# 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. This is somtimes supplemented with a symbol of the form ${\displaystyle \Gamma _{x}^{y}}$  which specifies the bravais lattice. Here ${\displaystyle x\in \{t,m,o,q,rh,h,c\}}$  is the lattice system, and ${\displaystyle y\in \{\emptyset ,b,v,f\}}$  is the centering type.[1]

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
1 1 ${\displaystyle 1}$  P1 P 1 ${\displaystyle \Gamma _{t}C_{1}^{1}}$  1s ${\displaystyle (a/b/c)\cdot 1}$  ${\displaystyle (\circ )}$
2 1 ${\displaystyle \times }$  P1 P 1 ${\displaystyle \Gamma _{t}C_{i}^{1}}$  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)
3 2 ${\displaystyle 22}$  P2 P 1 2 1 P 1 1 2 ${\displaystyle \Gamma _{m}C_{2}^{1}}$  3s ${\displaystyle (b:(c/a)):2}$  ${\displaystyle (2_{0}2_{0}2_{0}2_{0})}$  ${\displaystyle ({*}_{0}{*}_{0})}$
4 P21 P 1 21 1 P 1 1 21 ${\displaystyle \Gamma _{m}C_{2}^{2}}$  1a ${\displaystyle (b:(c/a)):2_{1}}$  ${\displaystyle (2_{1}2_{1}2_{1}2_{1})}$  ${\displaystyle ({\bar {\times }}{\bar {\times }})}$
5 C2 C 1 2 1 B 1 1 2 ${\displaystyle \Gamma _{m}^{b}C_{2}^{3}}$  4s ${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right):2}$  ${\displaystyle (2_{0}2_{0}2_{1}2_{1})}$  ${\displaystyle ({*}_{1}{*}_{1})}$ , ${\displaystyle ({*}{\bar {\times }})}$
6 m ${\displaystyle *}$  Pm P 1 m 1 P 1 1 m ${\displaystyle \Gamma _{m}C_{s}^{1}}$  5s ${\displaystyle (b:(c/a))\cdot m}$  ${\displaystyle [\circ _{0}]}$  ${\displaystyle ({*}{\cdot }{*}{\cdot })}$
7 Pc P 1 c 1 P 1 1 b ${\displaystyle \Gamma _{m}C_{s}^{2}}$  1h ${\displaystyle (b:(c/a))\cdot {\tilde {c}}}$  ${\displaystyle ({\bar {\circ }}_{0})}$  ${\displaystyle ({*}{:}{*}{:})}$ , ${\displaystyle ({\times }{\times }_{0})}$
8 Cm C 1 m 1 B 1 1 m ${\displaystyle \Gamma _{m}^{b}C_{s}^{3}}$  6s ${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot m}$  ${\displaystyle [\circ _{1}]}$  ${\displaystyle ({*}{\cdot }{*}{:})}$ , ${\displaystyle ({*}{\cdot }{\times })}$
9 Cc C 1 c 1 B 1 1 b ${\displaystyle \Gamma _{m}^{b}C_{s}^{4}}$  2h ${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot {\tilde {c}}}$  ${\displaystyle ({\bar {\circ }}_{1})}$  ${\displaystyle ({*}{:}{\times })}$ , ${\displaystyle ({\times }{\times }_{1})}$
10 2/m ${\displaystyle 2*}$  P2/m P 1 2/m 1 P 1 1 2/m ${\displaystyle \Gamma _{m}C_{2h}^{1}}$  7s ${\displaystyle (b:(c/a))\cdot m:2}$  ${\displaystyle [2_{0}2_{0}2_{0}2_{0}]}$  ${\displaystyle [*2{\cdot }22{\cdot }2)}$
11 P21/m P 1 21/m 1 P 1 1 21/m ${\displaystyle \Gamma _{m}C_{2h}^{2}}$  2a ${\displaystyle (b:(c/a))\cdot m:2_{1}}$  ${\displaystyle [2_{1}2_{1}2_{1}2_{1}]}$  ${\displaystyle (22{*}{\cdot })}$
12 C2/m C 1 2/m 1 B 1 1 2/m ${\displaystyle \Gamma _{m}^{b}C_{2h}^{3}}$  8s ${\displaystyle \left({\tfrac {a+b}{2}}/b:(c/a)\right)\cdot m:2}$  ${\displaystyle [2_{0}2_{0}2_{1}2_{1}]}$  ${\displaystyle (*2{\cdot }22{:}2)}$ , ${\displaystyle (2{\bar {*}}2{\cdot }2)}$
13 P2/c P 1 2/c 1 P 1 1 2/b ${\displaystyle \Gamma _{m}C_{2h}^{4}}$  3h ${\displaystyle (b:(c/a))\cdot {\tilde {c}}:2}$  ${\displaystyle (2_{0}2_{0}22)}$  ${\displaystyle (*2{:}22{:}2)}$ , ${\displaystyle (22{*}_{0})}$
14 P21/c P 1 21/c 1 P 1 1 21/b ${\displaystyle \Gamma _{m}C_{2h}^{5}}$  3a ${\displaystyle (b:(c/a))\cdot {\tilde {c}}:2_{1}}$  ${\displaystyle (2_{1}2_{1}22)}$  ${\displaystyle (22{*}{:})}$ , ${\displaystyle (22{\times })}$
15 C2/c C 1 2/c 1 B 1 1 2/b ${\displaystyle \Gamma _{m}^{b}C_{2h}^{6}}$  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 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 ${\displaystyle 222}$  P222 P 2 2 2 ${\displaystyle \Gamma _{o}D_{2}^{1}}$  9s ${\displaystyle (c:a:b):2:2}$  ${\displaystyle (*2_{0}2_{0}2_{0}2_{0})}$
17 P2221 P 2 2 21 ${\displaystyle \Gamma _{o}D_{2}^{2}}$  4a ${\displaystyle (c:a:b):2_{1}:2}$  ${\displaystyle (*2_{1}2_{1}2_{1}2_{1})}$  ${\displaystyle (2_{0}2_{0}{*})}$
