List of space groups explained

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.

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

A gyration point can be replaced by a screw axis denoted by a number, n, where the angle of rotation is

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

Wherever there is both a rotation or screw axis n and a mirror or glide plane m along the same crystallographic direction, they are represented as a fraction $\backslash frac$ 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 ^{y} _{x} which specifies the Bravais lattice. Here x \in \{t, m, o, q, rh, h, c\} is the lattice system, and y \in \{\empty, b, v, f\} is the centering type.^[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

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

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

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

Triclinic crystal system! Number! Point group! Orbifold! Short name! Full name! Schoenflies! Fedorov! Shubnikov! Fibrifold

1	1	1	P1	P 1	\Gamma _{tC} ^{1} ^{1}	1s	(a/b/c) ⋅ 1	(\circ)
2		x	P	P	\Gamma _{tC} ^{1} ^{i}	2s	(a/b/c) ⋅ \tilde 2	( 2222 )

List of monoclinic

Full name(s)! Schoenflies! Fedorov! Shubnikov! Fibrifold (primary)! Fibrifold (secondary)

3	2	22	P2	P 1 2 1	P 1 1 2	\Gamma _{mC} ^{1} ^{2}	3s	(b:(c/a)): 2	( 2 _{02} _{02} _{02} _{0)}	({*} _{0{*}} _{0)}
4			P21	P 1 21 1	P 1 1 21	\Gamma _{mC} ^{2} ^{2}	1a	(b:(c/a)): 2 _{1}	( 2 _{12} _{12} _{12} _{1)}	(\bar{ x }\bar{ x })
5			C2	C 1 2 1	B 1 1 2	\Gamma ^{3} _{2}	4s	\left ( \tfrac{a + b}{ 2 }/b:(c/a)\right ) : 2	( 2 _{02} _{02} _{12} _{1)}	({*} _{1{*}} _{1)} , ({*}\bar{ x })
6	m	*	Pm	P 1 m 1	P 1 1 m	\Gamma _{mC} ^{1} ^{s}	5s	(b:(c/a)) ⋅ m	[\circ _{0]}	({*}{ ⋅ }{*}{ ⋅ })
7			Pc	P 1 c 1	P 1 1 b	\Gamma _{mC} ^{2} ^{s}	1h	(b:(c/a)) ⋅ \tilde c	(\bar\circ _{0)}	({*}{:}{*}{:}) , ({ x }{ x } _{0)}
8			Cm	C 1 m 1	B 1 1 m	\Gamma ^{3} _{s}	6s	\left ( \tfrac{a + b}{ 2 }/b:(c/a)\right ) ⋅ m	[\circ _{1]}	({*}{ ⋅ }{*}{:}) , ({*}{ ⋅ }{ x })
9			Cc	C 1 c 1	B 1 1 b	\Gamma ^{4} _{s}	2h	\left ( \tfrac{a + b}{ 2 }/b:(c/a)\right ) ⋅ \tilde c	(\bar\circ _{1)}	({*}{:}{ x }) , ({ x }{ x } _{1)}
10	2/m	2 *	P2/m	P 1 2/m 1	P 1 1 2/m	\Gamma _{mC} ^{1} ^{2 h}	7s	(b:(c/a)) ⋅ m: 2	[ 2 _{02} _{02} _{02} _{0]}	[* 2 { ⋅ } 22 { ⋅ } 2 ) |- align=center|11||P21/m||P 1 21/m 1||P 1 1 21/m || \Gamma _{mC} ^{2} ^{2 h} || 2a || (b:(c/a)) ⋅ m: 2 _{1} || [ 2 _{12} _{12} _{12} _{1]} || ( 22 {*}{ ⋅ }) |- align=center|12||C2/m||C 1 2/m 1||B 1 1 2/m || \Gamma ^{3} _{2 h} || 8s || \left ( \tfrac{a + b}{ 2 }/b:(c/a)\right ) ⋅ m: 2 || [ 2 _{02} _{02} _{12} _{1]} || (* 2 { ⋅ } 22 {:} 2 ) , ( 2 \bar{*} 2 { ⋅ } 2 ) |- align=center|13||P2/c||P 1 2/c 1||P 1 1 2/b || \Gamma _{mC} ^{4} ^{2 h} || 3h || (b:(c/a)) ⋅ \tilde c: 2 || ( 2 _{02} _{022)} || (* 2 {:} 22 {:} 2 ) , ( 22 {*} _{0)} |- align=center|14||P21/c||P 1 21/c 1||P 1 1 21/b || \Gamma _{mC} ^{5} ^{2 h} || 3a || (b:(c/a)) ⋅ \tilde c: 2 _{1} || ( 2 _{12} _{122)} || ( 22 {*}{:}) , ( 22 { x }) |- align=center|15||C2/c||C 1 2/c 1||B 1 1 2/b || \Gamma ^{6} _{2 h} || 4h || \left ( \tfrac{a + b}{ 2 }/b:(c/a)\right ) ⋅ \tilde c: 2 || ( 2 _{02} _{122)} || ( 2 \bar{*} 2 {:} 2 ) , ( 22 {*} _{1)} |} List of orthorhombic 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 _{oD} ^{1} ^{2} 9s (c:a:b): 2 : 2 (* 2 _{02} _{02} _{02} _{0)} 17 P2221 P 2 2 21 \Gamma _{oD} ^{2} ^{2} 4a (c:a:b): 2 _{1:2} (* 2 _{12} _{12} _{12} _{1)} ( 2 _{02} _{0{*})} 18 P21212 P 21 21 2 \Gamma _{oD} ^{3} ^{2} 7a (c:a:b): 2 2 _{1} ( 2 _{02} _{0\bar{ x })} ( 2 _{12} _{1{*})} 19 P212121 P 21 21 21 \Gamma _{oD} ^{4} ^{2} 8a (c:a:b): 2 _{1} 2 _{1} ( 2 _{12} _{1\bar{ x })} 20 C2221 C 2 2 21 \Gamma ^{5} _{2} 5a \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) : 2 _{1:2} ( 2 _{1{*}2} _{12} _{1)} ( 2 _{02} _{1{*})} 