List_of_space_groups Knowpia

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 edit

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)
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.^{[note 1]}

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 ½ of the lattice vector. 3₁ is a 120° (threefold) rotation followed by a translation of ⅓ 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 ${\textstyle {\frac {n}{m}}}$ 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 $y\in \{\emptyset ,b,v,f\}$ is the centering type.^[2]

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 edit

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, for example, the space groups 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 edit

The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Hemisymmorphic space groups contain the axial combination 422, which are 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 edit

The remaining 103 space groups are asymmorphic, for example, those derived from the point group 4/mmm ( ${\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ).

List of triclinic edit

Triclinic Bravais lattice

Triclinic crystal system
Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
1	1	$1$	P1	P 1	$\Gamma _{t}C_{1}^{1}$	1s	$(a/b/c)\cdot 1$	$(\circ )$
2	1	$\times$	P1	P 1	$\Gamma _{t}C_{i}^{1}$	2s	$(a/b/c)\cdot {\tilde {2}}$	$(2222)$

List of monoclinic edit

Monoclinic Bravais lattice
Simple (P)	Base (C)

Monoclinic crystal system
Number	Point group	Orbifold	Short name	Full name(s)		Schoenflies	Fedorov	Shubnikov	Fibrifold (primary)	Fibrifold (secondary)
3	2	$22$	P2	P 1 2 1	P 1 1 2	$\Gamma _{m}C_{2}^{1}$	3s	$(b:(c/a)):2$	$(2_{0}2_{0}2_{0}2_{0})$	$({}_{0}{}_{0})$
4			P2₁	P 1 2₁ 1	P 1 1 2₁	$\Gamma _{m}C_{2}^{2}$	1a	$(b:(c/a)):2_{1}$	$(2_{1}2_{1}2_{1}2_{1})$	$({\bar {\times }}{\bar {\times }})$
5			C2	C 1 2 1	B 1 1 2	$\Gamma _{m}^{b}C_{2}^{3}$	4s	$\left({\tfrac {a+b}{2}}/b:(c/a)\right):2$	$(2_{0}2_{0}2_{1}2_{1})$	$({}_{1}{}_{1})$ , $({*}{\bar {\times }})$
6	m	$*$	Pm	P 1 m 1	P 1 1 m	$\Gamma _{m}C_{s}^{1}$	5s	$(b:(c/a))\cdot m$	$[\circ _{0}]$	$({}{\cdot }{}{\cdot })$
7			Pc	P 1 c 1	P 1 1 b	$\Gamma _{m}C_{s}^{2}$	1h	$(b:(c/a))\cdot {\tilde {c}}$	$({\bar {\circ }}_{0})$	$({}{:}{}{:})$ , $({\times }{\times }_{0})$
8			Cm	C 1 m 1	B 1 1 m	$\Gamma _{m}^{b}C_{s}^{3}$	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	$\Gamma _{m}^{b}C_{s}^{4}$	2h	$\left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot {\tilde {c}}$	$({\bar {\circ }}_{1})$	$({*}{:}{\times })$ , $({\times }{\times }_{1})$
10	2/m	$2*$	P2/m	P 1 2/m 1	P 1 1 2/m	$\Gamma _{m}C_{2h}^{1}$	7s	$(b:(c/a))\cdot m:2$	$[2_{0}2_{0}2_{0}2_{0}]$	$[*2{\cdot }22{\cdot }2)$
11			P2₁/m	P 1 2₁/m 1	P 1 1 2₁/m	$\Gamma _{m}C_{2h}^{2}$	2a	$(b:(c/a))\cdot m:2_{1}$	$[2_{1}2_{1}2_{1}2_{1}]$	$(22{*}{\cdot })$
12			C2/m	C 1 2/m 1	B 1 1 2/m	$\Gamma _{m}^{b}C_{2h}^{3}$	8s	$\left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot m:2$	$[2_{0}2_{0}2_{1}2_{1}]$	$(2{\cdot }22{:}2)$ , $(2{\bar {}}2{\cdot }2)$
13			P2/c	P 1 2/c 1	P 1 1 2/b	$\Gamma _{m}C_{2h}^{4}$	3h	$(b:(c/a))\cdot {\tilde {c}}:2$	$(2_{0}2_{0}22)$	$(2{:}22{:}2)$ , $(22{}_{0})$
14			P2₁/c	P 1 2₁/c 1	P 1 1 2₁/b	$\Gamma _{m}C_{2h}^{5}$	3a	$(b:(c/a))\cdot {\tilde {c}}:2_{1}$	$(2_{1}2_{1}22)$	$(22{*}{:})$ , $(22{\times })$
15			C2/c	C 1 2/c 1	B 1 1 2/b	$\Gamma _{m}^{b}C_{2h}^{6}$	4h	$\left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot {\tilde {c}}:2$	$(2_{0}2_{1}22)$	$(2{\bar {}}2{:}2)$ , $(22{}_{1})$

List of orthorhombic edit

Orthorhombic Bravais lattice
Simple (P)	Body (I)	Face (F)	Base (A or C)

