# List of space groups

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")
• 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.

• ${\displaystyle a}$, ${\displaystyle b}$, or ${\displaystyle c}$ glide translation along half the lattice vector of this face
• ${\displaystyle n}$ glide translation along with half a face diagonal
• ${\displaystyle d}$ glide planes with translation along a quarter of a face diagonal.
• ${\displaystyle e}$ 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 ${\displaystyle \color {Black}{\tfrac {360^{\circ }}{n}}}$. 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.

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.

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 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. Hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. All the other space groups are asymmorphic. Example for point group 4/mmm (${\displaystyle {\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$): the symmorphic space groups are P4/mmm (${\displaystyle P{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$, 36s) and I4/mmm (${\displaystyle I{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$, 37s); hemisymmorphic space groups should contain axial combination 422, these are P4/mcc (${\displaystyle P{\tfrac {4}{m}}{\tfrac {2}{c}}{\tfrac {2}{c}}}$, 35h), P4/nbm (${\displaystyle P{\tfrac {4}{n}}{\tfrac {2}{b}}{\tfrac {2}{m}}}$, 36h), P4/nnc (${\displaystyle P{\tfrac {4}{n}}{\tfrac {2}{n}}{\tfrac {2}{c}}}$, 37h), and I4/mcm (${\displaystyle I{\tfrac {4}{m}}{\tfrac {2}{c}}{\tfrac {2}{m}}}$, 38h).

## List of Triclinic

Triclinic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
11${\displaystyle 1}$P1P 1${\displaystyle C_{1}^{1}}$1s${\displaystyle (a/b/c)\cdot 1}$${\displaystyle (\circ )}$
21${\displaystyle \times }$P1P 1${\displaystyle C_{i}^{1}}$2s${\displaystyle (a/b/c)\cdot {\tilde {2}}}$${\displaystyle (2222)}$

## List of Monoclinic

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)
32${\displaystyle 22}$P2P 1 2 1P 1 1 2${\displaystyle C_{2}^{1}}$3s${\displaystyle (b:(c/a)):2}$${\displaystyle (2_{0}2_{0}2_{0}2_{0})}$${\displaystyle ({*}_{0}{*}_{0})}$
4P21P 1 21 1P 1 1 21${\displaystyle C_{2}^{2}}$1a${\displaystyle (b:(c/a)):2_{1}}$${\displaystyle (2_{1}2_{1}2_{1}2_{1})}$${\displaystyle ({\bar {\times }}{\bar {\times }})}$
5C2C 1 2 1B 1 1 2${\displaystyle C_{2}^{3}}$4s${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right):2}$${\displaystyle (2_{0}2_{0}2_{1}2_{1})}$${\displaystyle ({*}_{1}{*}_{1})}$, ${\displaystyle ({*}{\bar {\times }})}$
6m${\displaystyle *}$PmP 1 m 1P 1 1 m${\displaystyle C_{s}^{1}}$5s${\displaystyle (b:(c/a))\cdot m}$${\displaystyle [\circ _{0}]}$${\displaystyle ({*}{\cdot }{*}{\cdot })}$
7PcP 1 c 1P 1 1 b${\displaystyle C_{s}^{2}}$1h${\displaystyle (b:(c/a))\cdot {\tilde {c}}}$${\displaystyle ({\bar {\circ }}_{0})}$${\displaystyle ({*}{:}{*}{:})}$, ${\displaystyle ({\times }{\times }_{0})}$
8CmC 1 m 1B 1 1 m${\displaystyle C_{s}^{3}}$6s${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot m}$${\displaystyle [\circ _{1}]}$${\displaystyle ({*}{\cdot }{*}{:})}$, ${\displaystyle ({*}{\cdot }{\times })}$
9CcC 1 c 1B 1 1 b${\displaystyle C_{s}^{4}}$2h${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot {\tilde {c}}}$${\displaystyle ({\bar {\circ }}_{1})}$${\displaystyle ({*}{:}{\times })}$, ${\displaystyle ({\times }{\times }_{1})}$
102/m${\displaystyle 2*}$P2/mP 1 2/m 1P 1 1 2/m${\displaystyle C_{2h}^{1}}$7s${\displaystyle (b:(c/a))\cdot m:2}$${\displaystyle [2_{0}2_{0}2_{0}2_{0}]}$${\displaystyle [*2{\cdot }22{\cdot }2)}$
11P21/mP 1 21/m 1P 1 1 21/m${\displaystyle C_{2h}^{2}}$2a${\displaystyle (b:(c/a))\cdot m:2_{1}}$${\displaystyle [2_{1}2_{1}2_{1}2_{1}]}$${\displaystyle (22{*}{\cdot })}$
12C2/mC 1 2/m 1B 1 1 2/m${\displaystyle C_{2h}^{3}}$8s${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot m:2}$${\displaystyle [2_{0}2_{0}2_{1}2_{1}]}$${\displaystyle (*2{\cdot }22{:}2)}$, ${\displaystyle (2{\bar {*}}2{\cdot }2)}$
13P2/cP 1 2/c 1P 1 1 2/b${\displaystyle C_{2h}^{4}}$3h${\displaystyle (b:(c/a))\cdot {\tilde {c}}:2}$${\displaystyle (2_{0}2_{0}22)}$${\displaystyle (*2{:}22{:}2)}$, ${\displaystyle (22{*}_{0})}$
14P21/cP 1 21/c 1P 1 1 21/b${\displaystyle C_{2h}^{5}}$3a${\displaystyle (b:(c/a))\cdot {\tilde {c}}:2_{1}}$${\displaystyle (2_{1}2_{1}22)}$${\displaystyle (22{*}{:})}$, ${\displaystyle (22{\times })}$
15C2/cC 1 2/c 1B 1 1 2/b${\displaystyle C_{2h}^{6}}$4h${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot {\tilde {c}}:2}$${\displaystyle (2_{0}2_{1}22)}$${\displaystyle (2{\bar {*}}2{:}2)}$, ${\displaystyle (22{*}_{1})}$