21 C222 C 2 2 2 \Gamma ^{6} _{2} 10s \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) : 2 : 2 ( 2 _{0{*}2} _{02} _{0)} (* 2 _{02} _{02} _{12} _{1)} 22 F222 F 2 2 2 \Gamma ^{7} _{2} 12s \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:c:a:b\right ) : 2 : 2 (* 2 _{02} _{12} _{02} _{1)} 23 I222 I 2 2 2 \Gamma ^{8} _{2} 11s \left ( \tfrac{a + b + c}{ 2 }/c:a:b\right ) : 2 : 2 ( 2 _{1{*}2} _{02} _{0)} 24 I212121 I 21 21 21 \Gamma ^{9} _{2} 6a \left ( \tfrac{a + b + c}{ 2 }/c:a:b \right ) : 2 : 2 _{1} ( 2 _{0{*}2} _{12} _{1)} 25 mm2 * 22 Pmm2 P m m 2 \Gamma _{oC} ^{1} ^{2 v} 13s (c:a:b):m ⋅ 2 (*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 { ⋅ } 2 ) [{*} _{0{ ⋅ }{*}} _{0{ ⋅ }]} 26 Pmc21 P m c 21 \Gamma _{oC} ^{2} ^{2 v} 9a (c:a:b): \tilde c ⋅ 2 _{1} (*{ ⋅ } 2 {:} 2 { ⋅ } 2 {:} 2 ) (\bar{*}{ ⋅ }\bar{*}{ ⋅ }) , [{ x _{0}{ x} _{0}]} 27 Pcc2 P c c 2 \Gamma _{oC} ^{3} ^{2 v} 5h (c:a:b): \tilde c ⋅ 2 (*{:} 2 {:} 2 {:} 2 {:} 2 ) (\bar{*} _{0\bar{*}} _{0)} 28 Pma2 P m a 2 \Gamma _{oC} ^{4} ^{2 v} 6h (c:a:b): \tilde a ⋅ 2 ( 2 _{02} _{0{*}{ ⋅ })} [{*} _{0{:}{*}} _{0{:}]} , (*{ ⋅ }{*} _{0)} 29 Pca21 P c a 21 \Gamma _{oC} ^{5} ^{2 v} 11a (c:a:b): \tilde a ⋅ 2 _{1} ( 2 _{12} _{1{*}{:})} (\bar{*}{:}\bar{*}{:}) 30 Pnc2 P n c 2 \Gamma _{oC} ^{6} ^{2 v} 7h (c:a:b): \tilde c \odot 2 ( 2 _{02} _{0{*}{:})} (\bar{*} _{1\bar{*}} _{1)} , ({*} _{0{ x }} _{0)} 31 Pmn21 P m n 21 \Gamma _{oC} ^{7} ^{2 v} 10a (c:a:b): \widetilde{ac} ⋅ 2 _{1} ( 2 _{12} _{1{*}{ ⋅ })} (*{ ⋅ }\bar{ x }) , [{ x } _{0{ x }} _{1]} 32 Pba2 P b a 2 \Gamma _{oC} ^{8} ^{2 v} 9h (c:a:b): \tilde a \odot 2 ( 2 _{02} _{0{ x }} _{0)} (*{:}{*} _{0)} 33 Pna21 P n a 21 \Gamma _{oC} ^{9} ^{2 v} 12a (c:a:b): \tilde a \odot 2 _{1} ( 2 _{12} _{1{ x })} (*{:}{ x }) , ({ x }{ x } _{1)} 34 Pnn2 P n n 2 \Gamma _{oC} ^{10} ^{2 v} 8h (c:a:b): \widetilde{ac} \odot 2 ( 2 _{02} _{0{ x }} _{1)} (* _{0{ x }} _{1)} 35 Cmm2 C m m 2 \Gamma ^{11} _{2 v} 14s \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) :m ⋅ 2 ( 2 _{0{*}{ ⋅ }2{ ⋅ }2)} [* _{0{ ⋅ }{*}} _{0{:}]} 36 Cmc21 C m c 21 \Gamma ^{12} _{2 v} 13a \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) :\tilde c ⋅ 2 _{1} ( 2 _{1{*}{ ⋅ }2{:}2)} (\bar{*}{ ⋅ }\bar{*}{:}) , [{ x } _{1{ x }} _{1]} 37 Ccc2 C c c 2 \Gamma ^{13} _{2 v} 10h \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) : \tilde c ⋅ 2 ( 2 _{0{*}{:}2{:}2)} (\bar{*} _{0\bar{*}} _{1)} 38 Amm2 A m m 2 \Gamma ^{14} _{2 v} 15s \left ( \tfrac{b + c}{ 2 }/c:a:b\right ):m ⋅ 2 (*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 {:} 2 ) [{*} _{1{ ⋅ }{*}} _{1{ ⋅ }]} , [*{ ⋅ }{ x } _{0]} 39 Aem2 A b m 2 \Gamma ^{15} _{2 v} 11h \left ( \tfrac{b + c}{ 2 }/c:a:b\right ) :m ⋅ 2 _{1} (*{ ⋅ } 2 {:} 2 {:} 2 {:} 2 ) [{*} _{1{:}{*}} _{1{:}]} , (\bar{*}{ ⋅ }\bar{*} _{0)} 40 Ama2 A m a 2 \Gamma ^{16} _{2 v} 12h \left ( \tfrac{b + c}{ 2 }/c:a:b\right ) : \tilde a ⋅ 2 ( 2 _{02} _{1{*}{ ⋅ })} (*{ ⋅ }{*} _{1)} , [*{:}{ x } _{1]} 41 Aea2 A b a 2 \Gamma ^{17} _{2 v} 13h \left ( \tfrac{b + c}{ 2 }/c:a:b\right ) : \tilde a ⋅ 2 _{1} ( 2 _{02} _{1{*}{:})} (*{:}{*} _{1)} , (\bar{*}{:}\bar{*} _{1)} 42 Fmm2 F m m 2 \Gamma ^{18} _{2 v} 17s \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:c:a:b\right ) :m ⋅ 2 (*{ ⋅ } 2 { ⋅ } 2 {:} 2 {:} 2 ) [{*} _{1{ ⋅ }{*}} _{1{:}]} 43 Fdd2 F d d 2 \Gamma ^{19} _{2 v} 16h \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:c:a:b \right ) : \tfrac{ 1 }{ 2 } \widetilde{ac} \odot 2 ( 2 _{02} _{1{ x })} ({*} _{1{ x })} 44 Imm2 I m m 2 \Gamma ^{20} _{2 v} 16s \left ( \tfrac{a + b + c}{ 2 }/c:a:b \right ) :m ⋅ 2 ( 2 _{1{*}{ ⋅ }2{ ⋅ }2)} [*{ ⋅ }{ x } _{1]} 45 Iba2 I b a 2 \Gamma ^{21} _{2 v} 15h \left ( \tfrac{a + b + c}{ 2 }/c:a:b \right ) : \tilde c ⋅ 2 ( 2 _{1{*}{:}2{:}2)} (\bar{*}{:}\bar{*} _{0)} 46 Ima2 I m a 2 \Gamma ^{22} _{2 v} 14h \left ( \tfrac{a + b + c}{ 2 }/c:a:b \right ) : \tilde a ⋅ 2 ( 2 _{0{*}{ ⋅ }2{:}2)} (\bar{*}{ ⋅ }\bar{*} _{1)} , [*{:}{ x } _{0]} 47 \tfrac{ 2 }{m}\tfrac{ 2 }{m}\tfrac{ 2 }{m} * 222 Pmmm P 2/m 2/m 2/m \Gamma _{oD} ^{1} ^{2 h} 18s \left ( c:a:b \right ) ⋅ m: 2 ⋅ m [*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 { ⋅ } 2 ] 48 Pnnn P 2/n 2/n 2/n \Gamma _{oD} ^{2} ^{2 h} 19h \left ( c:a:b \right ) ⋅ \widetilde{ab}: 2 \odot \widetilde{ac} ( 2 \bar{*} _{12} _{02} _{0)} 49 Pccm P 2/c 2/c 2/m \Gamma _{oD} ^{3} ^{2 h} 17h \left ( c:a:b \right ) ⋅ m: 2 ⋅ \tilde c [*{:} 2 {:} 2 {:} 2 {:} 2 ] (* 2 _{02} _{02{ ⋅ }2)} 50 Pban P 2/b 2/a 2/n \Gamma _{oD} ^{4} ^{2 h} 18h \left ( c:a:b \right ) ⋅ \widetilde{ab}: 2 \odot \tilde a ( 2 \bar{*} _{02} _{02} _{0)} (* 2 _{02} _{02{:}2)} 51 Pmma P 21/m 2/m 2/a \Gamma _{oD} ^{5} ^{2 h} 14a \left ( c:a:b \right ) ⋅ \tilde a : 2 ⋅ m [ 2 _{02} _{0{*}{ ⋅ }]} [*{ ⋅ } 2 {:} 2 { ⋅ } 2 {:} 2 ] , [* 2 { ⋅ } 2 { ⋅ } 2 { ⋅ } 2 ] 52 Pnna P 2/n 21/n 2/a \Gamma _{oD} ^{6} ^{2 h} 17a \left ( c:a:b \right ) ⋅ \tilde a: 2 \odot \widetilde{ac} ( 2 _{02\bar{*}} _{1)} ( 2 _{0{*}2{:}2)} , ( 2 \bar{*} 2 _{12} _{1)} 53 Pmna P 2/m 2/n 21/a \Gamma _{oD} ^{7} ^{2 h} 15a \left ( c:a:b \right ) ⋅ \tilde a: 2 _{1} ⋅ \widetilde{ac} [ 2 _{02} _{0{*}{:}]} (* 2 _{12} _{12{ ⋅ }2)} , ( 2 _{0{*}2{ ⋅ }2)} 54 Pcca P 21/c 2/c 2/a \Gamma _{oD} ^{8} ^{2 h} 16a \left ( c:a:b \right ) ⋅ \tilde a: 2 ⋅ \tilde c ( 2 _{02\bar{*}} _{0)} (* 2 {:} 2 {:} 2 {:} 2 ) , (* 2 _{12} _{12{:}2)} 55 Pbam P 21/b 21/a 2/m \Gamma _{oD} ^{9} ^{2 h} 22a \left ( c:a:b \right ) ⋅ m: 2 \odot \tilde a [ 2 _{02} _{0{ x }} _{0]} (* 2 { ⋅ } 2 {:} 2 { ⋅ } 2 ) 56 Pccn P 21/c 21/c 2/n \Gamma _{oD} ^{10} ^{2 h} 27a \left ( c:a:b \right ) ⋅ \widetilde{ab}: 2 ⋅ \tilde c ( 2 \bar{*}{:} 2 {:} 2 ) ( 2 _{12\bar{*}} _{0)} 57 Pbcm P 2/b 21/c 21/m \Gamma _{oD} ^{11} ^{2 h} 23a \left ( c:a:b \right ) ⋅ m: 2 _{1} \odot \tilde c ( 2 _{02\bar{*}{ ⋅ })} (* 2 {:} 2 { ⋅ } 2 {:} 2 ) , [ 2 _{12} _{1{*}{:}]} 58 Pnnm P 21/n 21/n 2/m \Gamma _{oD} ^{12} ^{2 h} 25a \left ( c:a:b \right ) ⋅ m: 2 \odot \widetilde{ac} [ 2 _{02} _{0{ x }} _{1]} ( 2 _{1{*}2{ ⋅ }2)} 59 Pmmn P 21/m 21/m 2/n \Gamma _{oD} ^{13} ^{2 h} 24a \left ( c:a:b \right ) ⋅ \widetilde{ab}: 2 ⋅ m ( 2 \bar{*}{ ⋅ } 2 { ⋅ } 2 ) [ 2 _{12} _{1{*}{ ⋅ }]} 60 Pbcn P 21/b 2/c 21/n \Gamma _{oD} ^{14} ^{2 h} 26a \left ( c:a:b \right ) ⋅ \widetilde{ab}: 2 _{1} \odot \tilde c ( 2 _{02\bar{*}{:})} ( 2 _{1{*}2{:}2)} , ( 2 _{12\bar{*}} _{1)} 61 Pbca P 21/b 21/c 21/a \Gamma _{oD} ^{15} ^{2 h} 29a \left ( c:a:b \right ) ⋅ \tilde a: 2 _{1} \odot \tilde c ( 2 _{12\bar{*}{:})} 62 Pnma P 21/n 21/m 21/a \Gamma _{oD} ^{16} ^{2 h} 28a \left ( c:a:b \right ) ⋅ \tilde a: 2 _{1} \odot m ( 2 _{12\bar{*}{ ⋅ })} ( 2 \bar{*}{ ⋅ } 2 {:} 2 ) , [ 2 _{12} _{1{ x }]} 63 Cmcm C 2/m 2/c 21/m \Gamma ^{17} _{2 h} 18a \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ m: 2 _{1} ⋅ \tilde c [ 2 _{02} _{1{*}{ ⋅ }]} (* 2 { ⋅ } 2 { ⋅ } 2 {:} 2 ) , [ 2 _{1{*}{ ⋅ }2{:}2]} 64 Cmce C 2/m 2/c 21/a \Gamma ^{18} _{2 h} 19a \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ \tilde a : 2 _{1} ⋅ \tilde c [ 2 _{02} _{1{*}{:}]} (* 2 { ⋅ } 2 {:} 2 {:} 2 ) , (* 2 _{12{ ⋅ }2{:}2)} 65 Cmmm C 2/m 2/m 2/m \Gamma ^{19} _{2 h} 19s \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ m: 2 ⋅ m [ 2 _{0{*}{ ⋅ }2{ ⋅ }2]} [*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 {:} 2 ] 66 Cccm C 2/c 2/c 2/m \Gamma ^{20} _{2 h} 20h \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ m: 2 ⋅ \tilde c [ 2 _{0{*}{:}2{:}2]} (* 2 _{02} _{12{ ⋅ }2)} 67 Cmme C 2/m 2/m 2/e \Gamma ^{21} _{2 h} 21h \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ \tilde a : 2 ⋅ m (* 2 _{02{ ⋅ }2{ ⋅ }2)} [*{ ⋅ } 2 {:} 2 {:} 2 {:} 2 ] 68 Ccce C 2/c 2/c 2/e \Gamma ^{22} _{2 h} 22h \left ( \tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ \tilde a : 2 ⋅ \tilde c (* 2 _{02{:}2{:}2)} (* 2 _{02} _{12{:}2)} 69 Fmmm F 2/m 2/m 2/m \Gamma ^{23} _{2 h} 21s \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ m: 2 ⋅ m [*{ ⋅ } 2 { ⋅ } 2 {:} 2 {:} 2 ] 70 Fddd F 2/d 2/d 2/d \Gamma ^{24} _{2 h} 24h \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:c:a:b\right ) ⋅ \tfrac{ 1 }{ 2 }\widetilde{ab}: 2 \odot \tfrac{ 1 }{ 2 }\widetilde{ac} ( 2 \bar{*} 2 _{02} _{1)} 71 Immm I 2/m 2/m 2/m \Gamma ^{25} _{2 h} 20s \left ( \tfrac{a + b + c}{ 2 }/c:a:b\right ) ⋅ m: 2 ⋅ m [ 2 _{1{*}{ ⋅ }2{ ⋅ }2]} 72 Ibam I 2/b 2/a 2/m \Gamma ^{26} _{2 h} 23h \left ( \tfrac{a + b + c}{ 2 }/c:a:b\right ) ⋅ m: 2 ⋅ \tilde c [ 2 _{1{*}{:}2{:}2]} (* 2 _{02{ ⋅ }2{:}2)} 73 Ibca I 2/b 2/c 2/a \Gamma ^{27} _{2 h} 21a \left ( \tfrac{a + b + c}{ 2 }/c:a:b\right ) ⋅ \tilde a : 2 ⋅ \tilde c (* 2 _{12{:}2{:}2)} 74 Imma I 2/m 2/m 2/a \Gamma ^{28} _{2 h} 20a \left ( \tfrac{a + b + c}{ 2 }/c:a:b\right ) ⋅ \tilde a : 2 ⋅ m (* 2 _{12{ ⋅ }2{ ⋅ }2)} [ 2 _{0{*}{ ⋅ }2{:}2]} List of tetragonal Tetragonal crystal system!Number! Point group! Orbifold! Short name! Full name! Schoenflies! Fedorov! Shubnikov! Fibrifold 75 4 44 P4 P 4 \Gamma _{qC} ^{1} ^{4} 22s (c:a:a): 4 ( 4 _{04} _{02} _{0)} 76 P41 P 41 \Gamma _{qC} ^{2} ^{4} 30a (c:a:a) : 4 _{1} ( 4 _{14} _{12} _{1)} 77 P42 P 42 \Gamma _{qC} ^{3} ^{4} 33a (c:a:a) : 4 _{2} ( 4 _{24} _{22} _{0)} 78 P43 P 43 \Gamma _{qC} ^{4} ^{4} 31a (c:a:a) : 4 _{3} ( 4 _{14} _{12} _{1)} 79 I4 I 4 \Gamma ^{5} _{4} 23s \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 ( 4 _{24} _{02} _{1)} 80 I41 I 41 \Gamma ^{6} _{4} 32a \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 _{1} ( 4 _{34} _{12} _{0)} 81 2 x P P \Gamma _{qS} ^{1} ^{4} 26s (c:a:a):\tilde 4 ( 442 _{0)} 82 I I \Gamma ^{2} _{4} 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 _{qC} ^{1} ^{4 h} 28s (c:a:a) ⋅ m: 4 [ 4 _{04} _{02} _{0]} 84 P42/m P 42/m \Gamma _{qC} ^{2} ^{4 h} 41a (c:a:a) ⋅ m: 4 _{2} [ 4 _{24} _{22} _{0]} 85 P4/n P 4/n \Gamma _{qC} ^{3} ^{4 h} 29h (c:a:a) ⋅ \widetilde{ab}: 4 ( 44 _{02)} 86 P42/n P 42/n \Gamma _{qC} ^{4} ^{4 h} 42a (c:a:a) ⋅ \widetilde{ab}: 4 _{2} ( 44 _{22)} 87 I4/m I 4/m \Gamma ^{5} _{4 h} 29s \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) ⋅ m: 4 [ 4 _{24} _{02} _{1]} 88 I41/a I 41/a \Gamma ^{6} _{4 h} 40a \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) ⋅ \tilde a : 4 _{1} ( 44 _{12)} 89 422 224 P422 P 4 2 2 \Gamma _{qD} ^{1} ^{4} 30s (c:a:a): 4 : 2 (* 4 _{04} _{02} _{0)} 90 P4212 P4212 \Gamma _{qD} ^{2} ^{4} 43a (c:a:a): 4 2 _{1} ( 4 _{0{*}2} _{0)} 91 P4122 P 41 2 2 \Gamma _{qD} ^{3} ^{4} 44a (c:a:a): 4 _{1:2} (* 4 _{14} _{12} _{1)} 92 P41212 P 41 21 2 \Gamma _{qD} ^{4} ^{4} 48a (c:a:a): 4 _{1} 2 _{1} ( 4 _{1{*}2} _{1)} 93 P4222 P 42 2 2 \Gamma _{qD} ^{5} ^{4} 47a (c:a:a): 4 _{2:2} (* 4 _{24} _{22} _{0)} 94 P42212 P 42 21 2 \Gamma _{qD} ^{6} ^{4} 50a (c:a:a): 4 _{2} 2 _{1} ( 4 _{2{*}2} _{0)} 95 P4322 P 43 2 2 \Gamma _{qD} ^{7} ^{4} 45a (c:a:a): 4 _{3:2} (* 4 _{14} _{12} _{1)} 96 P43212 P 43 21 2 \Gamma _{qD} ^{8} ^{4} 49a (c:a:a): 4 _{3} 2 _{1} ( 4 _{1{*}2} _{1)} 97 I422 I 4 2 2 \Gamma ^{9} _{4} 31s \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 : 2 (* 4 _{24} _{02} _{1)} 98 I4122 I 41 2 2 \Gamma ^{10} _{4} 46a \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 : 2 _{1} (* 4 _{34} _{12} _{0)} 99 4mm * 44 P4mm P 4 m m \Gamma _{qC} ^{1} ^{4 v} 24s (c:a:a): 4 ⋅ m (*{ ⋅ } 4 { ⋅ } 4 { ⋅ } 2 ) 100 P4bm P 4 b m \Gamma _{qC} ^{2} ^{4 v} 26h (c:a:a): 4 \odot \tilde a ( 4 _{0{*}{ ⋅ }2)} 101 P42cm P 42 c m \Gamma _{qC} ^{3} ^{4 v} 37a (c:a:a): 4 _{2 ⋅} \tilde c (*{:} 4 { ⋅ } 4 {:} 2 ) 102 P42nm P 42 n m \Gamma _{qC} ^{4} ^{4 v} 38a (c:a:a): 4 _{2\odot} \widetilde{ac} ( 4 _{2{*}{ ⋅ }2)} 103 P4cc P 4 c c \Gamma _{qC} ^{5} ^{4 v} 25h (c:a:a): 4 ⋅ \tilde c (*{:} 4 {:} 4 {:} 2 ) 104 P4nc P 4 n c \Gamma _{qC} ^{6} ^{4 v} 27h (c:a:a): 4 \odot \widetilde{ac} ( 4 _{0{*}{:}2)} 105 P42mc P 42 m c \Gamma _{qC} ^{7} ^{4 v} 36a (c:a:a): 4 _{2 ⋅} m (*{ ⋅ } 4 {:} 4 { ⋅ } 2 ) 106 P42bc P 42 b c \Gamma _{qC} ^{8} ^{4 v} 39a (c:a:a): 4 \odot \tilde a ( 4 _{2{*}{:}2)} 107 I4mm I 4 m m \Gamma ^{9} _{4 v} 25s \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 ⋅ m (*{ ⋅ } 4 { ⋅ } 4 {:} 2 ) 108 I4cm I 4 c m \Gamma ^{10} _{4 v} 28h \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 ⋅ \tilde c (*{ ⋅ } 4 {:} 4 {:} 2 ) 109 I41md I 41 m d \Gamma ^{11} _{4 v} 34a \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 _{1\odot} m ( 4 _{1{*}{ ⋅ }2)} 110 I41cd I 41 c d \Gamma ^{12} _{4 v} 35a \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) : 4 _{1\odot} \tilde c ( 4 _{1{*}{:}2)} 111 2m 2 {*} 2 P2m P 2 m \Gamma _{qD} ^{1} ^{2 d} 32s (c:a:a):\tilde 4 : 2 (* 4 { ⋅ } 42 _{0)} 112 P2c P 2 c \Gamma _{qD} ^{2} ^{2 d} 30h (c:a:a):\tilde 4 2 (* 4 {:} 42 _{0)} 113 P21m P 21 m \Gamma _{qD} ^{3} ^{2 d} 52a (c:a:a):\tilde 4 ⋅ \widetilde{ab} ( 4 \bar{*}{ ⋅ } 2 ) 114 P21c P 21 c \Gamma _{qD} ^{4} ^{2 d} 53a (c:a:a):\tilde 4 ⋅ \widetilde{abc} ( 4 \bar{*}{:} 2 ) 115 Pm2 P m 2 \Gamma _{qD} ^{5} ^{2 d} 33s (c:a:a):\tilde 4 ⋅ m (*{ ⋅ } 44 { ⋅ } 2 ) 116 Pc2 P c 2 \Gamma _{qD} ^{6} ^{2 d} 31h (c:a:a):\tilde 4 ⋅ \tilde c (*{:} 44 {:} 2 ) 117 Pb2 P b 2 \Gamma _{qD} ^{7} ^{2 d} 32h (c:a:a):\tilde 4 \odot \tilde a ( 4 \bar{*} _{02} _{0)} 118 Pn2 P n 2 \Gamma _{qD} ^{8} ^{2 d} 33h (c:a:a):\tilde 4 ⋅ \widetilde{ac} ( 4 \bar{*} _{12} _{0)} 119 Im2 I m 2 \Gamma ^{9} _{2 d} 35s \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) :\tilde 4 ⋅ m (* 4 { ⋅ } 42 _{1)} 120 Ic2 I c 2 \Gamma ^{10} _{2 d} 34h \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) :\tilde 4 ⋅ \tilde c (* 4 {:} 42 _{1)} 121 I2m I 2 m \Gamma ^{11} _{2 d} 34s \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) :\tilde 4 : 2 (*{ ⋅ } 44 {:} 2 ) 122 I2d I 2 d \Gamma ^{12} _{2 d} 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 _{qD} ^{1} ^{4 h} 36s (c:a:a) ⋅ m: 4 ⋅ m [*{ ⋅ } 4 { ⋅ } 4 { ⋅ } 2 ] 124 P4/mcc P 4/m 2/c 2/c \Gamma _{qD} ^{2} ^{4 h} 35h (c:a:a) ⋅ m: 4 ⋅ \tilde c [*{:} 4 {:} 4 {:} 2 ] 125 P4/nbm P 4/n 2/b 2/m \Gamma _{qD} ^{3} ^{4 h} 36h (c:a:a) ⋅ \widetilde{ab}: 4 \odot \tilde a (* 4 _{04{ ⋅ }2)} 126 P4/nnc P 4/n 2/n 2/c \Gamma _{qD} ^{4} ^{4 h} 37h (c:a:a) ⋅ \widetilde{ab}: 4 \odot \widetilde{ac} (* 4 _{04{:}2)} 127 P4/mbm P 4/m 21/b 2/m \Gamma _{qD} ^{5} ^{4 h} 54a (c:a:a) ⋅ m: 4 \odot \tilde a [ 4 _{0{*}{ ⋅ }2]} 128 P4/mnc P 4/m 21/n 2/c \Gamma _{qD} ^{6} ^{4 h} 56a (c:a:a) ⋅ m: 4 \odot \widetilde{ac} [ 4 _{0{*}{:}2]} 129 P4/nmm P 4/n 21/m 2/m \Gamma _{qD} ^{7} ^{4 h} 55a (c:a:a) ⋅ \widetilde{ab}: 4 ⋅ m (* 4 { ⋅ } 4 { ⋅ } 2 ) 130 P4/ncc P 4/n 21/c 2/c \Gamma _{qD} ^{8} ^{4 h} 57a (c:a:a) ⋅ \widetilde{ab}: 4 ⋅ \tilde c (* 4 {:} 4 {:} 2 ) 131 P42/mmc P 42/m 2/m 2/c \Gamma _{qD} ^{9} ^{4 h} 60a (c:a:a) ⋅ m: 4 _{2 ⋅} m [*{ ⋅ } 4 {:} 4 { ⋅ } 2 ] 132 P42/mcm P 42/m 2/c 2/m \Gamma _{qD} ^{10} ^{4 h} 61a (c:a:a) ⋅ m: 4 _{2 ⋅} \tilde c [*{:} 4 { ⋅ } 4 {:} 2 ] 133 P42/nbc P 42/n 2/b 2/c \Gamma _{qD} ^{11} ^{4 h} 63a (c:a:a) ⋅ \widetilde{ab}: 4 _{2\odot} \tilde a (* 4 _{24{:}2)} 134 P42/nnm P 42/n 2/n 2/m \Gamma _{qD} ^{12} ^{4 h} 62a (c:a:a) ⋅ \widetilde{ab}: 4 _{2\odot} \widetilde{ac} (* 4 _{24{ ⋅ }2)} 135 P42/mbc P 42/m 21/b 2/c \Gamma _{qD} ^{13} ^{4 h} 66a (c:a:a) ⋅ m: 4 _{2\odot} \tilde a [ 4 _{2{*}{:}2]} 136 P42/mnm P 42/m 21/n 2/m \Gamma _{qD} ^{14} ^{4 h} 65a (c:a:a) ⋅ m: 4 _{2\odot} \widetilde{ac} [ 4 _{2{*}{ ⋅ }2]} 137 P42/nmc P 42/n 21/m 2/c \Gamma _{qD} ^{15} ^{4 h} 67a (c:a:a) ⋅ \widetilde{ab}: 4 _{2 ⋅} m (* 4 { ⋅ } 4 {:} 2 ) 138 P42/ncm P 42/n 21/c 2/m \Gamma _{qD} ^{16} ^{4 h} 65a (c:a:a) ⋅ \widetilde{ab}: 4 _{2 ⋅} \tilde c (* 4 {:} 4 { ⋅ } 2 ) 139 I4/mmm I 4/m 2/m 2/m \Gamma ^{17} _{4 h} 37s \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) ⋅ m: 4 ⋅ m [*{ ⋅ } 4 { ⋅ } 4 {:} 2 ] 140 I4/mcm I 4/m 2/c 2/m \Gamma ^{18} _{4 h} 38h \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) ⋅ m: 4 ⋅ \tilde c [*{ ⋅ } 4 {:} 4 {:} 2 ] 141 I41/amd I 41/a 2/m 2/d \Gamma ^{19} _{4 h} 59a \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) ⋅ \tilde a : 4 _{1\odot} m (* 4 _{14{ ⋅ }2)} 142 I41/acd I 41/a 2/c 2/d \Gamma ^{20} _{4 h} 58a \left ( \tfrac{a + b + c}{ 2 }/c:a:a\right ) ⋅ \tilde a : 4 _{1\odot} \tilde c (* 4 _{14{:}2)} List of trigonal Trigonal crystal system!Number! Point group! Orbifold! Short name! Full name! Schoenflies! Fedorov! Shubnikov! Fibrifold 143 3 33 P3 P 3 \Gamma _{hC} ^{1} ^{3} 38s (c:(a/a)): 3 ( 3 _{03} _{03} _{0)} 144 P31 P 31 \Gamma _{hC} ^{2} ^{3} 68a (c:(a/a)): 3 _{1} ( 3 _{13} _{13} _{1)} 145 P32 P 32 \Gamma _{hC} ^{3} ^{3} 69a (c:(a/a)): 3 _{2} ( 3 _{13} _{13} _{1)} 146 R3 R 3 \Gamma _{rh} C ^{4} _{3} 39s (a/a/a)/ 3 ( 3 _{03} _{13} _{2)} 147 3 x P P \Gamma _{hC} ^{1} ^{3 i} 51s (c:(a/a)):\tilde 6 ( 63 _{02)} 148 R R \Gamma _{rh} C ^{2} _{3 i} 52s (a/a/a)/\tilde 6 ( 63 _{12)} 149 32 223 P312 P 3 1 2 \Gamma _{hD} ^{1} ^{3} 45s (c:(a/a)): 2 : 3 (* 3 _{03} _{03} _{0)} 150 P321 P 3 2 1 \Gamma _{hD} ^{2} ^{3} 44s (c:(a/a)) ⋅ 2 : 3 ( 3 _{0{*}3} _{0)} 151 P3112 P 31 1 2 \Gamma _{hD} ^{3} ^{3} 72a (c:(a/a)): 2 : 3 _{1} (* 3 _{13} _{13} _{1)} 152 P3121 P 31 2 1 \Gamma _{hD} ^{4} ^{3} 70a (c:(a/a)) ⋅ 2 : 3 _{1} ( 3 _{1{*}3} _{1)} 153 P3212 P 32 1 2 \Gamma _{hD} ^{5} ^{3} 73a (c:(a/a)): 2 : 3 _{2} (* 3 _{13} _{13} _{1)} 154 P3221 P 32 2 1 \Gamma _{hD} ^{6} ^{3} 71a (c:(a/a)) ⋅ 2 : 3 _{2} ( 3 _{1{*}3} _{1)} 155 R32 R 3 2 \Gamma _{rh} D ^{7} _{3} 46s (a/a/a)/ 3 : 2 (* 3 _{03} _{13} _{2)} 156 3m * 33 P3m1 P 3 m 1 \Gamma _{hC} ^{1} ^{3 v} 40s (c:(a/a)):m ⋅ 3 (*{ ⋅ } 3 { ⋅ } 3 { ⋅ } 3 ) 157 P31m P 3 1 m \Gamma _{hC} ^{2} ^{3 v} 41s (c:(a/a)) ⋅ m ⋅ 3 ( 3 _{0{*}{ ⋅ }3)} 158 P3c1 P 3 c 1 \Gamma _{hC} ^{3} ^{3 v} 39h (c:(a/a)):\tilde c: 3 (*{:} 3 {:} 3 {:} 3 ) 159 P31c P 3 1 c \Gamma _{hC} ^{4} ^{3 v} 40h (c:(a/a)) ⋅ \tilde c : 3 ( 3 _{0{*}{:}3)} 160 R3m R 3 m \Gamma _{rh} C ^{5} _{3 v} 42s (a/a/a)/ 3 ⋅ m ( 3 _{1{*}{ ⋅ }3)} 161 R3c R 3 c \Gamma _{rh} C ^{6} _{3 v} 41h (a/a/a)/ 3 ⋅ \tilde c ( 3 _{1{*}{:}3)} 162 2/m 2 {*} 3 P1m P 1 2/m \Gamma _{hD} ^{1} ^{3 d} 56s (c:(a/a)) ⋅ m ⋅ \tilde 6 (*{ ⋅ } 63 _{02)} 163 P1c P 1 2/c \Gamma _{hD} ^{2} ^{3 d} 46h (c:(a/a)) ⋅ \tilde c ⋅ \tilde 6 (*{:} 63 _{02)} 164 Pm1 P 2/m 1 \Gamma _{hD} ^{3} ^{3 d} 55s (c:(a/a)):m ⋅ \tilde 6 (* 6 { ⋅ } 3 { ⋅ } 2 ) 165 Pc1 P 2/c 1 \Gamma _{hD} ^{4} ^{3 d} 45h (c:(a/a)):\tilde c ⋅ \tilde 6 (* 6 {:} 3 {:} 2 ) 166 Rm R 2/m \Gamma _{rh} D ^{5} _{3 d} 57s (a/a/a)/\tilde 6 ⋅ m (*{ ⋅ } 63 _{12)} 167 Rc R 2/c \Gamma _{rh} D ^{6} _{3 d} 47h (a/a/a)/\tilde 6 ⋅ \tilde c (*{:} 63 _{12)} List of hexagonal Hexagonal crystal system!Number! Point group! Orbifold! Short name! Full name! Schoenflies! Fedorov! Shubnikov! Fibrifold 168 6 66 P6 P 6 \Gamma _{hC} ^{1} ^{6} 49s (c:(a/a)): 6 ( 6 _{03} _{02} _{0)} 169 P61 P 61 \Gamma _{hC} ^{2} ^{6} 74a (c:(a/a)): 6 _{1} ( 6 _{13} _{12} _{1)} 170 P65 P 65 \Gamma _{hC} ^{3} ^{6} 75a (c:(a/a)): 6 _{5} ( 6 _{13} _{12} _{1)} 171 P62 P 62 \Gamma _{hC} ^{4} ^{6} 76a (c:(a/a)): 6 _{2} ( 6 _{23} _{22} _{0)} 172 P64 P 64 \Gamma _{hC} ^{5} ^{6} 77a (c:(a/a)): 6 _{4} ( 6 _{23} _{22} _{0)} 173 P63 P 63 \Gamma _{hC} ^{6} ^{6} 78a (c:(a/a)): 6 _{3} ( 6 _{33} _{02} _{1)} 174 3 * P P \Gamma _{hC} ^{1} ^{3 h} 43s (c:(a/a)): 3 :m [ 3 _{03} _{03} _{0]} 175 6/m 6 * P6/m P 6/m \Gamma _{hC} ^{1} ^{6 h} 53s (c:(a/a)) ⋅ m : 6 [ 6 _{03} _{02} _{0]} 176 P63/m P 63/m \Gamma _{hC} ^{2} ^{6 h} 81a (c:(a/a)) ⋅ m : 6 _{3} [ 6 _{33} _{02} _{1]} 177 622 226 P622 P 6 2 2 \Gamma _{hD} ^{1} ^{6} 54s (c:(a/a)) ⋅ 2 : 6 (* 6 _{03} _{02} _{0)} 178 P6122 P 61 2 2 \Gamma _{hD} ^{2} ^{6} 82a (c:(a/a)) ⋅ 2 : 6 _{1} (* 6 _{13} _{12} _{1)} 179 P6522 P 65 2 2 \Gamma _{hD} ^{3} ^{6} 83a (c:(a/a)) ⋅ 2 : 6 _{5} (* 6 _{13} _{12} _{1)} 180 P6222 P 62 2 2 \Gamma _{hD} ^{4} ^{6} 84a (c:(a/a)) ⋅ 2 : 6 _{2} (* 6 _{23} _{22} _{0)} 181 P6422 P 64 2 2 \Gamma _{hD} ^{5} ^{6} 85a (c:(a/a)) ⋅ 2 : 6 _{4} (* 6 _{23} _{22} _{0)} 182 P6322 P 63 2 2 \Gamma _{hD} ^{6} ^{6} 86a (c:(a/a)) ⋅ 2 : 6 _{3} (* 6 _{33} _{02} _{1)} 183 6mm * 66 P6mm P 6 m m \Gamma _{hC} ^{1} ^{6 v} 50s (c:(a/a)):m ⋅ 6 (*{ ⋅ } 6 { ⋅ } 3 { ⋅ } 2 ) 184 P6cc P 6 c c \Gamma _{hC} ^{2} ^{6 v} 44h (c:(a/a)):\tilde c ⋅ 6 (*{:} 6 {:} 3 {:} 2 ) 185 P63cm P 63 c m \Gamma _{hC} ^{3} ^{6 v} 80a (c:(a/a)):\tilde c ⋅ 6 _{3} (*{ ⋅ } 6 {:} 3 {:} 2 ) 186 P63mc P 63 m c \Gamma _{hC} ^{4} ^{6 v} 79a (c:(a/a)):m ⋅ 6 _{3} (*{:} 6 { ⋅ } 3 { ⋅ } 2 ) 187 m2 * 223 Pm2 P m 2 \Gamma _{hD} ^{1} ^{3 h} 48s (c:(a/a)):m ⋅ 3 :m [*{ ⋅ } 3 { ⋅ } 3 { ⋅ } 3 ] 188 Pc2 P c 2 \Gamma _{hD} ^{2} ^{3 h} 43h (c:(a/a)):\tilde c ⋅ 3 :m [*{:} 3 {:} 3 {:} 3 ] 189 P2m P 2 m \Gamma _{hD} ^{3} ^{3 h} 47s (c:(a/a)) ⋅ m: 3 ⋅ m [ 3 _{0{*}{ ⋅ }3]} 190 P2c P 2 c \Gamma _{hD} ^{4} ^{3 h} 42h (c:(a/a)) ⋅ m: 3 ⋅ \tilde c [ 3 _{0{*}{:}3]} 191 6/m 2/m 2/m * 226 P6/mmm P 6/m 2/m 2/m \Gamma _{hD} ^{1} ^{6 h} 58s (c:(a/a)) ⋅ m: 6 ⋅ m [*{ ⋅ } 6 { ⋅ } 3 { ⋅ } 2 ] 192 P6/mcc P 6/m 2/c 2/c \Gamma _{hD} ^{2} ^{6 h} 48h (c:(a/a)) ⋅ m: 6 ⋅ \tilde c [*{:} 6 {:} 3 {:} 2 ] 193 P63/mcm P 63/m 2/c 2/m \Gamma _{hD} ^{3} ^{6 h} 87a (c:(a/a)) ⋅ m: 6 _{3 ⋅ \tilde} c [*{ ⋅ } 6 {:} 3 {:} 2 ] 194 P63/mmc P 63/m 2/m 2/c \Gamma _{hD} ^{4} ^{6 h} 88a (c:(a/a)) ⋅ m: 6 _{3 ⋅} m [*{:} 6 { ⋅ } 3 { ⋅ } 2 ] List of cubic 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 ^{1} _{cT} 59s \left ( a:a:a\right ) : 2 / 3 2 ^{\circ} (* 2 _{02} _{02} _{02} _{0){:}3} (* 2 _{02} _{02} _{02} _{0){:}3} 196 F23 F 2 3 \Gamma ^{fT} _{c} ^{2} 61s \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) : 2 / 3 1 ^{\circ} (* 2 _{02} _{12} _{02} _{1){:}3} (* 2 _{02} _{12} _{02} _{1){:}3} 197 I23 I 2 3 \Gamma ^{vT} _{c} ^{3} 60s \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) : 2 / 3 4 ^{\circ\circ} ( 2 _{1{*}2} _{02} _{0){:}3} ( 2 _{1{*}2} _{02} _{0){:}3} 198 P213 P 21 3 \Gamma ^{4} _{cT} 89a \left ( a:a:a\right ) : 2 _{1/3} 1 ^{\circ/4} ( 2 _{12} _{1\bar{ x }){:}3} ( 2 _{12} _{1\bar{ x }){:}3} 199 I213 I 21 3 \Gamma ^{vT} _{c} ^{5} 90a \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) : 2 _{1/3} 2 ^{\circ/4} ( 2 _{0{*}2} _{12} _{1){:}3} ( 2 _{0{*}2} _{12} _{1){:}3} 200 2/m 3 {*} 2 Pm P 2/m \Gamma _{cT} ^{1} ^{h} 62s \left ( a:a:a\right ) ⋅ m/ \tilde 6 4 ^{-} [*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 { ⋅ } 2 ]{:} 3 [*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 { ⋅ } 2 ]{:} 3 201 Pn P 2/n \Gamma _{cT} ^{2} ^{h} 49h \left ( a:a:a\right ) ⋅ \widetilde{ab} / \tilde 6 4 ^{\circ+} ( 2 \bar{*} _{12} _{02} _{0){:}3} ( 2 \bar{*} _{12} _{02} _{0){:}3} 202 Fm F 2/m \Gamma ^{3} _{h} 64s \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) ⋅ m/ \tilde 6 2 ^{-} [*{ ⋅ } 2 { ⋅ } 2 {:} 2 {:} 2 ]{:} 3 [*{ ⋅ } 2 { ⋅ } 2 {:} 2 {:} 2 ]{:} 3 203 Fd F 2/d \Gamma ^{4} _{h} 50h \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) ⋅ \tfrac{ 1 }{ 2 }\widetilde{ab} / \tilde 6 2 ^{\circ+} ( 2 \bar{*} 2 _{02} _{1){:}3} ( 2 \bar{*} 2 _{02} _{1){:}3} 204 Im I 2/m \Gamma ^{5} _{h} 63s \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) ⋅ m/\tilde 6 8 ^{-\circ} [ 2 _{1{*}{ ⋅ }2{ ⋅ }2]{:}3} [ 2 _{1{*}{ ⋅ }2{ ⋅ }2]{:}3} 205 Pa P 21/a \Gamma _{cT} ^{6} ^{h} 91a \left ( a:a:a\right ) ⋅ \tilde a /\tilde 6 2 ^{-/4} ( 2 _{12\bar{*}{:}){:}3} ( 2 _{12\bar{*}{:}){:}3} 206 Ia I 21/a \Gamma ^{7} _{h} 92a \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) ⋅ \tilde a /\tilde 6 4 ^{-/4} (* 