18 P21212 P 21 21 2 ${\displaystyle \Gamma _{o}D_{2}^{3}}$  7a ${\displaystyle (c:a:b):2}$    ${\displaystyle 2_{1}}$  ${\displaystyle (2_{0}2_{0}{\bar {\times }})}$  ${\displaystyle (2_{1}2_{1}{*})}$
19 P212121 P 21 21 21 ${\displaystyle \Gamma _{o}D_{2}^{4}}$  8a ${\displaystyle (c:a:b):2_{1}}$    ${\displaystyle 2_{1}}$  ${\displaystyle (2_{1}2_{1}{\bar {\times }})}$
20 C2221 C 2 2 21 ${\displaystyle \Gamma _{o}^{b}D_{2}^{5}}$  5a ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):2_{1}:2}$  ${\displaystyle (2_{1}{*}2_{1}2_{1})}$  ${\displaystyle (2_{0}2_{1}{*})}$
21 C222 C 2 2 2 ${\displaystyle \Gamma _{o}^{b}D_{2}^{6}}$  10s ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):2:2}$  ${\displaystyle (2_{0}{*}2_{0}2_{0})}$  ${\displaystyle (*2_{0}2_{0}2_{1}2_{1})}$
22 F222 F 2 2 2 ${\displaystyle \Gamma _{o}^{f}D_{2}^{7}}$  12s ${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):2:2}$  ${\displaystyle (*2_{0}2_{1}2_{0}2_{1})}$
23 I222 I 2 2 2 ${\displaystyle \Gamma _{o}^{v}D_{2}^{8}}$  11s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):2:2}$  ${\displaystyle (2_{1}{*}2_{0}2_{0})}$
24 I212121 I 21 21 21 ${\displaystyle \Gamma _{o}^{v}D_{2}^{9}}$  6a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):2:2_{1}}$  ${\displaystyle (2_{0}{*}2_{1}2_{1})}$
25 mm2 ${\displaystyle *22}$  Pmm2 P m m 2 ${\displaystyle \Gamma _{o}C_{2v}^{1}}$  13s ${\displaystyle (c:a:b):m\cdot 2}$  ${\displaystyle (*{\cdot }2{\cdot }2{\cdot }2{\cdot }2)}$  ${\displaystyle [{*}_{0}{\cdot }{*}_{0}{\cdot }]}$
26 Pmc21 P m c 21 ${\displaystyle \Gamma _{o}C_{2v}^{2}}$  9a ${\displaystyle (c:a:b):{\tilde {c}}\cdot 2_{1}}$  ${\displaystyle (*{\cdot }2{:}2{\cdot }2{:}2)}$  ${\displaystyle ({\bar {*}}{\cdot }{\bar {*}}{\cdot })}$ , ${\displaystyle [{\times _{0}}{\times _{0}}]}$
27 Pcc2 P c c 2 ${\displaystyle \Gamma _{o}C_{2v}^{3}}$  5h ${\displaystyle (c:a:b):{\tilde {c}}\cdot 2}$  ${\displaystyle (*{:}2{:}2{:}2{:}2)}$  ${\displaystyle ({\bar {*}}_{0}{\bar {*}}_{0})}$
28 Pma2 P m a 2 ${\displaystyle \Gamma _{o}C_{2v}^{4}}$  6h ${\displaystyle (c:a:b):{\tilde {a}}\cdot 2}$  ${\displaystyle (2_{0}2_{0}{*}{\cdot })}$  ${\displaystyle [{*}_{0}{:}{*}_{0}{:}]}$ , ${\displaystyle (*{\cdot }{*}_{0})}$
29 Pca21 P c a 21 ${\displaystyle \Gamma _{o}C_{2v}^{5}}$  11a ${\displaystyle (c:a:b):{\tilde {a}}\cdot 2_{1}}$  ${\displaystyle (2_{1}2_{1}{*}{:})}$  ${\displaystyle ({\bar {*}}{:}{\bar {*}}{:})}$
30 Pnc2 P n c 2 ${\displaystyle \Gamma _{o}C_{2v}^{6}}$  7h ${\displaystyle (c:a:b):{\tilde {c}}\odot 2}$  ${\displaystyle (2_{0}2_{0}{*}{:})}$  ${\displaystyle ({\bar {*}}_{1}{\bar {*}}_{1})}$ , ${\displaystyle ({*}_{0}{\times }_{0})}$
31 Pmn21 P m n 21 ${\displaystyle \Gamma _{o}C_{2v}^{7}}$  10a ${\displaystyle (c:a:b):{\widetilde {ac}}\cdot 2_{1}}$  ${\displaystyle (2_{1}2_{1}{*}{\cdot })}$  ${\displaystyle (*{\cdot }{\bar {\times }})}$ , ${\displaystyle [{\times }_{0}{\times }_{1}]}$
32 Pba2 P b a 2 ${\displaystyle \Gamma _{o}C_{2v}^{8}}$  9h ${\displaystyle (c:a:b):{\tilde {a}}\odot 2}$  ${\displaystyle (2_{0}2_{0}{\times }_{0})}$  ${\displaystyle (*{:}{*}_{0})}$
33 Pna21 P n a 21 ${\displaystyle \Gamma _{o}C_{2v}^{9}}$  12a ${\displaystyle (c:a:b):{\tilde {a}}\odot 2_{1}}$  ${\displaystyle (2_{1}2_{1}{\times })}$  ${\displaystyle (*{:}{\times })}$ , ${\displaystyle ({\times }{\times }_{1})}$
34 Pnn2 P n n 2 ${\displaystyle \Gamma _{o}C_{2v}^{10}}$  8h ${\displaystyle (c:a:b):{\widetilde {ac}}\odot 2}$  ${\displaystyle (2_{0}2_{0}{\times }_{1})}$  ${\displaystyle (*_{0}{\times }_{1})}$
35 Cmm2 C m m 2 ${\displaystyle \Gamma _{o}^{b}C_{2v}^{11}}$  14s ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):m\cdot 2}$  ${\displaystyle (2_{0}{*}{\cdot }2{\cdot }2)}$  ${\displaystyle [*_{0}{\cdot }{*}_{0}{:}]}$
36 Cmc21 C m c 21 ${\displaystyle \Gamma _{o}^{b}C_{2v}^{12}}$  13a ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):{\tilde {c}}\cdot 2_{1}}$  ${\displaystyle (2_{1}{*}{\cdot }2{:}2)}$  ${\displaystyle ({\bar {*}}{\cdot }{\bar {*}}{:})}$ , ${\displaystyle [{\times }_{1}{\times }_{1}]}$