## List of Orthorhombic

Orthorhombic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)
16222${\displaystyle 222}$P222P 2 2 2${\displaystyle D_{2}^{1}}$9s${\displaystyle (c:a:b):2:2}$${\displaystyle (*2_{0}2_{0}2_{0}2_{0})}$
17P2221P 2 2 21${\displaystyle D_{2}^{2}}$4a${\displaystyle (c:a:b):2_{1}:2}$${\displaystyle (*2_{1}2_{1}2_{1}2_{1})}$${\displaystyle (2_{0}2_{0}{*})}$
18P21212P 21 21 2${\displaystyle D_{2}^{3}}$7a${\displaystyle (c:a:b):2}$ ${\displaystyle 2_{1}}$${\displaystyle (2_{0}2_{0}{\bar {\times }})}$${\displaystyle (2_{1}2_{1}{*})}$
19P212121P 21 21 21${\displaystyle D_{2}^{4}}$8a${\displaystyle (c:a:b):2_{1}}$ ${\displaystyle 2_{1}}$${\displaystyle (2_{1}2_{1}{\bar {\times }})}$
20C2221C 2 2 21${\displaystyle D_{2}^{5}}$5a${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):2_{1}:2}$${\displaystyle (2_{1}{*}2_{1}2_{1})}$${\displaystyle (2_{0}2_{1}{*})}$
21C222C 2 2 2${\displaystyle D_{2}^{6}}$10s${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):2:2}$${\displaystyle (2_{0}{*}2_{0}2_{0})}$${\displaystyle (*2_{0}2_{0}2_{1}2_{1})}$
22F222F 2 2 2${\displaystyle D_{2}^{7}}$12s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):2:2}$${\displaystyle (*2_{0}2_{1}2_{0}2_{1})}$
23I222I 2 2 2${\displaystyle D_{2}^{8}}$11s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):2:2}$${\displaystyle (2_{1}{*}2_{0}2_{0})}$
24I212121I 21 21 21${\displaystyle D_{2}^{9}}$6a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):2:2_{1}}$${\displaystyle (2_{0}{*}2_{1}2_{1})}$
25mm2${\displaystyle *22}$Pmm2P m m 2${\displaystyle C_{2v}^{1}}$13s${\displaystyle (c:a:b):m\cdot 2}$${\displaystyle (*{\cdot }2{\cdot }2{\cdot }2{\cdot }2)}$${\displaystyle [{*}_{0}{\cdot }{*}_{0}{\cdot }]}$
26Pmc21P m c 21${\displaystyle C_{2v}^{2}}$9a${\displaystyle (c:a:b):{\tilde {c}}\cdot 2_{1}}$${\displaystyle (*{\cdot }2{:}2{\cdot }2{:}2)}$${\displaystyle ({\bar {*}}{\cdot }{\bar {*}}{\cdot })}$, ${\displaystyle [{\times _{0}}{\times _{0}}]}$
27Pcc2P c c 2${\displaystyle C_{2v}^{3}}$5h${\displaystyle (c:a:b):{\tilde {c}}\cdot 2}$${\displaystyle (*{:}2{:}2{:}2{:}2)}$${\displaystyle ({\bar {*}}_{0}{\bar {*}}_{0})}$
28Pma2P m a 2${\displaystyle C_{2v}^{4}}$6h${\displaystyle (c:a:b):{\tilde {a}}\cdot 2}$${\displaystyle (2_{0}2_{0}{*}{\cdot })}$${\displaystyle [{*}_{0}{:}{*}_{0}{:}]}$, ${\displaystyle (*{\cdot }{*}_{0})}$
29Pca21P c a 21${\displaystyle C_{2v}^{5}}$11a${\displaystyle (c:a:b):{\tilde {a}}\cdot 2_{1}}$${\displaystyle (2_{1}2_{1}{*}{:})}$${\displaystyle ({\bar {*}}{:}{\bar {*}}{:})}$
30Pnc2P n c 2${\displaystyle C_{2v}^{6}}$7h${\displaystyle (c:a:b):{\tilde {c}}\odot 2}$${\displaystyle (2_{0}2_{0}{*}{:})}$${\displaystyle ({\bar {*}}_{1}{\bar {*}}_{1})}$, ${\displaystyle ({*}_{0}{\times }_{0})}$
31Pmn21P m n 21${\displaystyle C_{2v}^{7}}$10a${\displaystyle (c:a:b):{\widetilde {ac}}\cdot 2_{1}}$${\displaystyle (2_{1}2_{1}{*}{\cdot })}$${\displaystyle (*{\cdot }{\bar {\times }})}$, ${\displaystyle [{\times }_{0}{\times }_{1}]}$
32Pba2P b a 2${\displaystyle C_{2v}^{8}}$9h${\displaystyle (c:a:b):{\tilde {a}}\odot 2}$${\displaystyle (2_{0}2_{0}{\times }_{0})}$${\displaystyle (*{:}{*}_{0})}$
33Pna21P n a 21${\displaystyle C_{2v}^{9}}$12a${\displaystyle (c:a:b):{\tilde {a}}\odot 2_{1}}$${\displaystyle (2_{1}2_{1}{\times })}$${\displaystyle (*{:}{\times })}$, ${\displaystyle ({\times }{\times }_{1})}$
34Pnn2P n n 2${\displaystyle C_{2v}^{10}}$8h${\displaystyle (c:a:b):{\widetilde {ac}}\odot 2}$${\displaystyle (2_{0}2_{0}{\times }_{1})}$${\displaystyle (*_{0}{\times }_{1})}$
35Cmm2C m m 2${\displaystyle C_{2v}^{11}}$14s${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):m\cdot 2}$${\displaystyle (2_{0}{*}{\cdot }2{\cdot }2)}$${\displaystyle [*_{0}{\cdot }{*}_{0}{:}]}$
36Cmc21C m c 21${\displaystyle C_{2v}^{12}}$13a${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):{\tilde {c}}\cdot 2_{1}}$${\displaystyle (2_{1}{*}{\cdot }2{:}2)}$${\displaystyle ({\bar {*}}{\cdot }{\bar {*}}{:})}$, ${\displaystyle [{\times }_{1}{\times }_{1}]}$
37Ccc2C c c 2${\displaystyle C_{2v}^{13}}$10h${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):{\tilde {c}}\cdot 2}$${\displaystyle (2_{0}{*}{:}2{:}2)}$${\displaystyle ({\bar {*}}_{0}{\bar {*}}_{1})}$
38Amm2A m m 2${\displaystyle C_{2v}^{14}}$15s${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):m\cdot 2}$${\displaystyle (*{\cdot }2{\cdot }2{\cdot }2{:}2)}$${\displaystyle [{*}_{1}{\cdot }{*}_{1}{\cdot }]}$, ${\displaystyle [*{\cdot }{\times }_{0}]}$
39Aem2A b m 2${\displaystyle C_{2v}^{15}}$11h${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):m\cdot 2_{1}}$${\displaystyle (*{\cdot }2{:}2{:}2{:}2)}$${\displaystyle [{*}_{1}{:}{*}_{1}{:}]}$, ${\displaystyle ({\bar {*}}{\cdot }{\bar {*}}_{0})}$
40Ama2A m a 2${\displaystyle C_{2v}^{16}}$12h${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):{\tilde {a}}\cdot 2}$${\displaystyle (2_{0}2_{1}{*}{\cdot })}$${\displaystyle (*{\cdot }{*}_{1})}$, ${\displaystyle [*{:}{\times }_{1}]}$
41Aea2A b a 2${\displaystyle C_{2v}^{17}}$13h${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):{\tilde {a}}\cdot 2_{1}}$${\displaystyle (2_{0}2_{1}{*}{:})}$${\displaystyle (*{:}{*}_{1})}$, ${\displaystyle ({\bar {*}}{:}{\bar {*}}_{1})}$
42Fmm2F m m 2${\displaystyle C_{2v}^{18}}$17s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):m\cdot 2}$${\displaystyle (*{\cdot }2{\cdot }2{:}2{:}2)}$${\displaystyle [{*}_{1}{\cdot }{*}_{1}{:}]}$
43Fdd2F dd2${\displaystyle C_{2v}^{19}}$16h${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):{\tfrac {1}{2}}{\widetilde {ac}}\odot 2}$${\displaystyle (2_{0}2_{1}{\times })}$${\displaystyle ({*}_{1}{\times })}$
44Imm2I m m 2${\displaystyle C_{2v}^{20}}$16s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):m\cdot 2}$${\displaystyle (2_{1}{*}{\cdot }2{\cdot }2)}$${\displaystyle [*{\cdot }{\times }_{1}]}$
45Iba2I b a 2${\displaystyle C_{2v}^{21}}$15h${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):{\tilde {c}}\cdot 2}$${\displaystyle (2_{1}{*}{:}2{:}2)}$${\displaystyle ({\bar {*}}{:}{\bar {*}}_{0})}$