Orthorhombic crystal system
Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold (primary)	Fibrifold (secondary)
16	222	$222$	P222	P 2 2 2	$\Gamma _{o}D_{2}^{1}$	9s	$(c:a:b):2:2$	$(*2_{0}2_{0}2_{0}2_{0})$
17			P222₁	P 2 2 2₁	$\Gamma _{o}D_{2}^{2}$	4a	$(c:a:b):2_{1}:2$	$(*2_{1}2_{1}2_{1}2_{1})$	$(2_{0}2_{0}{*})$
18			P2₁2₁2	P 2₁ 2₁ 2	$\Gamma _{o}D_{2}^{3}$	7a	$(c:a:b):2$ $2_{1}$	$(2_{0}2_{0}{\bar {\times }})$	$(2_{1}2_{1}{*})$
19			P2₁2₁2₁	P 2₁ 2₁ 2₁	$\Gamma _{o}D_{2}^{4}$	8a	$(c:a:b):2_{1}$ $2_{1}$	$(2_{1}2_{1}{\bar {\times }})$
20			C222₁	C 2 2 2₁	$\Gamma _{o}^{b}D_{2}^{5}$	5a	$\left({\tfrac {a+b}{2}}:c:a:b\right):2_{1}:2$	$(2_{1}{*}2_{1}2_{1})$	$(2_{0}2_{1}{*})$
21			C222	C 2 2 2	$\Gamma _{o}^{b}D_{2}^{6}$	10s	$\left({\tfrac {a+b}{2}}:c:a:b\right):2:2$	$(2_{0}{*}2_{0}2_{0})$	$(*2_{0}2_{0}2_{1}2_{1})$
22			F222	F 2 2 2	$\Gamma _{o}^{f}D_{2}^{7}$	12s	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):2:2$	$(*2_{0}2_{1}2_{0}2_{1})$
23			I222	I 2 2 2	$\Gamma _{o}^{v}D_{2}^{8}$	11s	$\left({\tfrac {a+b+c}{2}}/c:a:b\right):2:2$	$(2_{1}{*}2_{0}2_{0})$
24			I2₁2₁2₁	I 2₁ 2₁ 2₁	$\Gamma _{o}^{v}D_{2}^{9}$	6a	$\left({\tfrac {a+b+c}{2}}/c:a:b\right):2:2_{1}$	$(2_{0}{*}2_{1}2_{1})$
25	mm2	$*22$	Pmm2	P m m 2	$\Gamma _{o}C_{2v}^{1}$	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	$\Gamma _{o}C_{2v}^{2}$	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	$\Gamma _{o}C_{2v}^{3}$	5h	$(c:a:b):{\tilde {c}}\cdot 2$	$(*{:}2{:}2{:}2{:}2)$	$({\bar {}}_{0}{\bar {}}_{0})$
28			Pma2	P m a 2	$\Gamma _{o}C_{2v}^{4}$	6h	$(c:a:b):{\tilde {a}}\cdot 2$	$(2_{0}2_{0}{*}{\cdot })$	$[{}_{0}{:}{}_{0}{:}]$ , $({\cdot }{}_{0})$
29			Pca2₁	P c a 2₁	$\Gamma _{o}C_{2v}^{5}$	11a	$(c:a:b):{\tilde {a}}\cdot 2_{1}$	$(2_{1}2_{1}{*}{:})$	$({\bar {}}{:}{\bar {}}{:})$
30			Pnc2	P n c 2	$\Gamma _{o}C_{2v}^{6}$	7h	$(c:a:b):{\tilde {c}}\odot 2$	$(2_{0}2_{0}{*}{:})$	$({\bar {}}_{1}{\bar {}}_{1})$ , $({*}_{0}{\times }_{0})$
31			Pmn2₁	P m n 2₁	$\Gamma _{o}C_{2v}^{7}$	10a	$(c:a:b):{\widetilde {ac}}\cdot 2_{1}$	$(2_{1}2_{1}{*}{\cdot })$	$(*{\cdot }{\bar {\times }})$ , $[{\times }_{0}{\times }_{1}]$
32			Pba2	P b a 2	$\Gamma _{o}C_{2v}^{8}$	9h	$(c:a:b):{\tilde {a}}\odot 2$	$(2_{0}2_{0}{\times }_{0})$	$({:}{}_{0})$
33			Pna2₁	P n a 2₁	$\Gamma _{o}C_{2v}^{9}$	12a	$(c:a:b):{\tilde {a}}\odot 2_{1}$	$(2_{1}2_{1}{\times })$	$(*{:}{\times })$ , $({\times }{\times }_{1})$
34			Pnn2	P n n 2	$\Gamma _{o}C_{2v}^{10}$	8h	$(c:a:b):{\widetilde {ac}}\odot 2$	$(2_{0}2_{0}{\times }_{1})$	$(*_{0}{\times }_{1})$
35			Cmm2	C m m 2	$\Gamma _{o}^{b}C_{2v}^{11}$	14s	$\left({\tfrac {a+b}{2}}:c:a:b\right):m\cdot 2$	$(2_{0}{*}{\cdot }2{\cdot }2)$	$[_{0}{\cdot }{}_{0}{:}]$
36			Cmc2₁	C m c 2₁	$\Gamma _{o}^{b}C_{2v}^{12}$	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	$\Gamma _{o}^{b}C_{2v}^{13}$	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	$\Gamma _{o}^{b}C_{2v}^{14}$	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	$\Gamma _{o}^{b}C_{2v}^{15}$	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	$\Gamma _{o}^{b}C_{2v}^{16}$	12h	$\left({\tfrac {b+c}{2}}/c:a:b\right):{\tilde {a}}\cdot 2$	$(2_{0}2_{1}{*}{\cdot })$	$({\cdot }{}_{1})$ , $[*{:}{\times }_{1}]$
41			Aea2	A b a 2	$\Gamma _{o}^{b}C_{2v}^{17}$	13h	$\left({\tfrac {b+c}{2}}/c:a:b\right):{\tilde {a}}\cdot 2_{1}$	$(2_{0}2_{1}{*}{:})$	$({:}{}_{1})$ , $({\bar {}}{:}{\bar {}}_{1})$
42			Fmm2	F m m 2	$\Gamma _{o}^{f}C_{2v}^{18}$	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 dd2	$\Gamma _{o}^{f}C_{2v}^{19}$	16h	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):{\tfrac {1}{2}}{\widetilde {ac}}\odot 2$	$(2_{0}2_{1}{\times })$	$({*}_{1}{\times })$
44			Imm2	I m m 2	$\Gamma _{o}^{v}C_{2v}^{20}$	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	$\Gamma _{o}^{v}C_{2v}^{21}$	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	$\Gamma _{o}^{v}C_{2v}^{22}$	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	${\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	$*222$	Pmmm	P 2/m 2/m 2/m	$\Gamma _{o}D_{2h}^{1}$	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	$\Gamma _{o}D_{2h}^{2}$	19h	$\left(c:a:b\right)\cdot {\widetilde {ab}}:2\odot {\widetilde {ac}}$	$(2{\bar {*}}_{1}2_{0}2_{0})$
49			Pccm	P 2/c 2/c 2/m	$\Gamma _{o}D_{2h}^{3}$	17h	$\left(c:a:b\right)\cdot m:2\cdot {\tilde {c}}$	$[*{:}2{:}2{:}2{:}2]$	$(*2_{0}2_{0}2{\cdot }2)$