37 Ccc2 C c c 2 ${\displaystyle \Gamma _{o}^{b}C_{2v}^{13}}$  10h ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right):{\tilde {c}}\cdot 2}$  ${\displaystyle (2_{0}{*}{:}2{:}2)}$  ${\displaystyle ({\bar {*}}_{0}{\bar {*}}_{1})}$
38 Amm2 A m m 2 ${\displaystyle \Gamma _{o}^{b}C_{2v}^{14}}$  15s ${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):m\cdot 2}$  ${\displaystyle (*{\cdot }2{\cdot }2{\cdot }2{:}2)}$  ${\displaystyle [{*}_{1}{\cdot }{*}_{1}{\cdot }]}$ , ${\displaystyle [*{\cdot }{\times }_{0}]}$
39 Aem2 A b m 2 ${\displaystyle \Gamma _{o}^{b}C_{2v}^{15}}$  11h ${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):m\cdot 2_{1}}$  ${\displaystyle (*{\cdot }2{:}2{:}2{:}2)}$  ${\displaystyle [{*}_{1}{:}{*}_{1}{:}]}$ , ${\displaystyle ({\bar {*}}{\cdot }{\bar {*}}_{0})}$
40 Ama2 A m a 2 ${\displaystyle \Gamma _{o}^{b}C_{2v}^{16}}$  12h ${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):{\tilde {a}}\cdot 2}$  ${\displaystyle (2_{0}2_{1}{*}{\cdot })}$  ${\displaystyle (*{\cdot }{*}_{1})}$ , ${\displaystyle [*{:}{\times }_{1}]}$
41 Aea2 A b a 2 ${\displaystyle \Gamma _{o}^{b}C_{2v}^{17}}$  13h ${\displaystyle \left({\tfrac {b+c}{2}}/c:a:b\right):{\tilde {a}}\cdot 2_{1}}$  ${\displaystyle (2_{0}2_{1}{*}{:})}$  ${\displaystyle (*{:}{*}_{1})}$ , ${\displaystyle ({\bar {*}}{:}{\bar {*}}_{1})}$
42 Fmm2 F m m 2 ${\displaystyle \Gamma _{o}^{f}C_{2v}^{18}}$  17s ${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):m\cdot 2}$  ${\displaystyle (*{\cdot }2{\cdot }2{:}2{:}2)}$  ${\displaystyle [{*}_{1}{\cdot }{*}_{1}{:}]}$
43 Fdd2 F dd2 ${\displaystyle \Gamma _{o}^{f}C_{2v}^{19}}$  16h ${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right):{\tfrac {1}{2}}{\widetilde {ac}}\odot 2}$  ${\displaystyle (2_{0}2_{1}{\times })}$  ${\displaystyle ({*}_{1}{\times })}$
44 Imm2 I m m 2 ${\displaystyle \Gamma _{o}^{v}C_{2v}^{20}}$  16s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):m\cdot 2}$  ${\displaystyle (2_{1}{*}{\cdot }2{\cdot }2)}$  ${\displaystyle [*{\cdot }{\times }_{1}]}$
45 Iba2 I b a 2 ${\displaystyle \Gamma _{o}^{v}C_{2v}^{21}}$  15h ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right):{\tilde {c}}\cdot 2}$  ${\displaystyle (2_{1}{*}{:}2{:}2)}$  ${\displaystyle ({\bar {*}}{:}{\bar {*}}_{0})}$
46 Ima2 I m a 2 ${\displaystyle \Gamma _{o}^{v}C_{2v}^{22}}$  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 ${\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$  ${\displaystyle *222}$  Pmmm P 2/m 2/m 2/m ${\displaystyle \Gamma _{o}D_{2h}^{1}}$  18s ${\displaystyle \left(c:a:b\right)\cdot m:2\cdot m}$  ${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{\cdot }2]}$
48 Pnnn P 2/n 2/n 2/n ${\displaystyle \Gamma _{o}D_{2h}^{2}}$  19h ${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\odot {\widetilde {ac}}}$  ${\displaystyle (2{\bar {*}}_{1}2_{0}2_{0}}$
49 Pccm P 2/c 2/c 2/m ${\displaystyle \Gamma _{o}D_{2h}^{3}}$  17h ${\displaystyle \left(c:a:b\right)\cdot m:2\cdot {\tilde {c}}}$  ${\displaystyle [*{:}2{:}2{:}2{:}2]}$  ${\displaystyle (*2_{0}2_{0}2{\cdot }2)}$
50 Pban P 2/b 2/a 2/n ${\displaystyle \Gamma _{o}D_{2h}^{4}}$  18h ${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\odot {\tilde {a}}}$  ${\displaystyle (2{\bar {*}}_{0}2_{0}2_{0})}$  ${\displaystyle (*2_{0}2_{0}2{:}2)}$
51 Pmma P 21/m 2/m 2/a ${\displaystyle \Gamma _{o}D_{2h}^{5}}$  14a ${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2\cdot m}$  ${\displaystyle [2_{0}2_{0}{*}{\cdot }]}$  ${\displaystyle [*{\cdot }2{:}2{\cdot }2{:}2]}$ , ${\displaystyle [*2{\cdot }2{\cdot }2{\cdot }2]}$
52 Pnna P 2/n 21/n 2/a ${\displaystyle \Gamma _{o}D_{2h}^{6}}$  17a ${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2\odot {\widetilde {ac}}}$  ${\displaystyle (2_{0}2{\bar {*}}_{1})}$  ${\displaystyle (2_{0}{*}2{:}2)}$ , ${\displaystyle (2{\bar {*}}2_{1}2_{1})}$