46Ima2I m a 2${\displaystyle C_{2v}^{22}}$14h${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):{\tilde {a}}\cdot 2}$${\displaystyle (2_{0}{*}{\cdot }2{:}2)}$${\displaystyle ({\bar {*}}{\cdot }{\bar {*}}_{1})}$, ${\displaystyle [*{:}{\times }_{0}]}$
47${\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$${\displaystyle *222}$PmmmP 2/m 2/m 2/m${\displaystyle D_{2h}^{1}}$18s${\displaystyle \left(c:a:b\right)\cdot m:2\cdot m}$${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{\cdot }2]}$
48PnnnP 2/n 2/n 2/n${\displaystyle D_{2h}^{2}}$19h${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\odot {\widetilde {ac}}}$${\displaystyle (2{\bar {*}}_{1}2_{0}2_{0}}$
49PccmP 2/c 2/c 2/m${\displaystyle D_{2h}^{3}}$17h${\displaystyle \left(c:a:b\right)\cdot m:2\cdot {\tilde {c}}}$${\displaystyle [*{:}2{:}2{:}2{:}2]}$${\displaystyle (*2_{0}2_{0}2{\cdot }2)}$
50PbanP 2/b 2/a 2/n${\displaystyle D_{2h}^{4}}$18h${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\odot {\tilde {a}}}$${\displaystyle (2{\bar {*}}_{0}2_{0}2_{0})}$${\displaystyle (*2_{0}2_{0}2{:}2)}$
51PmmaP 21/m 2/m 2/a${\displaystyle D_{2h}^{5}}$14a${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2\cdot m}$${\displaystyle [2_{0}2_{0}{*}{\cdot }]}$${\displaystyle [*{\cdot }2{:}2{\cdot }2{:}2]}$, ${\displaystyle [*2{\cdot }2{\cdot }2{\cdot }2]}$
52PnnaP 2/n 21/n 2/a${\displaystyle D_{2h}^{6}}$17a${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2\odot {\widetilde {ac}}}$${\displaystyle (2_{0}2{\bar {*}}_{1})}$${\displaystyle (2_{0}{*}2{:}2)}$, ${\displaystyle (2{\bar {*}}2_{1}2_{1})}$
53PmnaP 2/m 2/n 21/a${\displaystyle D_{2h}^{7}}$15a${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\cdot {\widetilde {ac}}}$${\displaystyle [2_{0}2_{0}{*}{:}]}$${\displaystyle (*2_{1}2_{1}2{\cdot }2)}$, ${\displaystyle (2_{0}{*}2{\cdot }2)}$
54PccaP 21/c 2/c 2/a${\displaystyle D_{2h}^{8}}$16a${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}}$${\displaystyle (2_{0}2{\bar {*}}_{0})}$${\displaystyle (*2{:}2{:}2{:}2)}$, ${\displaystyle (*2_{1}2_{1}2{:}2)}$
55PbamP 21/b 21/a 2/m${\displaystyle D_{2h}^{9}}$22a${\displaystyle \left(c:a:b\right)\cdot m:2\odot {\tilde {a}}}$${\displaystyle [2_{0}2_{0}{\times }_{0}]}$${\displaystyle (*2{\cdot }2{:}2{\cdot }2)}$
56PccnP 21/c 21/c 2/n${\displaystyle D_{2h}^{10}}$27a${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\cdot {\tilde {c}}}$${\displaystyle (2{\bar {*}}{:}2{:}2)}$${\displaystyle (2_{1}2{\bar {*}}_{0})}$
57PbcmP 2/b 21/c 21/m${\displaystyle D_{2h}^{11}}$23a${\displaystyle \left(c:a:b\right)\cdot m:2_{1}\odot {\tilde {c}}}$${\displaystyle (2_{0}2{\bar {*}}{\cdot })}$${\displaystyle (*2{:}2{\cdot }2{:}2)}$, ${\displaystyle [2_{1}2_{1}{*}{:}]}$
58PnnmP 21/n 21/n 2/m${\displaystyle D_{2h}^{12}}$25a${\displaystyle \left(c:a:b\right)\cdot m:2\odot {\widetilde {ac}}}$${\displaystyle [2_{0}2_{0}{\times }_{1}]}$${\displaystyle (2_{1}{*}2{\cdot }2)}$
59PmmnP 21/m 21/m 2/n${\displaystyle D_{2h}^{13}}$24a${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\cdot m}$${\displaystyle (2{\bar {*}}{\cdot }2{\cdot }2)}$${\displaystyle [2_{1}2_{1}{*}{\cdot }]}$
60PbcnP 21/b 2/c 21/n${\displaystyle D_{2h}^{14}}$26a${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2_{1}\odot {\tilde {c}}}$${\displaystyle (2_{0}2{\bar {*}}{:})}$${\displaystyle (2_{1}{*}2{:}2)}$, ${\displaystyle (2_{1}2{\bar {*}}_{1})}$
61PbcaP 21/b 21/c 21/a${\displaystyle D_{2h}^{15}}$29a${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\odot {\tilde {c}}}$${\displaystyle (2_{1}2{\bar {*}}{:})}$
62PnmaP 21/n 21/m 21/a${\displaystyle D_{2h}^{16}}$28a${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\odot m}$${\displaystyle (2_{1}2{\bar {*}}{\cdot })}$${\displaystyle (2{\bar {*}}{\cdot }2{:}2)}$, ${\displaystyle [2_{1}2_{1}{\times }]}$
63CmcmC 2/m 2/c 21/m${\displaystyle D_{2h}^{17}}$18a${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2_{1}\cdot {\tilde {c}}}$${\displaystyle [2_{0}2_{1}{*}{\cdot }]}$${\displaystyle (*2{\cdot }2{\cdot }2{:}2)}$, ${\displaystyle [2_{1}{*}{\cdot }2{:}2]}$
64CmcaC 2/m 2/c 21/a${\displaystyle D_{2h}^{18}}$19a${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2_{1}\cdot {\tilde {c}}}$${\displaystyle [2_{0}2_{1}{*}{:}]}$${\displaystyle (*2{\cdot }2{:}2{:}2)}$, ${\displaystyle (*2_{1}2{\cdot }2{:}2)}$
65CmmmC 2/m 2/m 2/m${\displaystyle D_{2h}^{19}}$19s${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2\cdot m}$${\displaystyle [2_{0}{*}{\cdot }2{\cdot }2]}$${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{:}2]}$
66CccmC 2/c 2/c 2/m${\displaystyle D_{2h}^{20}}$20h${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2\cdot {\tilde {c}}}$${\displaystyle [2_{0}{*}{:}2{:}2]}$${\displaystyle (*2_{0}2_{1}2{\cdot }2)}$
67CmmeC 2/m 2/m 2/e${\displaystyle D_{2h}^{21}}$21h${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2\cdot m}$${\displaystyle (*2_{0}2{\cdot }2{\cdot }2)}$${\displaystyle [*{\cdot }2{:}2{:}2{:}2]}$
68CcceC 2/c 2/c 2/e${\displaystyle D_{2h}^{22}}$22h${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}}$${\displaystyle (*2_{0}2{:}2{:}2)}$${\displaystyle (*2_{0}2_{1}2{:}2)}$
69FmmmF 2/m 2/m 2/m${\displaystyle D_{2h}^{23}}$21s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2\cdot m}$${\displaystyle [*{\cdot }2{\cdot }2{:}2{:}2]}$
70FdddF 2/d 2/d 2/d${\displaystyle D_{2h}^{24}}$24h${\displaystyle \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}}}$${\displaystyle (2{\bar {*}}2_{0}2_{1})}$
71ImmmI 2/m 2/m 2/m${\displaystyle D_{2h}^{25}}$20s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot m:2\cdot m}$${\displaystyle [2_{1}{*}{\cdot }2{\cdot }2]}$
72IbamI 2/b 2/a 2/m${\displaystyle D_{2h}^{26}}$23h${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot m:2\cdot {\tilde {c}}}$${\displaystyle [2_{1}{*}{:}2{:}2]}$${\displaystyle (*2_{0}2{\cdot }2{:}2)}$
73IbcaI 2/b 2/c 2/a${\displaystyle D_{2h}^{27}}$21a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}}$${\displaystyle (*2_{1}2{:}2{:}2)}$
74ImmaI 2/m 2/m 2/a${\displaystyle D_{2h}^{28}}$20a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot {\tilde {a}}:2\cdot m}$${\displaystyle (*2_{1}2{\cdot }2{\cdot }2)}$${\displaystyle [2_{0}{*}{\cdot }2{:}2]}$