50			Pban	P 2/b 2/a 2/n	$\Gamma _{o}D_{2h}^{4}$	18h	$\left(c:a:b\right)\cdot {\widetilde {ab}}:2\odot {\tilde {a}}$	$(2{\bar {*}}_{0}2_{0}2_{0})$	$(*2_{0}2_{0}2{:}2)$
51			Pmma	P 2₁/m 2/m 2/a	$\Gamma _{o}D_{2h}^{5}$	14a	$\left(c:a:b\right)\cdot {\tilde {a}}:2\cdot m$	$[2_{0}2_{0}{*}{\cdot }]$	$[{\cdot }2{:}2{\cdot }2{:}2]$ , $[2{\cdot }2{\cdot }2{\cdot }2]$
52			Pnna	P 2/n 2₁/n 2/a	$\Gamma _{o}D_{2h}^{6}$	17a	$\left(c:a:b\right)\cdot {\tilde {a}}:2\odot {\widetilde {ac}}$	$(2_{0}2{\bar {*}}_{1})$	$(2_{0}{}2{:}2)$ , $(2{\bar {}}2_{1}2_{1})$
53			Pmna	P 2/m 2/n 2₁/a	$\Gamma _{o}D_{2h}^{7}$	15a	$\left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\cdot {\widetilde {ac}}$	$[2_{0}2_{0}{*}{:}]$	$(2_{1}2_{1}2{\cdot }2)$ , $(2_{0}{}2{\cdot }2)$
54			Pcca	P 2₁/c 2/c 2/a	$\Gamma _{o}D_{2h}^{8}$	16a	$\left(c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}$	$(2_{0}2{\bar {*}}_{0})$	$(2{:}2{:}2{:}2)$ , $(2_{1}2_{1}2{:}2)$
55			Pbam	P 2₁/b 2₁/a 2/m	$\Gamma _{o}D_{2h}^{9}$	22a	$\left(c:a:b\right)\cdot m:2\odot {\tilde {a}}$	$[2_{0}2_{0}{\times }_{0}]$	$(*2{\cdot }2{:}2{\cdot }2)$
56			Pccn	P 2₁/c 2₁/c 2/n	$\Gamma _{o}D_{2h}^{10}$	27a	$\left(c:a:b\right)\cdot {\widetilde {ab}}:2\cdot {\tilde {c}}$	$(2{\bar {*}}{:}2{:}2)$	$(2_{1}2{\bar {*}}_{0})$
57			Pbcm	P 2/b 2₁/c 2₁/m	$\Gamma _{o}D_{2h}^{11}$	23a	$\left(c:a:b\right)\cdot m:2_{1}\odot {\tilde {c}}$	$(2_{0}2{\bar {*}}{\cdot })$	$(2{:}2{\cdot }2{:}2)$ , $[2_{1}2_{1}{}{:}]$
58			Pnnm	P 2₁/n 2₁/n 2/m	$\Gamma _{o}D_{2h}^{12}$	25a	$\left(c:a:b\right)\cdot m:2\odot {\widetilde {ac}}$	$[2_{0}2_{0}{\times }_{1}]$	$(2_{1}{*}2{\cdot }2)$
59			Pmmn	P 2₁/m 2₁/m 2/n	$\Gamma _{o}D_{2h}^{13}$	24a	$\left(c:a:b\right)\cdot {\widetilde {ab}}:2\cdot m$	$(2{\bar {*}}{\cdot }2{\cdot }2)$	$[2_{1}2_{1}{*}{\cdot }]$
60			Pbcn	P 2₁/b 2/c 2₁/n	$\Gamma _{o}D_{2h}^{14}$	26a	$\left(c:a:b\right)\cdot {\widetilde {ab}}:2_{1}\odot {\tilde {c}}$	$(2_{0}2{\bar {*}}{:})$	$(2_{1}{}2{:}2)$ , $(2_{1}2{\bar {}}_{1})$
61			Pbca	P 2₁/b 2₁/c 2₁/a	$\Gamma _{o}D_{2h}^{15}$	29a	$\left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\odot {\tilde {c}}$	$(2_{1}2{\bar {*}}{:})$
62			Pnma	P 2₁/n 2₁/m 2₁/a	$\Gamma _{o}D_{2h}^{16}$	28a	$\left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\odot m$	$(2_{1}2{\bar {*}}{\cdot })$	$(2{\bar {*}}{\cdot }2{:}2)$ , $[2_{1}2_{1}{\times }]$
63			Cmcm	C 2/m 2/c 2₁/m	$\Gamma _{o}^{b}D_{2h}^{17}$	18a	$\left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2_{1}\cdot {\tilde {c}}$	$[2_{0}2_{1}{*}{\cdot }]$	$(2{\cdot }2{\cdot }2{:}2)$ , $[2_{1}{}{\cdot }2{:}2]$
64			Cmce	C 2/m 2/c 2₁/a	$\Gamma _{o}^{b}D_{2h}^{18}$	19a	$\left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2_{1}\cdot {\tilde {c}}$	$[2_{0}2_{1}{*}{:}]$	$(2{\cdot }2{:}2{:}2)$ , $(2_{1}2{\cdot }2{:}2)$
65			Cmmm	C 2/m 2/m 2/m	$\Gamma _{o}^{b}D_{2h}^{19}$	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	$\Gamma _{o}^{b}D_{2h}^{20}$	20h	$\left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2\cdot {\tilde {c}}$	$[2_{0}{*}{:}2{:}2]$	$(*2_{0}2_{1}2{\cdot }2)$
67			Cmme	C 2/m 2/m 2/e	$\Gamma _{o}^{b}D_{2h}^{21}$	21h	$\left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2\cdot m$	$(*2_{0}2{\cdot }2{\cdot }2)$	$[*{\cdot }2{:}2{:}2{:}2]$
68			Ccce	C 2/c 2/c 2/e	$\Gamma _{o}^{b}D_{2h}^{22}$	22h	$\left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}$	$(*2_{0}2{:}2{:}2)$	$(*2_{0}2_{1}2{:}2)$
69			Fmmm	F 2/m 2/m 2/m	$\Gamma _{o}^{f}D_{2h}^{23}$	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	$\Gamma _{o}^{f}D_{2h}^{24}$	24h	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tfrac {1}{2}}{\widetilde {ab}}:2\odot {\tfrac {1}{2}}{\widetilde {ac}}$	$(2{\bar {*}}2_{0}2_{1})$
71			Immm	I 2/m 2/m 2/m	$\Gamma _{o}^{v}D_{2h}^{25}$	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	$\Gamma _{o}^{v}D_{2h}^{26}$	23h	$\left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot m:2\cdot {\tilde {c}}$	$[2_{1}{*}{:}2{:}2]$	$(*2_{0}2{\cdot }2{:}2)$
73			Ibca	I 2/b 2/c 2/a	$\Gamma _{o}^{v}D_{2h}^{27}$	21a	$\left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}$	$(*2_{1}2{:}2{:}2)$
74			Imma	I 2/m 2/m 2/a	$\Gamma _{o}^{v}D_{2h}^{28}$	20a	$\left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot {\tilde {a}}:2\cdot m$	$(*2_{1}2{\cdot }2{\cdot }2)$	$[2_{0}{*}{\cdot }2{:}2]$