53 Pmna P 2/m 2/n 21/a ${\displaystyle \Gamma _{o}D_{2h}^{7}}$  15a ${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\cdot {\widetilde {ac}}}$  ${\displaystyle [2_{0}2_{0}{*}{:}]}$  ${\displaystyle (*2_{1}2_{1}2{\cdot }2)}$ , ${\displaystyle (2_{0}{*}2{\cdot }2)}$
54 Pcca P 21/c 2/c 2/a ${\displaystyle \Gamma _{o}D_{2h}^{8}}$  16a ${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}}$  ${\displaystyle (2_{0}2{\bar {*}}_{0})}$  ${\displaystyle (*2{:}2{:}2{:}2)}$ , ${\displaystyle (*2_{1}2_{1}2{:}2)}$
55 Pbam P 21/b 21/a 2/m ${\displaystyle \Gamma _{o}D_{2h}^{9}}$  22a ${\displaystyle \left(c:a:b\right)\cdot m:2\odot {\tilde {a}}}$  ${\displaystyle [2_{0}2_{0}{\times }_{0}]}$  ${\displaystyle (*2{\cdot }2{:}2{\cdot }2)}$
56 Pccn P 21/c 21/c 2/n ${\displaystyle \Gamma _{o}D_{2h}^{10}}$  27a ${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\cdot {\tilde {c}}}$  ${\displaystyle (2{\bar {*}}{:}2{:}2)}$  ${\displaystyle (2_{1}2{\bar {*}}_{0})}$
57 Pbcm P 2/b 21/c 21/m ${\displaystyle \Gamma _{o}D_{2h}^{11}}$  23a ${\displaystyle \left(c:a:b\right)\cdot m:2_{1}\odot {\tilde {c}}}$  ${\displaystyle (2_{0}2{\bar {*}}{\cdot })}$  ${\displaystyle (*2{:}2{\cdot }2{:}2)}$ , ${\displaystyle [2_{1}2_{1}{*}{:}]}$
58 Pnnm P 21/n 21/n 2/m ${\displaystyle \Gamma _{o}D_{2h}^{12}}$  25a ${\displaystyle \left(c:a:b\right)\cdot m:2\odot {\widetilde {ac}}}$  ${\displaystyle [2_{0}2_{0}{\times }_{1}]}$  ${\displaystyle (2_{1}{*}2{\cdot }2)}$
59 Pmmn P 21/m 21/m 2/n ${\displaystyle \Gamma _{o}D_{2h}^{13}}$  24a ${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2\cdot m}$  ${\displaystyle (2{\bar {*}}{\cdot }2{\cdot }2)}$  ${\displaystyle [2_{1}2_{1}{*}{\cdot }]}$
60 Pbcn P 21/b 2/c 21/n ${\displaystyle \Gamma _{o}D_{2h}^{14}}$  26a ${\displaystyle \left(c:a:b\right)\cdot {\widetilde {ab}}:2_{1}\odot {\tilde {c}}}$  ${\displaystyle (2_{0}2{\bar {*}}{:})}$  ${\displaystyle (2_{1}{*}2{:}2)}$ , ${\displaystyle (2_{1}2{\bar {*}}_{1})}$
61 Pbca P 21/b 21/c 21/a ${\displaystyle \Gamma _{o}D_{2h}^{15}}$  29a ${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\odot {\tilde {c}}}$  ${\displaystyle (2_{1}2{\bar {*}}{:})}$
62 Pnma P 21/n 21/m 21/a ${\displaystyle \Gamma _{o}D_{2h}^{16}}$  28a ${\displaystyle \left(c:a:b\right)\cdot {\tilde {a}}:2_{1}\odot m}$  ${\displaystyle (2_{1}2{\bar {*}}{\cdot })}$  ${\displaystyle (2{\bar {*}}{\cdot }2{:}2)}$ , ${\displaystyle [2_{1}2_{1}{\times }]}$
63 Cmcm C 2/m 2/c 21/m ${\displaystyle \Gamma _{o}^{b}D_{2h}^{17}}$  18a ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2_{1}\cdot {\tilde {c}}}$  ${\displaystyle [2_{0}2_{1}{*}{\cdot }]}$  ${\displaystyle (*2{\cdot }2{\cdot }2{:}2)}$ , ${\displaystyle [2_{1}{*}{\cdot }2{:}2]}$
64 Cmca C 2/m 2/c 21/a ${\displaystyle \Gamma _{o}^{b}D_{2h}^{18}}$  19a ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2_{1}\cdot {\tilde {c}}}$  ${\displaystyle [2_{0}2_{1}{*}{:}]}$  ${\displaystyle (*2{\cdot }2{:}2{:}2)}$ , ${\displaystyle (*2_{1}2{\cdot }2{:}2)}$
65 Cmmm C 2/m 2/m 2/m ${\displaystyle \Gamma _{o}^{b}D_{2h}^{19}}$  19s ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2\cdot m}$  ${\displaystyle [2_{0}{*}{\cdot }2{\cdot }2]}$  ${\displaystyle [*{\cdot }2{\cdot }2{\cdot }2{:}2]}$
66 Cccm C 2/c 2/c 2/m ${\displaystyle \Gamma _{o}^{b}D_{2h}^{20}}$  20h ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2\cdot {\tilde {c}}}$  ${\displaystyle [2_{0}{*}{:}2{:}2]}$  ${\displaystyle (*2_{0}2_{1}2{\cdot }2)}$
67 Cmme C 2/m 2/m 2/e ${\displaystyle \Gamma _{o}^{b}D_{2h}^{21}}$  21h ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2\cdot m}$  ${\displaystyle (*2_{0}2{\cdot }2{\cdot }2)}$  ${\displaystyle [*{\cdot }2{:}2{:}2{:}2]}$
68 Ccce C 2/c 2/c 2/e ${\displaystyle \Gamma _{o}^{b}D_{2h}^{22}}$  22h ${\displaystyle \left({\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}}$  ${\displaystyle (*2_{0}2{:}2{:}2)}$  ${\displaystyle (*2_{0}2_{1}2{:}2)}$