## List of Tetragonal

Tetragonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
754${\displaystyle 44}$P4P 4${\displaystyle C_{4}^{1}}$22s${\displaystyle (c:a:a):4}$${\displaystyle (4_{0}4_{0}2_{0})}$
76P41P 41${\displaystyle C_{4}^{2}}$30a${\displaystyle (c:a:a):4_{1}}$${\displaystyle (4_{1}4_{1}2_{1})}$
77P42P 42${\displaystyle C_{4}^{3}}$33a${\displaystyle (c:a:a):4_{2}}$${\displaystyle (4_{2}4_{2}2_{0})}$
78P43P 43${\displaystyle C_{4}^{4}}$31a${\displaystyle (c:a:a):4_{3}}$${\displaystyle (4_{1}4_{1}2_{1})}$
79I4I 4${\displaystyle C_{4}^{5}}$23s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4}$${\displaystyle (4_{2}4_{0}2_{1})}$
80I41I 41${\displaystyle C_{4}^{6}}$32a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}}$${\displaystyle (4_{3}4_{1}2_{0})}$
814${\displaystyle 2\times }$P4P 4${\displaystyle S_{4}^{1}}$26s${\displaystyle (c:a:a):{\tilde {4}}}$${\displaystyle (442_{0})}$
82I4I 4${\displaystyle S_{4}^{2}}$27s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}}$${\displaystyle (442_{1})}$
834/m${\displaystyle 4*}$P4/mP 4/m${\displaystyle C_{4h}^{1}}$28s${\displaystyle (c:a:a)\cdot m:4}$${\displaystyle [4_{0}4_{0}2_{0}]}$
84P42/mP 42/m${\displaystyle C_{4h}^{2}}$41a${\displaystyle (c:a:a)\cdot m:4_{2}}$${\displaystyle [4_{2}4_{2}2_{0}]}$
85P4/nP 4/n${\displaystyle C_{4h}^{3}}$29h${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4}$${\displaystyle (44_{0}2)}$
86P42/nP 42/n${\displaystyle C_{4h}^{4}}$42a${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}}$${\displaystyle (44_{2}2)}$
87I4/mI 4/m${\displaystyle C_{4h}^{5}}$29s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot m:4}$${\displaystyle [4_{2}4_{0}2_{1}]}$
88I41/aI 41/a${\displaystyle C_{4h}^{6}}$40a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}}$${\displaystyle (44_{1}2)}$
89422${\displaystyle 224}$P422P 4 2 2${\displaystyle D_{4}^{1}}$30s${\displaystyle (c:a:a):4:2}$${\displaystyle (*4_{0}4_{0}2_{0})}$
90P4212P4212${\displaystyle D_{4}^{2}}$43a${\displaystyle (c:a:a):4}$ ${\displaystyle 2_{1}}$${\displaystyle (4_{0}{*}2_{0})}$
91P4122P 41 2 2${\displaystyle D_{4}^{3}}$44a${\displaystyle (c:a:a):4_{1}:2}$${\displaystyle (*4_{1}4_{1}2_{1})}$
92P41212P 41 21 2${\displaystyle D_{4}^{4}}$48a${\displaystyle (c:a:a):4_{1}}$ ${\displaystyle 2_{1}}$${\displaystyle (4_{1}{*}2_{1})}$
93P4222P 42 2 2${\displaystyle D_{4}^{5}}$47a${\displaystyle (c:a:a):4_{2}:2}$${\displaystyle (*4_{2}4_{2}2_{0})}$
94P42212P 42 21 2${\displaystyle D_{4}^{6}}$50a${\displaystyle (c:a:a):4_{2}}$ ${\displaystyle 2_{1}}$${\displaystyle (4_{2}{*}2_{0})}$
95P4322P 43 2 2${\displaystyle D_{4}^{7}}$45a${\displaystyle (c:a:a):4_{3}:2}$${\displaystyle (*4_{1}4_{1}2_{1})}$
96P43212P 43 21 2${\displaystyle D_{4}^{8}}$49a${\displaystyle (c:a:a):4_{3}}$ ${\displaystyle 2_{1}}$${\displaystyle (4_{1}{*}2_{1})}$
97I422I 4 2 2${\displaystyle D_{4}^{9}}$31s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4:2}$${\displaystyle (*4_{2}4_{0}2_{1})}$
98I4122I 41 2 2${\displaystyle D_{4}^{10}}$46a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4:2_{1}}$${\displaystyle (*4_{3}4_{1}2_{0})}$
994mm${\displaystyle *44}$P4mmP 4 m m${\displaystyle C_{4v}^{1}}$24s${\displaystyle (c:a:a):4\cdot m}$${\displaystyle (*{\cdot }4{\cdot }4{\cdot }2)}$
100P4bmP 4 b m${\displaystyle C_{4v}^{2}}$26h${\displaystyle (c:a:a):4\odot {\tilde {a}}}$${\displaystyle (4_{0}{*}{\cdot }2)}$
101P42cmP 42 c m${\displaystyle C_{4v}^{3}}$37a${\displaystyle (c:a:a):4_{2}\cdot {\tilde {c}}}$${\displaystyle (*{:}4{\cdot }4{:}2)}$
102P42nmP 42 n m${\displaystyle C_{4v}^{4}}$38a${\displaystyle (c:a:a):4_{2}\odot {\widetilde {ac}}}$${\displaystyle (4_{2}{*}{\cdot }2)}$
103P4ccP 4 c c${\displaystyle C_{4v}^{5}}$25h${\displaystyle (c:a:a):4\cdot {\tilde {c}}}$${\displaystyle (*{:}4{:}4{:}2)}$
104P4ncP 4 n c${\displaystyle C_{4v}^{6}}$27h${\displaystyle (c:a:a):4\odot {\widetilde {ac}}}$${\displaystyle (4_{0}{*}{:}2)}$
105P42mcP 42 m c${\displaystyle C_{4v}^{7}}$36a${\displaystyle (c:a:a):4_{2}\cdot m}$${\displaystyle (*{\cdot }4{:}4{\cdot }2)}$
106P42bcP 42 b c${\displaystyle C_{4v}^{8}}$39a${\displaystyle (c:a:a):4\odot {\tilde {a}}}$${\displaystyle (4_{2}{*}{:}2)}$
107I4mmI 4 m m${\displaystyle C_{4v}^{9}}$25s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4\cdot m}$${\displaystyle (*{\cdot }4{\cdot }4{:}2)}$
108I4cmI 4 c m${\displaystyle C_{4v}^{10}}$28h${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4\cdot {\tilde {c}}}$${\displaystyle (*{\cdot }4{:}4{:}2)}$
109I41mdI 41 m d${\displaystyle C_{4v}^{11}}$34a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}\odot m}$${\displaystyle (4_{1}{*}{\cdot }2)}$
110I41cdI 41 c d${\displaystyle C_{4v}^{12}}$35a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}\odot {\tilde {c}}}$${\displaystyle (4_{1}{*}{:}2)}$