List of tetragonal edit

Tetragonal Bravais lattice
Simple (P)	Body (I)

Tetragonal crystal system
Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
75	4	$44$	P4	P 4	$\Gamma _{q}C_{4}^{1}$	22s	$(c:a:a):4$	$(4_{0}4_{0}2_{0})$
76			P4₁	P 4₁	$\Gamma _{q}C_{4}^{2}$	30a	$(c:a:a):4_{1}$	$(4_{1}4_{1}2_{1})$
77			P4₂	P 4₂	$\Gamma _{q}C_{4}^{3}$	33a	$(c:a:a):4_{2}$	$(4_{2}4_{2}2_{0})$
78			P4₃	P 4₃	$\Gamma _{q}C_{4}^{4}$	31a	$(c:a:a):4_{3}$	$(4_{1}4_{1}2_{1})$
79			I4	I 4	$\Gamma _{q}^{v}C_{4}^{5}$	23s	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):4$	$(4_{2}4_{0}2_{1})$
80			I4₁	I 4₁	$\Gamma _{q}^{v}C_{4}^{6}$	32a	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}$	$(4_{3}4_{1}2_{0})$
81	4	$2\times$	P4	P 4	$\Gamma _{q}S_{4}^{1}$	26s	$(c:a:a):{\tilde {4}}$	$(442_{0})$
82	4	$2\times$	I4	I 4	$\Gamma _{q}^{v}S_{4}^{2}$	27s	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}$	$(442_{1})$
83	4/m	$4*$	P4/m	P 4/m	$\Gamma _{q}C_{4h}^{1}$	28s	$(c:a:a)\cdot m:4$	$[4_{0}4_{0}2_{0}]$
84			P4₂/m	P 4₂/m	$\Gamma _{q}C_{4h}^{2}$	41a	$(c:a:a)\cdot m:4_{2}$	$[4_{2}4_{2}2_{0}]$
85			P4/n	P 4/n	$\Gamma _{q}C_{4h}^{3}$	29h	$(c:a:a)\cdot {\widetilde {ab}}:4$	$(44_{0}2)$
86			P4₂/n	P 4₂/n	$\Gamma _{q}C_{4h}^{4}$	42a	$(c:a:a)\cdot {\widetilde {ab}}:4_{2}$	$(44_{2}2)$
87			I4/m	I 4/m	$\Gamma _{q}^{v}C_{4h}^{5}$	29s	$\left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot m:4$	$[4_{2}4_{0}2_{1}]$
88			I4₁/a	I 4₁/a	$\Gamma _{q}^{v}C_{4h}^{6}$	40a	$\left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}$	$(44_{1}2)$
89	422	$224$	P422	P 4 2 2	$\Gamma _{q}D_{4}^{1}$	30s	$(c:a:a):4:2$	$(*4_{0}4_{0}2_{0})$
90			P42₁2	P42₁2	$\Gamma _{q}D_{4}^{2}$	43a	$(c:a:a):4$ $2_{1}$	$(4_{0}{*}2_{0})$
91			P4₁22	P 4₁ 2 2	$\Gamma _{q}D_{4}^{3}$	44a	$(c:a:a):4_{1}:2$	$(*4_{1}4_{1}2_{1})$
92			P4₁2₁2	P 4₁ 2₁ 2	$\Gamma _{q}D_{4}^{4}$	48a	$(c:a:a):4_{1}$ $2_{1}$	$(4_{1}{*}2_{1})$
93			P4₂22	P 4₂ 2 2	$\Gamma _{q}D_{4}^{5}$	47a	$(c:a:a):4_{2}:2$	$(*4_{2}4_{2}2_{0})$
94			P4₂2₁2	P 4₂ 2₁ 2	$\Gamma _{q}D_{4}^{6}$	50a	$(c:a:a):4_{2}$ $2_{1}$	$(4_{2}{*}2_{0})$
95			P4₃22	P 4₃ 2 2	$\Gamma _{q}D_{4}^{7}$	45a	$(c:a:a):4_{3}:2$	$(*4_{1}4_{1}2_{1})$
96			P4₃2₁2	P 4₃ 2₁ 2	$\Gamma _{q}D_{4}^{8}$	49a	$(c:a:a):4_{3}$ $2_{1}$	$(4_{1}{*}2_{1})$
97			I422	I 4 2 2	$\Gamma _{q}^{v}D_{4}^{9}$	31s	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):4:2$	$(*4_{2}4_{0}2_{1})$
98			I4₁22	I 4₁ 2 2	$\Gamma _{q}^{v}D_{4}^{10}$	46a	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):4:2_{1}$	$(*4_{3}4_{1}2_{0})$
99	4mm	$*44$	P4mm	P 4 m m	$\Gamma _{q}C_{4v}^{1}$	24s	$(c:a:a):4\cdot m$	$(*{\cdot }4{\cdot }4{\cdot }2)$
100			P4bm	P 4 b m	$\Gamma _{q}C_{4v}^{2}$	26h	$(c:a:a):4\odot {\tilde {a}}$	$(4_{0}{*}{\cdot }2)$
101			P4₂cm	P 4₂ c m	$\Gamma _{q}C_{4v}^{3}$	37a	$(c:a:a):4_{2}\cdot {\tilde {c}}$	$(*{:}4{\cdot }4{:}2)$
102			P4₂nm	P 4₂ n m	$\Gamma _{q}C_{4v}^{4}$	38a	$(c:a:a):4_{2}\odot {\widetilde {ac}}$	$(4_{2}{*}{\cdot }2)$
103			P4cc	P 4 c c	$\Gamma _{q}C_{4v}^{5}$	25h	$(c:a:a):4\cdot {\tilde {c}}$	$(*{:}4{:}4{:}2)$
104			P4nc	P 4 n c	$\Gamma _{q}C_{4v}^{6}$	27h	$(c:a:a):4\odot {\widetilde {ac}}$	$(4_{0}{*}{:}2)$
105			P4₂mc	P 4₂ m c	$\Gamma _{q}C_{4v}^{7}$	36a	$(c:a:a):4_{2}\cdot m$	$(*{\cdot }4{:}4{\cdot }2)$
106			P4₂bc	P 4₂ b c	$\Gamma _{q}C_{4v}^{8}$	39a	$(c:a:a):4\odot {\tilde {a}}$	$(4_{2}{*}{:}2)$
107			I4mm	I 4 m m	$\Gamma _{q}^{v}C_{4v}^{9}$	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	$\Gamma _{q}^{v}C_{4v}^{10}$	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	$\Gamma _{q}^{v}C_{4v}^{11}$	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	$\Gamma _{q}^{v}C_{4v}^{12}$	35a	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}\odot {\tilde {c}}$	$(4_{1}{*}{:}2)$