69 Fmmm F 2/m 2/m 2/m ${\displaystyle \Gamma _{o}^{f}D_{2h}^{23}}$  21s ${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right)\cdot m:2\cdot m}$  ${\displaystyle [*{\cdot }2{\cdot }2{:}2{:}2]}$
70 Fddd F 2/d 2/d 2/d ${\displaystyle \Gamma _{o}^{f}D_{2h}^{24}}$  24h ${\displaystyle \left({\tfrac {a+c}{2}}/{\tfrac {b+c}{2}}/{\tfrac {a+b}{2}}:c:a:b\right)\cdot {\tfrac {1}{2}}{\widetilde {ab}}:2\odot {\tfrac {1}{2}}{\widetilde {ac}}}$  ${\displaystyle (2{\bar {*}}2_{0}2_{1})}$
71 Immm I 2/m 2/m 2/m ${\displaystyle \Gamma _{o}^{v}D_{2h}^{25}}$  20s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot m:2\cdot m}$  ${\displaystyle [2_{1}{*}{\cdot }2{\cdot }2]}$
72 Ibam I 2/b 2/a 2/m ${\displaystyle \Gamma _{o}^{v}D_{2h}^{26}}$  23h ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot m:2\cdot {\tilde {c}}}$  ${\displaystyle [2_{1}{*}{:}2{:}2]}$  ${\displaystyle (*2_{0}2{\cdot }2{:}2)}$
73 Ibca I 2/b 2/c 2/a ${\displaystyle \Gamma _{o}^{v}D_{2h}^{27}}$  21a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:b\right)\cdot {\tilde {a}}:2\cdot {\tilde {c}}}$  ${\displaystyle (*2_{1}2{:}2{:}2)}$
74 Imma I 2/m 2/m 2/a ${\displaystyle \Gamma _{o}^{v}D_{2h}^{28}}$  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 Bravais lattice
Simple
(P)
Body
(I)

Tetragonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
75 4 ${\displaystyle 44}$  P4 P 4 ${\displaystyle \Gamma _{q}C_{4}^{1}}$  22s ${\displaystyle (c:a:a):4}$  ${\displaystyle (4_{0}4_{0}2_{0})}$
76 P41 P 41 ${\displaystyle \Gamma _{q}C_{4}^{2}}$  30a ${\displaystyle (c:a:a):4_{1}}$  ${\displaystyle (4_{1}4_{1}2_{1})}$
77 P42 P 42 ${\displaystyle \Gamma _{q}C_{4}^{3}}$  33a ${\displaystyle (c:a:a):4_{2}}$  ${\displaystyle (4_{2}4_{2}2_{0})}$
78 P43 P 43 ${\displaystyle \Gamma _{q}C_{4}^{4}}$  31a ${\displaystyle (c:a:a):4_{3}}$  ${\displaystyle (4_{1}4_{1}2_{1})}$
79 I4 I 4 ${\displaystyle \Gamma _{q}^{v}C_{4}^{5}}$  23s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4}$  ${\displaystyle (4_{2}4_{0}2_{1})}$
80 I41 I 41 ${\displaystyle \Gamma _{q}^{v}C_{4}^{6}}$  32a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}}$  ${\displaystyle (4_{3}4_{1}2_{0})}$
81 4 ${\displaystyle 2\times }$  P4 P 4 ${\displaystyle \Gamma _{q}S_{4}^{1}}$  26s ${\displaystyle (c:a:a):{\tilde {4}}}$  ${\displaystyle (442_{0})}$
82 I4 I 4 ${\displaystyle \Gamma _{q}^{v}S_{4}^{2}}$  27s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}}$  ${\displaystyle (442_{1})}$
83 4/m ${\displaystyle 4*}$  P4/m P 4/m ${\displaystyle \Gamma _{q}C_{4h}^{1}}$  28s ${\displaystyle (c:a:a)\cdot m:4}$  ${\displaystyle [4_{0}4_{0}2_{0}]}$
84 P42/m P 42/m ${\displaystyle \Gamma _{q}C_{4h}^{2}}$  41a ${\displaystyle (c:a:a)\cdot m:4_{2}}$  ${\displaystyle [4_{2}4_{2}2_{0}]}$
85 P4/n P 4/n ${\displaystyle \Gamma _{q}C_{4h}^{3}}$  29h ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4}$  ${\displaystyle (44_{0}2)}$
86 P42/n P 42/n ${\displaystyle \Gamma _{q}C_{4h}^{4}}$  42a ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}}$  ${\displaystyle (44_{2}2)}$
87 I4/m I 4/m ${\displaystyle \Gamma _{q}^{v}C_{4h}^{5}}$  29s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot m:4}$  ${\displaystyle [4_{2}4_{0}2_{1}]}$
88 I41/a I 41/a ${\displaystyle \Gamma _{q}^{v}C_{4h}^{6}}$  40a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}}$  ${\displaystyle (44_{1}2)}$
89 422 ${\displaystyle 224}$  P422 P 4 2 2 ${\displaystyle \Gamma _{q}D_{4}^{1}}$  30s ${\displaystyle (c:a:a):4:2}$  ${\displaystyle (*4_{0}4_{0}2_{0})}$
90 P4212 P4212 ${\displaystyle \Gamma _{q}D_{4}^{2}}$  43a ${\displaystyle (c:a:a):4}$    ${\displaystyle 2_{1}}$  ${\displaystyle (4_{0}{*}2_{0})}$
91 P4122 P 41 2 2 ${\displaystyle \Gamma _{q}D_{4}^{3}}$  44a ${\displaystyle (c:a:a):4_{1}:2}$  ${\displaystyle (*4_{1}4_{1}2_{1})}$