11142m${\displaystyle 2{*}2}$P42mP 4 2 m${\displaystyle D_{2d}^{1}}$32s${\displaystyle (c:a:a):{\tilde {4}}:2}$${\displaystyle (*4{\cdot }42_{0})}$
112P42cP 4 2 c${\displaystyle D_{2d}^{2}}$30h${\displaystyle (c:a:a):{\tilde {4}}}$ ${\displaystyle 2}$${\displaystyle (*4{:}42_{0})}$
113P421mP 4 21 m${\displaystyle D_{2d}^{3}}$52a${\displaystyle (c:a:a):{\tilde {4}}\cdot {\widetilde {ab}}}$${\displaystyle (4{\bar {*}}{\cdot }2)}$
114P421cP 4 21 c${\displaystyle D_{2d}^{4}}$53a${\displaystyle (c:a:a):{\tilde {4}}\cdot {\widetilde {abc}}}$${\displaystyle (4{\bar {*}}{:}2)}$
115P4m2P 4 m 2${\displaystyle D_{2d}^{5}}$33s${\displaystyle (c:a:a):{\tilde {4}}\cdot m}$${\displaystyle (*{\cdot }44{\cdot }2)}$
116P4c2P 4 c 2${\displaystyle D_{2d}^{6}}$31h${\displaystyle (c:a:a):{\tilde {4}}\cdot {\tilde {c}}}$${\displaystyle (*{:}44{:}2)}$
117P4b2P 4 b 2${\displaystyle D_{2d}^{7}}$32h${\displaystyle (c:a:a):{\tilde {4}}\odot {\tilde {a}}}$${\displaystyle (4{\bar {*}}_{0}2_{0})}$
118P4n2P 4 n 2${\displaystyle D_{2d}^{8}}$33h${\displaystyle (c:a:a):{\tilde {4}}\cdot {\widetilde {ac}}}$${\displaystyle (4{\bar {*}}_{1}2_{0})}$
119I4m2I 4 m 2${\displaystyle D_{2d}^{9}}$35s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\cdot m}$${\displaystyle (*4{\cdot }42_{1})}$
120I4c2I 4 c 2${\displaystyle D_{2d}^{10}}$34h${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\cdot {\tilde {c}}}$${\displaystyle (*4{:}42_{1})}$
121I42mI 4 2 m${\displaystyle D_{2d}^{11}}$34s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}:2}$${\displaystyle (*{\cdot }44{:}2)}$
122I42dI 4 2 d${\displaystyle D_{2d}^{12}}$51a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\odot {\tfrac {1}{2}}{\widetilde {abc}}}$${\displaystyle (4{\bar {*}}2_{1})}$
1234/m 2/m 2/m${\displaystyle *224}$P4/mmmP 4/m 2/m 2/m${\displaystyle D_{4h}^{1}}$36s${\displaystyle (c:a:a)\cdot m:4\cdot m}$${\displaystyle [*{\cdot }4{\cdot }4{\cdot }2]}$
124P4/mccP 4/m 2/c 2/c${\displaystyle D_{4h}^{2}}$35h${\displaystyle (c:a:a)\cdot m:4\cdot {\tilde {c}}}$${\displaystyle [*{:}4{:}4{:}2]}$
125P4/nbmP 4/n 2/b 2/m${\displaystyle D_{4h}^{3}}$36h${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\odot {\tilde {a}}}$${\displaystyle (*4_{0}4{\cdot }2)}$
126P4/nncP 4/n 2/n 2/c${\displaystyle D_{4h}^{4}}$37h${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\odot {\widetilde {ac}}}$${\displaystyle (*4_{0}4{:}2)}$
127P4/mbmP 4/m 21/b 2/m${\displaystyle D_{4h}^{5}}$54a${\displaystyle (c:a:a)\cdot m:4\odot {\tilde {a}}}$${\displaystyle [4_{0}{*}{\cdot }2]}$
128P4/mncP 4/m 21/n 2/c${\displaystyle D_{4h}^{6}}$56a${\displaystyle (c:a:a)\cdot m:4\odot {\widetilde {ac}}}$${\displaystyle [4_{0}{*}{:}2]}$
129P4/nmmP 4/n 21/m 2/m${\displaystyle D_{4h}^{7}}$55a${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\cdot m}$${\displaystyle (*4{\cdot }4{\cdot }2)}$
130P4/nccP 4/n 21/c 2/c${\displaystyle D_{4h}^{8}}$57a${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\cdot {\tilde {c}}}$${\displaystyle (*4{:}4{:}2)}$
131P42/mmcP 42/m 2/m 2/c${\displaystyle D_{4h}^{9}}$60a${\displaystyle (c:a:a)\cdot m:4_{2}\cdot m}$${\displaystyle [*{\cdot }4{:}4{\cdot }2]}$
132P42/mcmP 42/m 2/c 2/m${\displaystyle D_{4h}^{10}}$61a${\displaystyle (c:a:a)\cdot m:4_{2}\cdot {\tilde {c}}}$${\displaystyle [*{:}4{\cdot }4{:}2]}$
133P42/nbcP 42/n 2/b 2/c${\displaystyle D_{4h}^{11}}$63a${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\odot {\tilde {a}}}$${\displaystyle (*4_{2}4{:}2)}$
134P42/nnmP 42/n 2/n 2/m${\displaystyle D_{4h}^{12}}$62a${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\odot {\widetilde {ac}}}$${\displaystyle (*4_{2}4{\cdot }2)}$
135P42/mbcP 42/m 21/b 2/c${\displaystyle D_{4h}^{13}}$66a${\displaystyle (c:a:a)\cdot m:4_{2}\odot {\tilde {a}}}$${\displaystyle [4_{2}{*}{:}2]}$
136P42/mnmP 42/m 21/n 2/m${\displaystyle D_{4h}^{14}}$65a${\displaystyle (c:a:a)\cdot m:4_{2}\odot {\widetilde {ac}}}$${\displaystyle [4_{2}{*}{\cdot }2]}$
137P42/nmcP 42/n 21/m 2/c${\displaystyle D_{4h}^{15}}$67a${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\cdot m}$${\displaystyle (*4{\cdot }4{:}2)}$
138P42/ncmP 42/n 21/c 2/m${\displaystyle D_{4h}^{16}}$65a${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\cdot {\tilde {c}}}$${\displaystyle (*4{:}4{\cdot }2)}$
139I4/mmmI 4/m 2/m 2/m${\displaystyle D_{4h}^{17}}$37s${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot m:4\cdot m}$${\displaystyle [*{\cdot }4{\cdot }4{:}2]}$
140I4/mcmI 4/m 2/c 2/m${\displaystyle D_{4h}^{18}}$38h${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot m:4\cdot {\tilde {c}}}$${\displaystyle [*{\cdot }4{:}4{:}2]}$
141I41/amdI 41/a 2/m 2/d${\displaystyle D_{4h}^{19}}$59a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}\odot m}$${\displaystyle (*4_{1}4{\cdot }2)}$
142I41/acdI 41/a 2/c 2/d${\displaystyle D_{4h}^{20}}$58a${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}\odot {\tilde {c}}}$${\displaystyle (*4_{1}4{:}2)}$