111	42m	$2{*}2$	P42m	P 4 2 m	$\Gamma _{q}D_{2d}^{1}$	32s	$(c:a:a):{\tilde {4}}:2$	$(*4{\cdot }42_{0})$
112			P42c	P 4 2 c	$\Gamma _{q}D_{2d}^{2}$	30h	$(c:a:a):{\tilde {4}}$ $2$	$(*4{:}42_{0})$
113			P42₁m	P 4 2₁ m	$\Gamma _{q}D_{2d}^{3}$	52a	$(c:a:a):{\tilde {4}}\cdot {\widetilde {ab}}$	$(4{\bar {*}}{\cdot }2)$
114			P42₁c	P 4 2₁ c	$\Gamma _{q}D_{2d}^{4}$	53a	$(c:a:a):{\tilde {4}}\cdot {\widetilde {abc}}$	$(4{\bar {*}}{:}2)$
115			P4m2	P 4 m 2	$\Gamma _{q}D_{2d}^{5}$	33s	$(c:a:a):{\tilde {4}}\cdot m$	$(*{\cdot }44{\cdot }2)$
116			P4c2	P 4 c 2	$\Gamma _{q}D_{2d}^{6}$	31h	$(c:a:a):{\tilde {4}}\cdot {\tilde {c}}$	$(*{:}44{:}2)$
117			P4b2	P 4 b 2	$\Gamma _{q}D_{2d}^{7}$	32h	$(c:a:a):{\tilde {4}}\odot {\tilde {a}}$	$(4{\bar {*}}_{0}2_{0})$
118			P4n2	P 4 n 2	$\Gamma _{q}D_{2d}^{8}$	33h	$(c:a:a):{\tilde {4}}\cdot {\widetilde {ac}}$	$(4{\bar {*}}_{1}2_{0})$
119			I4m2	I 4 m 2	$\Gamma _{q}^{v}D_{2d}^{9}$	35s	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\cdot m$	$(*4{\cdot }42_{1})$
120			I4c2	I 4 c 2	$\Gamma _{q}^{v}D_{2d}^{10}$	34h	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\cdot {\tilde {c}}$	$(*4{:}42_{1})$
121			I42m	I 4 2 m	$\Gamma _{q}^{v}D_{2d}^{11}$	34s	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}:2$	$(*{\cdot }44{:}2)$
122			I42d	I 4 2 d	$\Gamma _{q}^{v}D_{2d}^{12}$	51a	$\left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\odot {\tfrac {1}{2}}{\widetilde {abc}}$	$(4{\bar {*}}2_{1})$
123	4/m 2/m 2/m	$*224$	P4/mmm	P 4/m 2/m 2/m	$\Gamma _{q}D_{4h}^{1}$	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	$\Gamma _{q}D_{4h}^{2}$	35h	$(c:a:a)\cdot m:4\cdot {\tilde {c}}$	$[*{:}4{:}4{:}2]$
125			P4/nbm	P 4/n 2/b 2/m	$\Gamma _{q}D_{4h}^{3}$	36h	$(c:a:a)\cdot {\widetilde {ab}}:4\odot {\tilde {a}}$	$(*4_{0}4{\cdot }2)$
126			P4/nnc	P 4/n 2/n 2/c	$\Gamma _{q}D_{4h}^{4}$	37h	$(c:a:a)\cdot {\widetilde {ab}}:4\odot {\widetilde {ac}}$	$(*4_{0}4{:}2)$
127			P4/mbm	P 4/m 2₁/b 2/m	$\Gamma _{q}D_{4h}^{5}$	54a	$(c:a:a)\cdot m:4\odot {\tilde {a}}$	$[4_{0}{*}{\cdot }2]$
128			P4/mnc	P 4/m 2₁/n 2/c	$\Gamma _{q}D_{4h}^{6}$	56a	$(c:a:a)\cdot m:4\odot {\widetilde {ac}}$	$[4_{0}{*}{:}2]$
129			P4/nmm	P 4/n 2₁/m 2/m	$\Gamma _{q}D_{4h}^{7}$	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	$\Gamma _{q}D_{4h}^{8}$	57a	$(c:a:a)\cdot {\widetilde {ab}}:4\cdot {\tilde {c}}$	$(*4{:}4{:}2)$
131			P4₂/mmc	P 4₂/m 2/m 2/c	$\Gamma _{q}D_{4h}^{9}$	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	$\Gamma _{q}D_{4h}^{10}$	61a	$(c:a:a)\cdot m:4_{2}\cdot {\tilde {c}}$	$[*{:}4{\cdot }4{:}2]$
133			P4₂/nbc	P 4₂/n 2/b 2/c	$\Gamma _{q}D_{4h}^{11}$	63a	$(c:a:a)\cdot {\widetilde {ab}}:4_{2}\odot {\tilde {a}}$	$(*4_{2}4{:}2)$
134			P4₂/nnm	P 4₂/n 2/n 2/m	$\Gamma _{q}D_{4h}^{12}$	62a	$(c:a:a)\cdot {\widetilde {ab}}:4_{2}\odot {\widetilde {ac}}$	$(*4_{2}4{\cdot }2)$
135			P4₂/mbc	P 4₂/m 2₁/b 2/c	$\Gamma _{q}D_{4h}^{13}$	66a	$(c:a:a)\cdot m:4_{2}\odot {\tilde {a}}$	$[4_{2}{*}{:}2]$
136			P4₂/mnm	P 4₂/m 2₁/n 2/m	$\Gamma _{q}D_{4h}^{14}$	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	$\Gamma _{q}D_{4h}^{15}$	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	$\Gamma _{q}D_{4h}^{16}$	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	$\Gamma _{q}^{v}D_{4h}^{17}$	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	$\Gamma _{q}^{v}D_{4h}^{18}$	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	$\Gamma _{q}^{v}D_{4h}^{19}$	59a	$\left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}\odot m$	$(*4_{1}4{\cdot }2)$
142			I4₁/acd	I 4₁/a 2/c 2/d	$\Gamma _{q}^{v}D_{4h}^{20}$	58a	$\left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}\odot {\tilde {c}}$	$(*4_{1}4{:}2)$