92 P41212 P 41 21 2 ${\displaystyle \Gamma _{q}D_{4}^{4}}$  48a ${\displaystyle (c:a:a):4_{1}}$    ${\displaystyle 2_{1}}$  ${\displaystyle (4_{1}{*}2_{1})}$
93 P4222 P 42 2 2 ${\displaystyle \Gamma _{q}D_{4}^{5}}$  47a ${\displaystyle (c:a:a):4_{2}:2}$  ${\displaystyle (*4_{2}4_{2}2_{0})}$
94 P42212 P 42 21 2 ${\displaystyle \Gamma _{q}D_{4}^{6}}$  50a ${\displaystyle (c:a:a):4_{2}}$    ${\displaystyle 2_{1}}$  ${\displaystyle (4_{2}{*}2_{0})}$
95 P4322 P 43 2 2 ${\displaystyle \Gamma _{q}D_{4}^{7}}$  45a ${\displaystyle (c:a:a):4_{3}:2}$  ${\displaystyle (*4_{1}4_{1}2_{1})}$
96 P43212 P 43 21 2 ${\displaystyle \Gamma _{q}D_{4}^{8}}$  49a ${\displaystyle (c:a:a):4_{3}}$    ${\displaystyle 2_{1}}$  ${\displaystyle (4_{1}{*}2_{1})}$
97 I422 I 4 2 2 ${\displaystyle \Gamma _{q}^{v}D_{4}^{9}}$  31s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4:2}$  ${\displaystyle (*4_{2}4_{0}2_{1})}$
98 I4122 I 41 2 2 ${\displaystyle \Gamma _{q}^{v}D_{4}^{10}}$  46a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4:2_{1}}$  ${\displaystyle (*4_{3}4_{1}2_{0})}$
99 4mm ${\displaystyle *44}$  P4mm P 4 m m ${\displaystyle \Gamma _{q}C_{4v}^{1}}$  24s ${\displaystyle (c:a:a):4\cdot m}$  ${\displaystyle (*{\cdot }4{\cdot }4{\cdot }2)}$
100 P4bm P 4 b m ${\displaystyle \Gamma _{q}C_{4v}^{2}}$  26h ${\displaystyle (c:a:a):4\odot {\tilde {a}}}$  ${\displaystyle (4_{0}{*}{\cdot }2)}$
101 P42cm P 42 c m ${\displaystyle \Gamma _{q}C_{4v}^{3}}$  37a ${\displaystyle (c:a:a):4_{2}\cdot {\tilde {c}}}$  ${\displaystyle (*{:}4{\cdot }4{:}2)}$
102 P42nm P 42 n m ${\displaystyle \Gamma _{q}C_{4v}^{4}}$  38a ${\displaystyle (c:a:a):4_{2}\odot {\widetilde {ac}}}$  ${\displaystyle (4_{2}{*}{\cdot }2)}$
103 P4cc P 4 c c ${\displaystyle \Gamma _{q}C_{4v}^{5}}$  25h ${\displaystyle (c:a:a):4\cdot {\tilde {c}}}$  ${\displaystyle (*{:}4{:}4{:}2)}$
104 P4nc P 4 n c ${\displaystyle \Gamma _{q}C_{4v}^{6}}$  27h ${\displaystyle (c:a:a):4\odot {\widetilde {ac}}}$  ${\displaystyle (4_{0}{*}{:}2)}$
105 P42mc P 42 m c ${\displaystyle \Gamma _{q}C_{4v}^{7}}$  36a ${\displaystyle (c:a:a):4_{2}\cdot m}$  ${\displaystyle (*{\cdot }4{:}4{\cdot }2)}$
106 P42bc P 42 b c ${\displaystyle \Gamma _{q}C_{4v}^{8}}$  39a ${\displaystyle (c:a:a):4\odot {\tilde {a}}}$  ${\displaystyle (4_{2}{*}{:}2)}$
107 I4mm I 4 m m ${\displaystyle \Gamma _{q}^{v}C_{4v}^{9}}$  25s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4\cdot m}$  ${\displaystyle (*{\cdot }4{\cdot }4{:}2)}$
108 I4cm I 4 c m ${\displaystyle \Gamma _{q}^{v}C_{4v}^{10}}$  28h ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4\cdot {\tilde {c}}}$  ${\displaystyle (*{\cdot }4{:}4{:}2)}$
109 I41md I 41 m d ${\displaystyle \Gamma _{q}^{v}C_{4v}^{11}}$  34a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}\odot m}$  ${\displaystyle (4_{1}{*}{\cdot }2)}$
110 I41cd I 41 c d ${\displaystyle \Gamma _{q}^{v}C_{4v}^{12}}$  35a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):4_{1}\odot {\tilde {c}}}$  ${\displaystyle (4_{1}{*}{:}2)}$
111 42m ${\displaystyle 2{*}2}$  P42m P 4 2 m ${\displaystyle \Gamma _{q}D_{2d}^{1}}$  32s ${\displaystyle (c:a:a):{\tilde {4}}:2}$  ${\displaystyle (*4{\cdot }42_{0})}$
112 P42c P 4 2 c ${\displaystyle \Gamma _{q}D_{2d}^{2}}$  30h ${\displaystyle (c:a:a):{\tilde {4}}}$    ${\displaystyle 2}$  ${\displaystyle (*4{:}42_{0})}$
113 P421m P 4 21 m ${\displaystyle \Gamma _{q}D_{2d}^{3}}$  52a ${\displaystyle (c:a:a):{\tilde {4}}\cdot {\widetilde {ab}}}$  ${\displaystyle (4{\bar {*}}{\cdot }2)}$
114 P421c P 4 21 c ${\displaystyle \Gamma _{q}D_{2d}^{4}}$  53a ${\displaystyle (c:a:a):{\tilde {4}}\cdot {\widetilde {abc}}}$  ${\displaystyle (4{\bar {*}}{:}2)}$
115 P4m2 P 4 m 2 ${\displaystyle \Gamma _{q}D_{2d}^{5}}$  33s ${\displaystyle (c:a:a):{\tilde {4}}\cdot m}$  ${\displaystyle (*{\cdot }44{\cdot }2)}$
116 P4c2 P 4 c 2 ${\displaystyle \Gamma _{q}D_{2d}^{6}}$  31h ${\displaystyle (c:a:a):{\tilde {4}}\cdot {\tilde {c}}}$  ${\displaystyle (*{:}44{:}2)}$