## List of Trigonal

Unit cells for trigonal crystal system
Rhombohedral
(R)
Hexagonal
(P)
Trigonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1433${\displaystyle 33}$P3P 3${\displaystyle C_{3}^{1}}$38s${\displaystyle (c:(a/a)):3}$${\displaystyle (3_{0}3_{0}3_{0})}$
144P31P 31${\displaystyle C_{3}^{2}}$68a${\displaystyle (c:(a/a)):3_{1}}$${\displaystyle (3_{1}3_{1}3_{1})}$
145P32P 32${\displaystyle C_{3}^{3}}$69a${\displaystyle (c:(a/a)):3_{2}}$${\displaystyle (3_{1}3_{1}3_{1})}$
146R3R 3${\displaystyle C_{3}^{4}}$39s${\displaystyle (a/a/a)/3}$${\displaystyle (3_{0}3_{1}3_{2})}$
1473${\displaystyle 3\times }$P3P 3${\displaystyle C_{3i}^{1}}$51s${\displaystyle (c:(a/a)):{\tilde {6}}}$${\displaystyle (63_{0}2)}$
148R3R 3${\displaystyle C_{3i}^{2}}$52s${\displaystyle (a/a/a)/{\tilde {6}}}$${\displaystyle (63_{1}2)}$
14932${\displaystyle 223}$P312P 3 1 2${\displaystyle D_{3}^{1}}$45s${\displaystyle (c:(a/a)):2:3}$${\displaystyle (*3_{0}3_{0}3_{0})}$
150P321P 3 2 1${\displaystyle D_{3}^{2}}$44s${\displaystyle (c:(a/a))\cdot 2:3}$${\displaystyle (3_{0}{*}3_{0})}$
151P3112P 31 1 2${\displaystyle D_{3}^{3}}$72a${\displaystyle (c:(a/a)):2:3_{1}}$${\displaystyle (*3_{1}3_{1}3_{1})}$
152P3121P 31 2 1${\displaystyle D_{3}^{4}}$70a${\displaystyle (c:(a/a))\cdot 2:3_{1}}$${\displaystyle (3_{1}{*}3_{1})}$
153P3212P 32 1 2${\displaystyle D_{3}^{5}}$73a${\displaystyle (c:(a/a)):2:3_{2}}$${\displaystyle (*3_{1}3_{1}3_{1})}$
154P3221P 32 2 1${\displaystyle D_{3}^{6}}$71a${\displaystyle (c:(a/a))\cdot 2:3_{2}}$${\displaystyle (3_{1}{*}3_{1})}$
155R32R 3 2${\displaystyle D_{3}^{7}}$46s${\displaystyle (a/a/a)/3:2}$${\displaystyle (*3_{0}3_{1}3_{2})}$
1563m${\displaystyle *33}$P3m1P 3 m 1${\displaystyle C_{3v}^{1}}$40s${\displaystyle (c:(a/a)):m\cdot 3}$${\displaystyle (*{\cdot }3{\cdot }3{\cdot }3)}$
157P31mP 3 1 m${\displaystyle C_{3v}^{2}}$41s${\displaystyle (c:(a/a))\cdot m\cdot 3}$${\displaystyle (3_{0}{*}{\cdot }3)}$
158P3c1P 3 c 1${\displaystyle C_{3v}^{3}}$39h${\displaystyle (c:(a/a)):{\tilde {c}}:3}$${\displaystyle (*{:}3{:}3{:}3)}$
159P31cP 3 1 c${\displaystyle C_{3v}^{4}}$40h${\displaystyle (c:(a/a))\cdot {\tilde {c}}:3}$${\displaystyle (3_{0}{*}{:}3)}$
160R3mR 3 m${\displaystyle C_{3v}^{5}}$42s${\displaystyle (a/a/a)/3\cdot m}$${\displaystyle (3_{1}{*}{\cdot }3)}$
161R3cR 3 c${\displaystyle C_{3v}^{6}}$41h${\displaystyle (a/a/a)/3\cdot {\tilde {c}}}$${\displaystyle (3_{1}{*}{:}3)}$
1623 2/m${\displaystyle 2{*}3}$P31mP 3 1 2/m${\displaystyle D_{3d}^{1}}$56s${\displaystyle (c:(a/a))\cdot m\cdot {\tilde {6}}}$${\displaystyle (*{\cdot }63_{0}2)}$
163P31cP 3 1 2/c${\displaystyle D_{3d}^{2}}$46h${\displaystyle (c:(a/a))\cdot {\tilde {c}}\cdot {\tilde {6}}}$${\displaystyle (*{:}63_{0}2)}$
164P3m1P 3 2/m 1${\displaystyle D_{3d}^{3}}$55s${\displaystyle (c:(a/a)):m\cdot {\tilde {6}}}$${\displaystyle (*6{\cdot }3{\cdot }2)}$
165P3c1P 3 2/c 1${\displaystyle D_{3d}^{4}}$45h${\displaystyle (c:(a/a)):{\tilde {c}}\cdot {\tilde {6}}}$${\displaystyle (*6{:}3{:}2)}$
166R3mR 3 2/m${\displaystyle D_{3d}^{5}}$57s${\displaystyle (a/a/a)/{\tilde {6}}\cdot m}$${\displaystyle (*{\cdot }63_{1}2)}$
167R3cR 3 2/c${\displaystyle D_{3d}^{6}}$47h${\displaystyle (a/a/a)/{\tilde {6}}\cdot {\tilde {c}}}$${\displaystyle (*{:}63_{1}2)}$