List of trigonal edit

Trigonal Bravais lattice
Rhombohedral (R)	Hexagonal (P)

Trigonal crystal system
Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
143	3	$33$	P3	P 3	$\Gamma _{h}C_{3}^{1}$	38s	$(c:(a/a)):3$	$(3_{0}3_{0}3_{0})$
144			P3₁	P 3₁	$\Gamma _{h}C_{3}^{2}$	68a	$(c:(a/a)):3_{1}$	$(3_{1}3_{1}3_{1})$
145			P3₂	P 3₂	$\Gamma _{h}C_{3}^{3}$	69a	$(c:(a/a)):3_{2}$	$(3_{1}3_{1}3_{1})$
146			R3	R 3	$\Gamma _{rh}C_{3}^{4}$	39s	$(a/a/a)/3$	$(3_{0}3_{1}3_{2})$
147	3	$3\times$	P3	P 3	$\Gamma _{h}C_{3i}^{1}$	51s	$(c:(a/a)):{\tilde {6}}$	$(63_{0}2)$
148	3	$3\times$	R3	R 3	$\Gamma _{rh}C_{3i}^{2}$	52s	$(a/a/a)/{\tilde {6}}$	$(63_{1}2)$
149	32	$223$	P312	P 3 1 2	$\Gamma _{h}D_{3}^{1}$	45s	$(c:(a/a)):2:3$	$(*3_{0}3_{0}3_{0})$
150			P321	P 3 2 1	$\Gamma _{h}D_{3}^{2}$	44s	$(c:(a/a))\cdot 2:3$	$(3_{0}{*}3_{0})$
151			P3₁12	P 3₁ 1 2	$\Gamma _{h}D_{3}^{3}$	72a	$(c:(a/a)):2:3_{1}$	$(*3_{1}3_{1}3_{1})$
152			P3₁21	P 3₁ 2 1	$\Gamma _{h}D_{3}^{4}$	70a	$(c:(a/a))\cdot 2:3_{1}$	$(3_{1}{*}3_{1})$
153			P3₂12	P 3₂ 1 2	$\Gamma _{h}D_{3}^{5}$	73a	$(c:(a/a)):2:3_{2}$	$(*3_{1}3_{1}3_{1})$
154			P3₂21	P 3₂ 2 1	$\Gamma _{h}D_{3}^{6}$	71a	$(c:(a/a))\cdot 2:3_{2}$	$(3_{1}{*}3_{1})$
155			R32	R 3 2	$\Gamma _{rh}D_{3}^{7}$	46s	$(a/a/a)/3:2$	$(*3_{0}3_{1}3_{2})$
156	3m	$*33$	P3m1	P 3 m 1	$\Gamma _{h}C_{3v}^{1}$	40s	$(c:(a/a)):m\cdot 3$	$(*{\cdot }3{\cdot }3{\cdot }3)$
157			P31m	P 3 1 m	$\Gamma _{h}C_{3v}^{2}$	41s	$(c:(a/a))\cdot m\cdot 3$	$(3_{0}{*}{\cdot }3)$
158			P3c1	P 3 c 1	$\Gamma _{h}C_{3v}^{3}$	39h	$(c:(a/a)):{\tilde {c}}:3$	$(*{:}3{:}3{:}3)$
159			P31c	P 3 1 c	$\Gamma _{h}C_{3v}^{4}$	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	3 2/m	$2{*}3$	P31m	P 3 1 2/m	$\Gamma _{h}D_{3d}^{1}$	56s	$(c:(a/a))\cdot m\cdot {\tilde {6}}$	$(*{\cdot }63_{0}2)$
163			P31c	P 3 1 2/c	$\Gamma _{h}D_{3d}^{2}$	46h	$(c:(a/a))\cdot {\tilde {c}}\cdot {\tilde {6}}$	$(*{:}63_{0}2)$
164			P3m1	P 3 2/m 1	$\Gamma _{h}D_{3d}^{3}$	55s	$(c:(a/a)):m\cdot {\tilde {6}}$	$(*6{\cdot }3{\cdot }2)$
165			P3c1	P 3 2/c 1	$\Gamma _{h}D_{3d}^{4}$	45h	$(c:(a/a)):{\tilde {c}}\cdot {\tilde {6}}$	$(*6{:}3{:}2)$
166			R3m	R 3 2/m	$\Gamma _{rh}D_{3d}^{5}$	57s	$(a/a/a)/{\tilde {6}}\cdot m$	$(*{\cdot }63_{1}2)$
167			R3c	R 3 2/c	$\Gamma _{rh}D_{3d}^{6}$	47h	$(a/a/a)/{\tilde {6}}\cdot {\tilde {c}}$	$(*{:}63_{1}2)$

List of hexagonal edit

Hexagonal Bravais lattice

Hexagonal crystal system
Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
168	6	$66$	P6	P 6	$\Gamma _{h}C_{6}^{1}$	49s	$(c:(a/a)):6$	$(6_{0}3_{0}2_{0})$
169			P6₁	P 6₁	$\Gamma _{h}C_{6}^{2}$	74a	$(c:(a/a)):6_{1}$	$(6_{1}3_{1}2_{1})$
170			P6₅	P 6₅	$\Gamma _{h}C_{6}^{3}$	75a	$(c:(a/a)):6_{5}$	$(6_{1}3_{1}2_{1})$
171			P6₂	P 6₂	$\Gamma _{h}C_{6}^{4}$	76a	$(c:(a/a)):6_{2}$	$(6_{2}3_{2}2_{0})$
172			P6₄	P 6₄	$\Gamma _{h}C_{6}^{5}$	77a	$(c:(a/a)):6_{4}$	$(6_{2}3_{2}2_{0})$
173			P6₃	P 6₃	$\Gamma _{h}C_{6}^{6}$	78a	$(c:(a/a)):6_{3}$	$(6_{3}3_{0}2_{1})$
174	6	$3*$	P6	P 6	$\Gamma _{h}C_{3h}^{1}$	43s	$(c:(a/a)):3:m$	$[3_{0}3_{0}3_{0}]$
175	6/m	$6*$	P6/m	P 6/m	$\Gamma _{h}C_{6h}^{1}$	53s	$(c:(a/a))\cdot m:6$	$[6_{0}3_{0}2_{0}]$
176	6/m	$6*$	P6₃/m	P 6₃/m	$\Gamma _{h}C_{6h}^{2}$	81a	$(c:(a/a))\cdot m:6_{3}$	$[6_{3}3_{0}2_{1}]$
177	622	$226$	P622	P 6 2 2	$\Gamma _{h}D_{6}^{1}$	54s	$(c:(a/a))\cdot 2:6$	$(*6_{0}3_{0}2_{0})$
178			P6₁22	P 6₁ 2 2	$\Gamma _{h}D_{6}^{2}$	82a	$(c:(a/a))\cdot 2:6_{1}$	$(*6_{1}3_{1}2_{1})$
179			P6₅22	P 6₅ 2 2	$\Gamma _{h}D_{6}^{3}$	83a	$(c:(a/a))\cdot 2:6_{5}$	$(*6_{1}3_{1}2_{1})$
180			P6₂22	P 6₂ 2 2	$\Gamma _{h}D_{6}^{4}$	84a	$(c:(a/a))\cdot 2:6_{2}$	$(*6_{2}3_{2}2_{0})$
181			P6₄22	P 6₄ 2 2	$\Gamma _{h}D_{6}^{5}$	85a	$(c:(a/a))\cdot 2:6_{4}$	$(*6_{2}3_{2}2_{0})$
182			P6₃22	P 6₃ 2 2	$\Gamma _{h}D_{6}^{6}$	86a	$(c:(a/a))\cdot 2:6_{3}$	$(*6_{3}3_{0}2_{1})$
183	6mm	$*66$	P6mm	P 6 m m	$\Gamma _{h}C_{6v}^{1}$	50s	$(c:(a/a)):m\cdot 6$	$(*{\cdot }6{\cdot }3{\cdot }2)$
184			P6cc	P 6 c c	$\Gamma _{h}C_{6v}^{2}$	44h	$(c:(a/a)):{\tilde {c}}\cdot 6$	$(*{:}6{:}3{:}2)$
185			P6₃cm	P 6₃ c m	$\Gamma _{h}C_{6v}^{3}$	80a	$(c:(a/a)):{\tilde {c}}\cdot 6_{3}$	$(*{\cdot }6{:}3{:}2)$
186			P6₃mc	P 6₃ m c	$\Gamma _{h}C_{6v}^{4}$	79a	$(c:(a/a)):m\cdot 6_{3}$	$(*{:}6{\cdot }3{\cdot }2)$
187	6m2	$*223$	P6m2	P 6 m 2	$\Gamma _{h}D_{3h}^{1}$	48s	$(c:(a/a)):m\cdot 3:m$	$[*{\cdot }3{\cdot }3{\cdot }3]$
188			P6c2	P 6 c 2	$\Gamma _{h}D_{3h}^{2}$	43h	$(c:(a/a)):{\tilde {c}}\cdot 3:m$	$[*{:}3{:}3{:}3]$
189			P62m	P 6 2 m	$\Gamma _{h}D_{3h}^{3}$	47s	$(c:(a/a))\cdot m:3\cdot m$	$[3_{0}{*}{\cdot }3]$
190			P62c	P 6 2 c	$\Gamma _{h}D_{3h}^{4}$	42h	$(c:(a/a))\cdot m:3\cdot {\tilde {c}}$	$[3_{0}{*}{:}3]$
191	6/m 2/m 2/m	$*226$	P6/mmm	P 6/m 2/m 2/m	$\Gamma _{h}D_{6h}^{1}$	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	$\Gamma _{h}D_{6h}^{2}$	48h	$(c:(a/a))\cdot m:6\cdot {\tilde {c}}$	$[*{:}6{:}3{:}2]$
193			P6₃/mcm	P 6₃/m 2/c 2/m	$\Gamma _{h}D_{6h}^{3}$	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	$\Gamma _{h}D_{6h}^{4}$	88a	$(c:(a/a))\cdot m:6_{3}\cdot m$	$[*{:}6{\cdot }3{\cdot }2]$