117 P4b2 P 4 b 2 ${\displaystyle \Gamma _{q}D_{2d}^{7}}$  32h ${\displaystyle (c:a:a):{\tilde {4}}\odot {\tilde {a}}}$  ${\displaystyle (4{\bar {*}}_{0}2_{0})}$
118 P4n2 P 4 n 2 ${\displaystyle \Gamma _{q}D_{2d}^{8}}$  33h ${\displaystyle (c:a:a):{\tilde {4}}\cdot {\widetilde {ac}}}$  ${\displaystyle (4{\bar {*}}_{1}2_{0})}$
119 I4m2 I 4 m 2 ${\displaystyle \Gamma _{q}^{v}D_{2d}^{9}}$  35s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\cdot m}$  ${\displaystyle (*4{\cdot }42_{1})}$
120 I4c2 I 4 c 2 ${\displaystyle \Gamma _{q}^{v}D_{2d}^{10}}$  34h ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\cdot {\tilde {c}}}$  ${\displaystyle (*4{:}42_{1})}$
121 I42m I 4 2 m ${\displaystyle \Gamma _{q}^{v}D_{2d}^{11}}$  34s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}:2}$  ${\displaystyle (*{\cdot }44{:}2)}$
122 I42d I 4 2 d ${\displaystyle \Gamma _{q}^{v}D_{2d}^{12}}$  51a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right):{\tilde {4}}\odot {\tfrac {1}{2}}{\widetilde {abc}}}$  ${\displaystyle (4{\bar {*}}2_{1})}$
123 4/m 2/m 2/m ${\displaystyle *224}$  P4/mmm P 4/m 2/m 2/m ${\displaystyle \Gamma _{q}D_{4h}^{1}}$  36s ${\displaystyle (c:a:a)\cdot m:4\cdot m}$  ${\displaystyle [*{\cdot }4{\cdot }4{\cdot }2]}$
124 P4/mcc P 4/m 2/c 2/c ${\displaystyle \Gamma _{q}D_{4h}^{2}}$  35h ${\displaystyle (c:a:a)\cdot m:4\cdot {\tilde {c}}}$  ${\displaystyle [*{:}4{:}4{:}2]}$
125 P4/nbm P 4/n 2/b 2/m ${\displaystyle \Gamma _{q}D_{4h}^{3}}$  36h ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\odot {\tilde {a}}}$  ${\displaystyle (*4_{0}4{\cdot }2)}$
126 P4/nnc P 4/n 2/n 2/c ${\displaystyle \Gamma _{q}D_{4h}^{4}}$  37h ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\odot {\widetilde {ac}}}$  ${\displaystyle (*4_{0}4{:}2)}$
127 P4/mbm P 4/m 21/b 2/m ${\displaystyle \Gamma _{q}D_{4h}^{5}}$  54a ${\displaystyle (c:a:a)\cdot m:4\odot {\tilde {a}}}$  ${\displaystyle [4_{0}{*}{\cdot }2]}$
128 P4/mnc P 4/m 21/n 2/c ${\displaystyle \Gamma _{q}D_{4h}^{6}}$  56a ${\displaystyle (c:a:a)\cdot m:4\odot {\widetilde {ac}}}$  ${\displaystyle [4_{0}{*}{:}2]}$
129 P4/nmm P 4/n 21/m 2/m ${\displaystyle \Gamma _{q}D_{4h}^{7}}$  55a ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\cdot m}$  ${\displaystyle (*4{\cdot }4{\cdot }2)}$
130 P4/ncc P 4/n 21/c 2/c ${\displaystyle \Gamma _{q}D_{4h}^{8}}$  57a ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4\cdot {\tilde {c}}}$  ${\displaystyle (*4{:}4{:}2)}$
131 P42/mmc P 42/m 2/m 2/c ${\displaystyle \Gamma _{q}D_{4h}^{9}}$  60a ${\displaystyle (c:a:a)\cdot m:4_{2}\cdot m}$  ${\displaystyle [*{\cdot }4{:}4{\cdot }2]}$
132 P42/mcm P 42/m 2/c 2/m ${\displaystyle \Gamma _{q}D_{4h}^{10}}$  61a ${\displaystyle (c:a:a)\cdot m:4_{2}\cdot {\tilde {c}}}$  ${\displaystyle [*{:}4{\cdot }4{:}2]}$
133 P42/nbc P 42/n 2/b 2/c ${\displaystyle \Gamma _{q}D_{4h}^{11}}$  63a ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\odot {\tilde {a}}}$  ${\displaystyle (*4_{2}4{:}2)}$
134 P42/nnm P 42/n 2/n 2/m ${\displaystyle \Gamma _{q}D_{4h}^{12}}$  62a ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\odot {\widetilde {ac}}}$  ${\displaystyle (*4_{2}4{\cdot }2)}$
135 P42/mbc P 42/m 21/b 2/c ${\displaystyle \Gamma _{q}D_{4h}^{13}}$  66a ${\displaystyle (c:a:a)\cdot m:4_{2}\odot {\tilde {a}}}$  ${\displaystyle [4_{2}{*}{:}2]}$
136 P42/mnm P 42/m 21/n 2/m ${\displaystyle \Gamma _{q}D_{4h}^{14}}$  65a ${\displaystyle (c:a:a)\cdot m:4_{2}\odot {\widetilde {ac}}}$  ${\displaystyle [4_{2}{*}{\cdot }2]}$
137 P42/nmc P 42/n 21/m 2/c ${\displaystyle \Gamma _{q}D_{4h}^{15}}$  67a ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\cdot m}$  ${\displaystyle (*4{\cdot }4{:}2)}$
138 P42/ncm P 42/n 21/c 2/m ${\displaystyle \Gamma _{q}D_{4h}^{16}}$  65a ${\displaystyle (c:a:a)\cdot {\widetilde {ab}}:4_{2}\cdot {\tilde {c}}}$  ${\displaystyle (*4{:}4{\cdot }2)}$