## List of Hexagonal

Hexagonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1686${\displaystyle 66}$P6P 6${\displaystyle C_{6}^{1}}$49s${\displaystyle (c:(a/a)):6}$${\displaystyle (6_{0}3_{0}2_{0})}$
169P61P 61${\displaystyle C_{6}^{2}}$74a${\displaystyle (c:(a/a)):6_{1}}$${\displaystyle (6_{1}3_{1}2_{1})}$
170P65P 65${\displaystyle C_{6}^{3}}$75a${\displaystyle (c:(a/a)):6_{5}}$${\displaystyle (6_{1}3_{1}2_{1})}$
171P62P 62${\displaystyle C_{6}^{4}}$76a${\displaystyle (c:(a/a)):6_{2}}$${\displaystyle (6_{2}3_{2}2_{0})}$
172P64P 64${\displaystyle C_{6}^{5}}$77a${\displaystyle (c:(a/a)):6_{4}}$${\displaystyle (6_{2}3_{2}2_{0})}$
173P63P 63${\displaystyle C_{6}^{6}}$78a${\displaystyle (c:(a/a)):6_{3}}$${\displaystyle (6_{3}3_{0}2_{1})}$
1746${\displaystyle 3*}$P6P 6${\displaystyle C_{3h}^{1}}$43s${\displaystyle (c:(a/a)):3:m}$${\displaystyle [3_{0}3_{0}3_{0}]}$
1756/m${\displaystyle 6*}$P6/mP 6/m${\displaystyle C_{6h}^{1}}$53s${\displaystyle (c:(a/a))\cdot m:6}$${\displaystyle [6_{0}3_{0}2_{0}]}$
176P63/mP 63/m${\displaystyle C_{6h}^{2}}$81a${\displaystyle (c:(a/a))\cdot m:6_{3}}$${\displaystyle [6_{3}3_{0}2_{1}]}$
177622${\displaystyle 226}$P622P 6 2 2${\displaystyle D_{6}^{1}}$54s${\displaystyle (c:(a/a))\cdot 2:6}$${\displaystyle (*6_{0}3_{0}2_{0})}$
178P6122P 61 2 2${\displaystyle D_{6}^{2}}$82a${\displaystyle (c:(a/a))\cdot 2:6_{1}}$${\displaystyle (*6_{1}3_{1}2_{1})}$
179P6522P 65 2 2${\displaystyle D_{6}^{3}}$83a${\displaystyle (c:(a/a))\cdot 2:6_{5}}$${\displaystyle (*6_{1}3_{1}2_{1})}$
180P6222P 62 2 2${\displaystyle D_{6}^{4}}$84a${\displaystyle (c:(a/a))\cdot 2:6_{2}}$${\displaystyle (*6_{2}3_{2}2_{0})}$
181P6422P 64 2 2${\displaystyle D_{6}^{5}}$85a${\displaystyle (c:(a/a))\cdot 2:6_{4}}$${\displaystyle (*6_{2}3_{2}2_{0})}$
182P6322P 63 2 2${\displaystyle D_{6}^{6}}$86a${\displaystyle (c:(a/a))\cdot 2:6_{3}}$${\displaystyle (*6_{3}3_{0}2_{1})}$
1836mm${\displaystyle *66}$P6mmP 6 m m${\displaystyle C_{6v}^{1}}$50s${\displaystyle (c:(a/a)):m\cdot 6}$${\displaystyle (*{\cdot }6{\cdot }3{\cdot }2)}$
184P6ccP 6 c c${\displaystyle C_{6v}^{2}}$44h${\displaystyle (c:(a/a)):{\tilde {c}}\cdot 6}$${\displaystyle (*{:}6{:}3{:}2)}$
185P63cmP 63 c m${\displaystyle C_{6v}^{3}}$80a${\displaystyle (c:(a/a)):{\tilde {c}}\cdot 6_{3}}$${\displaystyle (*{\cdot }6{:}3{:}2)}$
186P63mcP 63 m c${\displaystyle C_{6v}^{4}}$79a${\displaystyle (c:(a/a)):m\cdot 6_{3}}$${\displaystyle (*{:}6{\cdot }3{\cdot }2)}$
1876m2${\displaystyle *223}$P6m2P 6 m 2${\displaystyle D_{3h}^{1}}$48s${\displaystyle (c:(a/a)):m\cdot 3:m}$${\displaystyle [*{\cdot }3{\cdot }3{\cdot }3]}$
188P6c2P 6 c 2${\displaystyle D_{3h}^{2}}$43h${\displaystyle (c:(a/a)):{\tilde {c}}\cdot 3:m}$${\displaystyle [*{:}3{:}3{:}3]}$
189P62mP 6 2 m${\displaystyle D_{3h}^{3}}$47s${\displaystyle (c:(a/a))\cdot m:3\cdot m}$${\displaystyle [3_{0}{*}{\cdot }3]}$
190P62cP 6 2 c${\displaystyle D_{3h}^{4}}$42h${\displaystyle (c:(a/a))\cdot m:3\cdot {\tilde {c}}}$${\displaystyle [3_{0}{*}{:}3]}$
1916/m 2/m 2/m${\displaystyle *226}$P6/mmmP 6/m 2/m 2/m${\displaystyle D_{6h}^{1}}$58s${\displaystyle (c:(a/a))\cdot m:6\cdot m}$${\displaystyle [*{\cdot }6{\cdot }3{\cdot }2]}$
192P6/mccP 6/m 2/c 2/c${\displaystyle D_{6h}^{2}}$48h${\displaystyle (c:(a/a))\cdot m:6\cdot {\tilde {c}}}$${\displaystyle [*{:}6{:}3{:}2]}$
193P63/mcmP 63/m 2/c 2/m${\displaystyle D_{6h}^{3}}$87a${\displaystyle (c:(a/a))\cdot m:6_{3}\cdot {\tilde {c}}}$${\displaystyle [*{\cdot }6{:}3{:}2]}$
194P63/mmcP 63/m 2/m 2/c${\displaystyle D_{6h}^{4}}$88a${\displaystyle (c:(a/a))\cdot m:6_{3}\cdot m}$${\displaystyle [*{:}6{\cdot }3{\cdot }2]}$