List of cubic edit

Cubic Bravais lattice
Simple (P)	Body centered (I)	Face centered (F)

Example cubic structures

(221) Caesium chloride. Different colors for the two atom types.
(216) Sphalerite
(223) Weaire–Phelan structure

Cubic crystal system
Number	Point group	Orbifold	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Conway	Fibrifold (preserving $z$ )	Fibrifold (preserving $x$ , $y$ , $z$ )
195	23	$332$	P23	P 2 3	$\Gamma _{c}T^{1}$	59s	$\left(a:a:a\right):2/3$	$2^{\circ }$	$(*2_{0}2_{0}2_{0}2_{0}){:}3$	$(*2_{0}2_{0}2_{0}2_{0}){:}3$
196			F23	F 2 3	$\Gamma _{c}^{f}T^{2}$	61s	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):2/3$	$1^{\circ }$	$(*2_{0}2_{1}2_{0}2_{1}){:}3$	$(*2_{0}2_{1}2_{0}2_{1}){:}3$
197			I23	I 2 3	$\Gamma _{c}^{v}T^{3}$	60s	$\left({\tfrac {a+b+c}{2}}/a:a:a\right):2/3$	$4^{\circ \circ }$	$(2_{1}{*}2_{0}2_{0}){:}3$	$(2_{1}{*}2_{0}2_{0}){:}3$
198			P2₁3	P 2₁ 3	$\Gamma _{c}T^{4}$	89a	$\left(a:a:a\right):2_{1}/3$	$1^{\circ }/4$	$(2_{1}2_{1}{\bar {\times }}){:}3$	$(2_{1}2_{1}{\bar {\times }}){:}3$
199			I2₁3	I 2₁ 3	$\Gamma _{c}^{v}T^{5}$	90a	$\left({\tfrac {a+b+c}{2}}/a:a:a\right):2_{1}/3$	$2^{\circ }/4$	$(2_{0}{*}2_{1}2_{1}){:}3$	$(2_{0}{*}2_{1}2_{1}){:}3$
200	2/m 3	$3{*}2$	Pm3	P 2/m 3	$\Gamma _{c}T_{h}^{1}$	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			Pn3	P 2/n 3	$\Gamma _{c}T_{h}^{2}$	49h	$\left(a:a:a\right)\cdot {\widetilde {ab}}/{\tilde {6}}$	$4^{\circ +}$	$(2{\bar {*}}_{1}2_{0}2_{0}){:}3$	$(2{\bar {*}}_{1}2_{0}2_{0}){:}3$
202			Fm3	F 2/m 3	$\Gamma _{c}^{f}T_{h}^{3}$	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			Fd3	F 2/d 3	$\Gamma _{c}^{f}T_{h}^{4}$	50h	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right)\cdot {\tfrac {1}{2}}{\widetilde {ab}}/{\tilde {6}}$	$2^{\circ +}$	$(2{\bar {*}}2_{0}2_{1}){:}3$	$(2{\bar {*}}2_{0}2_{1}){:}3$
204			Im3	I 2/m 3	$\Gamma _{c}^{v}T_{h}^{5}$	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			Pa3	P 2₁/a 3	$\Gamma _{c}T_{h}^{6}$	91a	$\left(a:a:a\right)\cdot {\tilde {a}}/{\tilde {6}}$	$2^{-}/4$	$(2_{1}2{\bar {*}}{:}){:}3$	$(2_{1}2{\bar {*}}{:}){:}3$
206			Ia3	I 2₁/a 3	$\Gamma _{c}^{v}T_{h}^{7}$	92a	$\left({\tfrac {a+b+c}{2}}/a:a:a\right)\cdot {\tilde {a}}/{\tilde {6}}$	$4^{-}/4$	$(*2_{1}2{:}2{:}2){:}3$	$(*2_{1}2{:}2{:}2){:}3$
207	432	$432$	P432	P 4 3 2	$\Gamma _{c}O^{1}$	68s	$\left(a:a:a\right):4/3$	$4^{\circ -}$	$(*4_{0}4_{0}2_{0}){:}3$	$(*2_{0}2_{0}2_{0}2_{0}){:}6$
208			P4₂32	P 4₂ 3 2	$\Gamma _{c}O^{2}$	98a	$\left(a:a:a\right):4_{2}//3$	$4^{+}$	$(*4_{2}4_{2}2_{0}){:}3$	$(*2_{0}2_{0}2_{0}2_{0}){:}6$
209			F432	F 4 3 2	$\Gamma _{c}^{f}O^{3}$	70s	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4/3$	$2^{\circ -}$	$(*4_{2}4_{0}2_{1}){:}3$	$(*2_{0}2_{1}2_{0}2_{1}){:}6$
210			F4₁32	F 4₁ 3 2	$\Gamma _{c}^{f}O^{4}$	97a	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4_{1}//3$	$2^{+}$	$(*4_{3}4_{1}2_{0}){:}3$	$(*2_{0}2_{1}2_{0}2_{1}){:}6$
211			I432	I 4 3 2	$\Gamma _{c}^{v}O^{5}$	69s	$\left({\tfrac {a+b+c}{2}}/a:a:a\right):4/3$	$8^{+\circ }$	$(4_{2}4_{0}2_{1}){:}3$	$(2_{1}{*}2_{0}2_{0}){:}6$
212			P4₃32	P 4₃ 3 2	$\Gamma _{c}O^{6}$	94a	$\left(a:a:a\right):4_{3}//3$	$2^{+}/4$	$(4_{1}{*}2_{1}){:}3$	$(2_{1}2_{1}{\bar {\times }}){:}6$
213			P4₁32	P 4₁ 3 2	$\Gamma _{c}O^{7}$	95a	$\left(a:a:a\right):4_{1}//3$	$2^{+}/4$	$(4_{1}{*}2_{1}){:}3$	$(2_{1}2_{1}{\bar {\times }}){:}6$
214			I4₁32	I 4₁ 3 2	$\Gamma _{c}^{v}O^{8}$	96a	$\left({\tfrac {a+b+c}{2}}/:a:a:a\right):4_{1}//3$	$4^{+}/4$	$(*4_{3}4_{1}2_{0}){:}3$	$(2_{0}{*}2_{1}2_{1}){:}6$