139 I4/mmm I 4/m 2/m 2/m ${\displaystyle \Gamma _{q}^{v}D_{4h}^{17}}$  37s ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot m:4\cdot m}$  ${\displaystyle [*{\cdot }4{\cdot }4{:}2]}$
140 I4/mcm I 4/m 2/c 2/m ${\displaystyle \Gamma _{q}^{v}D_{4h}^{18}}$  38h ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot m:4\cdot {\tilde {c}}}$  ${\displaystyle [*{\cdot }4{:}4{:}2]}$
141 I41/amd I 41/a 2/m 2/d ${\displaystyle \Gamma _{q}^{v}D_{4h}^{19}}$  59a ${\displaystyle \left({\tfrac {a+b+c}{2}}/c:a:a\right)\cdot {\tilde {a}}:4_{1}\odot m}$  ${\displaystyle (*4_{1}4{\cdot }2)}$
142 I41/acd I 41/a 2/c 2/d ${\displaystyle \Gamma _{q}^{v}D_{4h}^{20}}$  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

Trigonal Bravais lattice
Rhombohedral
(R)
Hexagonal
(P)

Trigonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
143 3 ${\displaystyle 33}$  P3 P 3 ${\displaystyle \Gamma _{h}C_{3}^{1}}$  38s ${\displaystyle (c:(a/a)):3}$  ${\displaystyle (3_{0}3_{0}3_{0})}$
144 P31 P 31 ${\displaystyle \Gamma _{h}C_{3}^{2}}$  68a ${\displaystyle (c:(a/a)):3_{1}}$  ${\displaystyle (3_{1}3_{1}3_{1})}$
145 P32 P 32 ${\displaystyle \Gamma _{h}C_{3}^{3}}$  69a ${\displaystyle (c:(a/a)):3_{2}}$  ${\displaystyle (3_{1}3_{1}3_{1})}$
146 R3 R 3 ${\displaystyle \Gamma _{rh}C_{3}^{4}}$  39s ${\displaystyle (a/a/a)/3}$  ${\displaystyle (3_{0}3_{1}3_{2})}$
147 3 ${\displaystyle 3\times }$  P3 P 3 ${\displaystyle \Gamma _{h}C_{3i}^{1}}$  51s ${\displaystyle (c:(a/a)):{\tilde {6}}}$  ${\displaystyle (63_{0}2)}$
148 R3 R 3 ${\displaystyle \Gamma _{rh}C_{3i}^{2}}$  52s ${\displaystyle (a/a/a)/{\tilde {6}}}$  ${\displaystyle (63_{1}2)}$
149 32 ${\displaystyle 223}$  P312 P 3 1 2 ${\displaystyle \Gamma _{h}D_{3}^{1}}$  45s ${\displaystyle (c:(a/a)):2:3}$  ${\displaystyle (*3_{0}3_{0}3_{0})}$
150 P321 P 3 2 1 ${\displaystyle \Gamma _{h}D_{3}^{2}}$  44s ${\displaystyle (c:(a/a))\cdot 2:3}$  ${\displaystyle (3_{0}{*}3_{0})}$
151 P3112 P 31 1 2 ${\displaystyle \Gamma _{h}D_{3}^{3}}$  72a ${\displaystyle (c:(a/a)):2:3_{1}}$  ${\displaystyle (*3_{1}3_{1}3_{1})}$
152 P3121 P 31 2 1 ${\displaystyle \Gamma _{h}D_{3}^{4}}$  70a ${\displaystyle (c:(a/a))\cdot 2:3_{1}}$  ${\displaystyle (3_{1}{*}3_{1})}$
153 P3212 P 32 1 2 ${\displaystyle \Gamma _{h}D_{3}^{5}}$  73a ${\displaystyle (c:(a/a)):2:3_{2}}$  ${\displaystyle (*3_{1}3_{1}3_{1})}$
154 P3221 P 32 2 1 ${\displaystyle \Gamma _{h}D_{3}^{6}}$  71a ${\displaystyle (c:(a/a))\cdot 2:3_{2}}$  ${\displaystyle (3_{1}{*}3_{1})}$
155 R32 R 3 2 ${\displaystyle \Gamma _{rh}D_{3}^{7}}$  46s ${\displaystyle (a/a/a)/3:2}$  ${\displaystyle (*3_{0}3_{1}3_{2})}$
156 3m ${\displaystyle *33}$  P3m1 P 3 m 1 ${\displaystyle \Gamma _{h}C_{3v}^{1}}$  40s ${\displaystyle (c:(a/a)):m\cdot 3}$  ${\displaystyle (*{\cdot }3{\cdot }3{\cdot }3)}$
157 P31m P 3 1 m ${\displaystyle \Gamma _{h}C_{3v}^{2}}$  41s ${\displaystyle (c:(a/a))\cdot m\cdot 3}$  ${\displaystyle (3_{0}{*}{\cdot }3)}$
158 P3c1 P 3 c 1 ${\displaystyle \Gamma _{h}C_{3v}^{3}}$  39h ${\displaystyle (c:(a/a)):{\tilde {c}}:3}$  ${\displaystyle (*{:}3{:}3{:}3)}$
159 P31c P 3 1 c ${\displaystyle \Gamma _{h}C_{3v}^{4}}$  40h ${\displaystyle (c:(a/a))\cdot {\tilde {c}}:3}$  ${\displaystyle (3_{0}{*}{:}3)}$
160 R3m R 3 m ${\displaystyle \Gamma _{rh}C_{3v}^{5}}$  42s ${\displaystyle (a/a/a)/3\cdot m}$  ${\displaystyle (3_{1}{*}{\cdot }3)}$
161 R3c R 3 c ${\displaystyle \Gamma _{rh}C_{3v}^{6}}$  41h ${\displaystyle (a/a/a)/3\cdot {\tilde {c}}}$  ${\displaystyle (3_{1}{*}{:}3)}$
162 3 2/m ${\displaystyle 2{*}3}$  P31m P 3 1 2/m ${\displaystyle \Gamma _{h}D_{3d}^{1}}$  56s ${\displaystyle (c:(a/a))\cdot m\cdot {\tilde {6}}}$  ${\displaystyle (*{\cdot }63_{0}2)}$
163 P31c P 3 1 2/c ${\displaystyle \Gamma _{h}D_{3d}^{2}}$  46h ${\displaystyle (c:(a/a))\cdot {\tilde {c}}\cdot {\tilde {6}}}$  ${\displaystyle (*{:}63_{0}2)}$
164 P3m1 P 3 2/m 1 ${\displaystyle \Gamma _{h}D_{3d}^{3}}$  55s