## List of Cubic

Cubic Bravais lattice
Simple
(P)
Body centered
(I)
Face centered
(F)
Cubic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Conway Fibrifold (preserving ${\displaystyle z}$) Fibrifold (preserving ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$)
19523${\displaystyle 332}$P23P 2 3${\displaystyle T^{1}}$59s${\displaystyle \left(a:a:a\right):2/3}$${\displaystyle 2^{\circ }}$${\displaystyle (*2_{0}2_{0}2_{0}2_{0}){:}3}$${\displaystyle (*2_{0}2_{0}2_{0}2_{0}){:}3}$
196F23F 2 3${\displaystyle T^{2}}$61s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):2/3}$${\displaystyle 1^{\circ }}$${\displaystyle (*2_{0}2_{1}2_{0}2_{1}){:}3}$${\displaystyle (*2_{0}2_{1}2_{0}2_{1}){:}3}$
197I23I 2 3${\displaystyle T^{3}}$60s${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right):2/3}$${\displaystyle 4^{\circ \circ }}$${\displaystyle (2_{1}{*}2_{0}2_{0}){:}3}$${\displaystyle (2_{1}{*}2_{0}2_{0}){:}3}$
198P213P 21 3${\displaystyle T^{4}}$89a${\displaystyle \left(a:a:a\right):2_{1}/3}$${\displaystyle 1^{\circ }/4}$${\displaystyle (2_{1}2_{1}{\bar {\times }}){:}3}$${\displaystyle (2_{1}2_{1}{\bar {\times }}){:}3}$
199I213I 21 3${\displaystyle T^{5}}$90a${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right):2_{1}/3}$${\displaystyle 2^{\circ }/4}$${\displaystyle (2_{0}{*}2_{1}2_{1}){:}3}$${\displaystyle (2_{0}{*}2_{1}2_{1}){:}3}$
2002/m 3${\displaystyle 3{*}2}$Pm3P 2/m 3${\displaystyle T_{h}^{1}}$62s${\displaystyle \left(a:a:a\right)\cdot m/{\tilde {6}}}$${\displaystyle 4^{-}}$${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{\cdot }2]{:}3}$${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{\cdot }2]{:}3}$
201Pn3P 2/n 3${\displaystyle T_{h}^{2}}$49h${\displaystyle \left(a:a:a\right)\cdot {\widetilde {ab}}/{\tilde {6}}}$${\displaystyle 4^{\circ +}}$${\displaystyle (2{\bar {*}}_{1}2_{0}2_{0}){:}3}$${\displaystyle (2{\bar {*}}_{1}2_{0}2_{0}){:}3}$
202Fm3F 2/m 3${\displaystyle T_{h}^{3}}$64s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right)\cdot m/{\tilde {6}}}$${\displaystyle 2^{-}}$${\displaystyle [*{\cdot }2{\cdot }2{:}2{:}2]{:}3}$${\displaystyle [*{\cdot }2{\cdot }2{:}2{:}2]{:}3}$
203Fd3F 2/d 3${\displaystyle T_{h}^{4}}$50h${\displaystyle \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}}}$${\displaystyle 2^{\circ +}}$${\displaystyle (2{\bar {*}}2_{0}2_{1}){:}3}$${\displaystyle (2{\bar {*}}2_{0}2_{1}){:}3}$
204Im3I 2/m 3${\displaystyle T_{h}^{5}}$63s${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right)\cdot m/{\tilde {6}}}$${\displaystyle 8^{-\circ }}$${\displaystyle [2_{1}{*}{\cdot }2{\cdot }2]{:}3}$${\displaystyle [2_{1}{*}{\cdot }2{\cdot }2]{:}3}$
205Pa3P 21/a 3${\displaystyle T_{h}^{6}}$91a${\displaystyle \left(a:a:a\right)\cdot {\tilde {a}}/{\tilde {6}}}$${\displaystyle 2^{-}/4}$${\displaystyle (2_{1}2{\bar {*}}{:}){:}3)}$${\displaystyle (2_{1}2{\bar {*}}{:}){:}3)}$
206Ia3I 21/a 3${\displaystyle T_{h}^{7}}$92a${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right)\cdot {\tilde {a}}/{\tilde {6}}}$${\displaystyle 4^{-}/4}$${\displaystyle (*2_{1}2{:}2{:}2){:}3}$${\displaystyle (*2_{1}2{:}2{:}2){:}3}$
207432${\displaystyle 432}$P432P 4 3 2${\displaystyle O^{1}}$68s${\displaystyle \left(a:a:a\right):4/3}$${\displaystyle 4^{\circ -}}$${\displaystyle (*4_{0}4_{0}2_{0}){:}3}$${\displaystyle (*2_{0}2_{0}2_{0}2_{0}){:}6}$
208P4232P 42 3 2${\displaystyle O^{2}}$98a${\displaystyle \left(a:a:a\right):4_{2}//3}$${\displaystyle 4^{+}}$${\displaystyle (*4_{2}4_{2}2_{0}){:}3}$${\displaystyle (*2_{0}2_{0}2_{0}2_{0}){:}6}$
209F432F 4 3 2${\displaystyle O^{3}}$70s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4/3}$${\displaystyle 2^{\circ -}}$${\displaystyle (*4_{2}4_{0}2_{1}){:}3}$${\displaystyle (*2_{0}2_{1}2_{0}2_{1}){:}6}$
210F4132F 41 3 2${\displaystyle O^{4}}$97a${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4_{1}//3}$${\displaystyle 2^{+}}$${\displaystyle (*4_{3}4_{1}2_{0}){:}3}$${\displaystyle (*2_{0}2_{1}2_{0}2_{1}){:}6}$
211I432I 4 3 2${\displaystyle O^{5}}$69s${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right):4/3}$${\displaystyle 8^{+\circ }}$${\displaystyle (4_{2}4_{0}2_{1}){:3}}$${\displaystyle (2_{1}{*}2_{0}2_{0}){:}6}$
212P4332P 43 3 2${\displaystyle O^{6}}$94a${\displaystyle \left(a:a:a\right):4_{3}//3}$${\displaystyle 2^{+}/4}$${\displaystyle (4_{1}{*}2_{1}){:}3}$${\displaystyle (2_{1}2_{1}{\bar {\times }}){:}6}$
213P4132P 41 3 2${\displaystyle O^{7}}$95a${\displaystyle \left(a:a:a\right):4_{1}//3}$${\displaystyle 2^{+}/4}$${\displaystyle (4_{1}{*}2_{1}){:}3}$${\displaystyle (2_{1}2_{1}{\bar {\times }}){:}6}$
214I4132I 41 3 2${\displaystyle O^{8}}$96a${\displaystyle \left({\tfrac {a+b+c}{2}}/:a:a:a\right):4_{1}//3}$${\displaystyle 4^{+}/4}$${\displaystyle (*4_{3}4_{1}2_{0}){:}3}$${\displaystyle (2_{0}{*}2_{1}2_{1}){:}6}$
21543m${\displaystyle *332}$P43mP 4 3 m${\displaystyle T_{d}^{1}}$65s${\displaystyle \left(a:a:a\right):{\tilde {4}}/3}$${\displaystyle 2^{\circ }{:}2}$${\displaystyle (*4{\cdot }42_{0}){:}3}$${\displaystyle (*2_{0}2_{0}2_{0}2_{0}){:}6}$
216F43mF 4 3 m${\displaystyle T_{d}^{2}}$67s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):{\tilde {4}}/3}$${\displaystyle 1^{\circ }{:}2}$${\displaystyle (*4{\cdot }42_{1}){:}3}$${\displaystyle (*2_{0}2_{1}2_{0}2_{1}){:}6}$
217I43mI 4 3 m${\displaystyle T_{d}^{3}}$66s${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right):{\tilde {4}}/3}$${\displaystyle 4^{\circ }{:}2}$${\displaystyle (*{\cdot }44{:}2){:}3}$${\displaystyle (2_{1}{*}2_{0}2_{0}){:}6}$
218P43nP 4 3 n${\displaystyle T_{d}^{4}}$51h${\displaystyle \left(a:a:a\right):{\tilde {4}}//3}$${\displaystyle 4^{\circ }}$${\displaystyle (*4{:}42_{0}){:}3}$${\displaystyle (*2_{0}2_{0}2_{0}2_{0}){:}6}$
219F43cF 4 3 c${\displaystyle T_{d}^{5}}$52h${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):{\tilde {4}}//3}$${\displaystyle 2^{\circ \circ }}$${\displaystyle (*4{:}42_{1}){:}3}$${\displaystyle (*2_{0}2_{1}2_{0}2_{1}){:}6}$
220I43dI 4 3 d${\displaystyle T_{d}^{6}}$93a${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right):{\tilde {4}}//3}$${\displaystyle 4^{\circ }/4}$${\displaystyle (4{\bar {*}}2_{1}){:}3}$${\displaystyle (2_{0}{*}2_{1}2_{1}){:}6}$
2214/m 3 2/m${\displaystyle *432}$Pm3mP 4/m 3 2/m${\displaystyle O_{h}^{1}}$71s${\displaystyle \left(a:a:a\right):4/{\tilde {6}}\cdot m}$${\displaystyle 4^{-}{:}2}$${\displaystyle [*{\cdot }4{\cdot }4{\cdot }2]{:}3}$${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{\cdot }2]{:}6}$
222Pn3nP 4/n 3 2/n${\displaystyle O_{h}^{2}}$53h${\displaystyle \left(a:a:a\right):4/{\tilde {6}}\cdot {\widetilde {abc}}}$${\displaystyle 8^{\circ \circ }}$${\displaystyle (*4_{0}4{:}2){:}3}$${\displaystyle (2{\bar {*}}_{1}2_{0}2_{0}){:}6}$
223Pm3nP 42/m 3 2/n${\displaystyle O_{h}^{3}}$102a${\displaystyle \left(a:a:a\right):4_{2}//{\tilde {6}}\cdot {\widetilde {abc}}}$${\displaystyle 8^{\circ }}$${\displaystyle [*{\cdot }4{:}4{\cdot }2]{:}3}$${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{\cdot }2]{:}6}$
224Pn3mP 42/n 3 2/m${\displaystyle O_{h}^{4}}$103a${\displaystyle \left(a:a:a\right):4_{2}//{\tilde {6}}\cdot m}$${\displaystyle 4^{+}{:}2}$${\displaystyle (*4_{2}4{\cdot }2){:}3}$${\displaystyle (2{\bar {*}}_{1}2_{0}2_{0}){:}6}$
225Fm3mF 4/m 3 2/m${\displaystyle O_{h}^{5}}$73s${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4/{\tilde {6}}\cdot m}$${\displaystyle 2^{-}{:}2}$${\displaystyle [*{\cdot }4{\cdot }4{:}2]{:}3}$${\displaystyle [*{\cdot }2{\cdot }2{:}2{:}2]{:}6}$
226Fm3cF 4/m 3 2/c${\displaystyle O_{h}^{6}}$54h${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4/{\tilde {6}}\cdot {\tilde {c}}}$${\displaystyle 4^{--}}$${\displaystyle [*{\cdot }4{:}4{:}2]{:}3}$${\displaystyle [*{\cdot }2{\cdot }2{:}2{:}2]{:}6}$
227Fd3mF 41/d 3 2/m${\displaystyle O_{h}^{7}}$100a${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4_{1}//{\tilde {6}}\cdot m}$${\displaystyle 2^{+}{:}2}$${\displaystyle (*4_{1}4{\cdot }2){:}3}$${\displaystyle (2{\bar {*}}2_{0}2_{1}){:}6}$
228Fd3cF 41/d 3 2/c${\displaystyle O_{h}^{8}}$101a${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:a:a:a\right):4_{1}//{\tilde {6}}\cdot {\tilde {c}}}$${\displaystyle 4^{++}}$${\displaystyle (*4_{1}4{:}2){:}3}$${\displaystyle (2{\bar {*}}2_{0}2_{1}){:}6}$
229Im3mI 4/m 3 2/m${\displaystyle O_{h}^{9}}$72s${\displaystyle \left({\tfrac {a+b+c}{2}}/a:a:a\right):4/{\tilde {6}}\cdot m}$