215	43m	$*332$	P43m	P 4 3 m	$\Gamma _{c}T_{d}^{1}$	65s	$\left(a:a:a\right):{\tilde {4}}/3$	$2^{\circ }{:}2$	$(*4{\cdot }42_{0}){:}3$	$(*2_{0}2_{0}2_{0}2_{0}){:}6$
216			F43m	F 4 3 m	$\Gamma _{c}^{f}T_{d}^{2}$	67s	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):{\tilde {4}}/3$	$1^{\circ }{:}2$	$(*4{\cdot }42_{1}){:}3$	$(*2_{0}2_{1}2_{0}2_{1}){:}6$
217			I43m	I 4 3 m	$\Gamma _{c}^{v}T_{d}^{3}$	66s	$\left({\tfrac {a+b+c}{2}}/a:a:a\right):{\tilde {4}}/3$	$4^{\circ }{:}2$	$(*{\cdot }44{:}2){:}3$	$(2_{1}{*}2_{0}2_{0}){:}6$
218			P43n	P 4 3 n	$\Gamma _{c}T_{d}^{4}$	51h	$\left(a:a:a\right):{\tilde {4}}//3$	$4^{\circ }$	$(*4{:}42_{0}){:}3$	$(*2_{0}2_{0}2_{0}2_{0}){:}6$
219			F43c	F 4 3 c	$\Gamma _{c}^{f}T_{d}^{5}$	52h	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):{\tilde {4}}//3$	$2^{\circ \circ }$	$(*4{:}42_{1}){:}3$	$(*2_{0}2_{1}2_{0}2_{1}){:}6$
220			I43d	I 4 3 d	$\Gamma _{c}^{v}T_{d}^{6}$	93a	$\left({\tfrac {a+b+c}{2}}/a:a:a\right):{\tilde {4}}//3$	$4^{\circ }/4$	$(4{\bar {*}}2_{1}){:}3$	$(2_{0}{*}2_{1}2_{1}){:}6$
221	4/m 3 2/m	$*432$	Pm3m	P 4/m 3 2/m	$\Gamma _{c}O_{h}^{1}$	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			Pn3n	P 4/n 3 2/n	$\Gamma _{c}O_{h}^{2}$	53h	$\left(a:a:a\right):4/{\tilde {6}}\cdot {\widetilde {abc}}$	$8^{\circ \circ }$	$(*4_{0}4{:}2){:}3$	$(2{\bar {*}}_{1}2_{0}2_{0}){:}6$
223			Pm3n	P 4₂/m 3 2/n	$\Gamma _{c}O_{h}^{3}$	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			Pn3m	P 4₂/n 3 2/m	$\Gamma _{c}O_{h}^{4}$	103a	$\left(a:a:a\right):4_{2}//{\tilde {6}}\cdot m$	$4^{+}{:}2$	$(*4_{2}4{\cdot }2){:}3$	$(2{\bar {*}}_{1}2_{0}2_{0}){:}6$
225			Fm3m	F 4/m 3 2/m	$\Gamma _{c}^{f}O_{h}^{5}$	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			Fm3c	F 4/m 3 2/c	$\Gamma _{c}^{f}O_{h}^{6}$	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			Fd3m	F 4₁/d 3 2/m	$\Gamma _{c}^{f}O_{h}^{7}$	100a	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4_{1}//{\tilde {6}}\cdot m$	$2^{+}{:}2$	$(*4_{1}4{\cdot }2){:}3$	$(2{\bar {*}}2_{0}2_{1}){:}6$
228			Fd3c	F 4₁/d 3 2/c	$\Gamma _{c}^{f}O_{h}^{8}$	101a	$\left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4_{1}//{\tilde {6}}\cdot {\tilde {c}}$	$4^{++}$	$(*4_{1}4{:}2){:}3$	$(2{\bar {*}}2_{0}2_{1}){:}6$
229			Im3m	I 4/m 3 2/m	$\Gamma _{c}^{v}O_{h}^{9}$	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			Ia3d	I 4₁/a 3 2/d	$\Gamma _{c}^{v}O_{h}^{10}$	99a	$\left({\tfrac {a+b+c}{2}}/a:a:a\right):4_{1}//{\tilde {6}}\cdot {\tfrac {1}{2}}{\widetilde {abc}}$	$8^{\circ }/4$	$(*4_{1}4{:}2){:}3$	$(*2_{1}2{:}2{:}2){:}6$

Notes edit

^ The symbol $e$ was introduced by the IUCR in 1992. Prior to this, the space groups Aem2 (No. 39), Aea2 (No. 41), Cmce (No. 64), Cmme (No. 67), and Ccce (No. 68) were known as Abm2 (No. 39), Aba2 (No. 41), Cmca (No. 64), Cmma (No. 67), and Ccca (No. 68) respectively. Historical literature may refer to the old names, but their meaning is unchanged.^[1]

References edit

^ de Wolff, P. M.; Billiet, Y.; Donnay, J. D. H.; Fischer, W.; Galiulin, R. B.; Glazer, A. M.; Hahn, T.; Senechal, M.; Shoemaker, D. P.; Wondratschek, H.; Wilson, A. J. C.; Abrahams, S. C. (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 Foundations of Crystallography. 48 (5). International Union of Crystallography (IUCr): 727–732. doi:10.1107/s0108767392003428. ISSN 0108-7673.
^ Bradley, C. J.; Cracknell, A. P. (2010). The mathematical theory of symmetry in solids: representation theory for point groups and space groups. Oxford New York: Clarendon Press. pp. 127–134. ISBN 978-0-19-958258-7. OCLC 859155300.