdot \tfrac{1}{2}\widetilde{abc}	8^\circ/4	(*4_14{:}2){:}3	(*2_12{:}2{:}2){:}6

Notes

References

(1992-09-01). "Symbols for symmetry elements and symmetry operations. Final report of the IUCr Ad-Hoc Committee on the Nomenclature of Symmetry". Acta Crystallographica Section A.
(2010). "The mathematical theory of symmetry in solids: representation theory for point groups and space groups". Clarendon Press.

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