2 _{12{:}2{:}2){:}3} (* 2 _{12{:}2{:}2){:}3} 207 432 432 P432 P 4 3 2 \Gamma ^{1} _{cO} 68s \left ( a:a:a\right ) : 4 / 3 4 ^{\circ-} (* 4 _{04} _{02} _{0){:}3} (* 2 _{02} _{02} _{02} _{0){:}6} 208 P4232 P 42 3 2 \Gamma ^{2} _{cO} 98a \left ( a:a:a\right ) : 4 _{2//3} 4 ^{+} (* 4 _{24} _{22} _{0){:}3} (* 2 _{02} _{02} _{02} _{0){:}6} 209 F432 F 4 3 2 \Gamma ^{fO} _{c} ^{3} 70s \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) : 4 / 3 2 ^{\circ-} (* 4 _{24} _{02} _{1){:}3} (* 2 _{02} _{12} _{02} _{1){:}6} 210 F4132 F 41 3 2 \Gamma ^{fO} _{c} ^{4} 97a \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) : 4 _{1//3} 2 ^{+} (* 4 _{34} _{12} _{0){:}3} (* 2 _{02} _{12} _{02} _{1){:}6} 211 I432 I 4 3 2 \Gamma ^{vO} _{c} ^{5} 69s \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) : 4 / 3 8 ^{+\circ} ( 4 _{24} _{02} _{1){:}3} ( 2 _{1{*}2} _{02} _{0){:}6} 212 P4332 P 43 3 2 \Gamma ^{6} _{cO} 94a \left ( a:a:a\right ) : 4 _{3//3} 2 ^{+/4} ( 4 _{1{*}2} _{1){:}3} ( 2 _{12} _{1\bar{ x }){:}6} 213 P4132 P 41 3 2 \Gamma ^{7} _{cO} 95a \left ( a:a:a\right ) : 4 _{1//3} 2 ^{+/4} ( 4 _{1{*}2} _{1){:}3} ( 2 _{12} _{1\bar{ x }){:}6} 214 I4132 I 41 3 2 \Gamma ^{vO} _{c} ^{8} 96a \left ( \tfrac{a + b + c}{ 2 }/:a:a:a\right ) : 4 _{1//3} 4 ^{+/4} (* 4 _{34} _{12} _{0){:}3} ( 2 _{0{*}2} _{12} _{1){:}6} 215 3m * 332 P3m P 3 m \Gamma _{cT} ^{1} ^{d} 65s \left ( a:a:a\right ) :\tilde 4 / 3 2 ^{\circ{:}2} (* 4 { ⋅ } 42 _{0){:}3} (* 2 _{02} _{02} _{02} _{0){:}6} 216 F3m F 3 m \Gamma ^{2} _{d} 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 { ⋅ } 42 _{1){:}3} (* 2 _{02} _{12} _{02} _{1){:}6} 217 I3m I 3 m \Gamma ^{3} _{d} 66s \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) :\tilde 4 / 3 4 ^{\circ{:}2} (*{ ⋅ } 44 {:} 2 ){:} 3 ( 2 _{1{*}2} _{02} _{0){:}6} 218 P3n P 3 n \Gamma _{cT} ^{4} ^{d} 51h \left ( a:a:a\right ) :\tilde 4 // 3 4 ^{\circ} (* 4 {:} 42 _{0){:}3} (* 2 _{02} _{02} _{02} _{0){:}6} 219 F3c F 3 c \Gamma ^{5} _{d} 52h \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) :\tilde 4 // 3 2 ^{\circ\circ} (* 4 {:} 42 _{1){:}3} (* 2 _{02} _{12} _{02} _{1){:}6} 220 I3d I 3 d \Gamma ^{6} _{d} 93a \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) :\tilde 4 // 3 4 ^{\circ/4} ( 4 \bar{*} 2 _{1){:}3} ( 2 _{0{*}2} _{12} _{1){:}6} 221 4/m 2/m * 432 Pmm P 4/m 2/m \Gamma _{cO} ^{1} ^{h} 71s \left ( a:a:a\right ) : 4 /\tilde 6 ⋅ m 4 ^{-{:}2} [*{ ⋅ } 4 { ⋅ } 4 { ⋅ } 2 ]{:} 3 [*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 { ⋅ } 2 ]{:} 6 222 Pnn P 4/n 2/n \Gamma _{cO} ^{2} ^{h} 53h \left ( a:a:a\right ) : 4 /\tilde 6 ⋅ \widetilde{abc} 8 ^{\circ\circ} (* 4 _{04{:}2){:}3} ( 2 \bar{*} _{12} _{02} _{0){:}6} 223 Pmn P 42/m 2/n \Gamma _{cO} ^{3} ^{h} 102a \left ( a:a:a\right ) : 4 _{2//\tilde} 6 ⋅ \widetilde{abc} 8 ^{\circ} [*{ ⋅ } 4 {:} 4 { ⋅ } 2 ]{:} 3 [*{ ⋅ } 2 { ⋅ } 2 { ⋅ } 2 { ⋅ } 2 ]{:} 6 224 Pnm P 42/n 2/m \Gamma _{cO} ^{4} ^{h} 103a \left ( a:a:a\right ) : 4 _{2//\tilde} 6 ⋅ m 4 ^{+{:}2} (* 4 _{24{ ⋅ }2){:}3} ( 2 \bar{*} _{12} _{02} _{0){:}6} 225 Fmm F 4/m 2/m \Gamma ^{5} _{h} 73s \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) : 4 /\tilde 6 ⋅ m 2 ^{-{:}2} [*{ ⋅ } 4 { ⋅ } 4 {:} 2 ]{:} 3 [*{ ⋅ } 2 { ⋅ } 2 {:} 2 {:} 2 ]{:} 6 226 Fmc F 4/m 2/c \Gamma ^{6} _{h} 54h \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) : 4 /\tilde 6 ⋅ \tilde c 4 ^{--} [*{ ⋅ } 4 {:} 4 {:} 2 ]{:} 3 [*{ ⋅ } 2 { ⋅ } 2 {:} 2 {:} 2 ]{:} 6 227 Fdm F 41/d 2/m \Gamma ^{7} _{h} 100a \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) : 4 _{1//\tilde} 6 ⋅ m 2 ^{+{:}2} (* 4 _{14{ ⋅ }2){:}3} ( 2 \bar{*} 2 _{02} _{1){:}6} 228 Fdc F 41/d 2/c \Gamma ^{8} _{h} 101a \left ( \tfrac{a + c}{ 2 }/\tfrac{b + c}{ 2 }/\tfrac{a + b}{ 2 }:a:a:a\right ) : 4 _{1//\tilde} 6 ⋅ \tilde c 4 ^{++} (* 4 _{14{:}2){:}3} ( 2 \bar{*} 2 _{02} _{1){:}6} 229 Imm I 4/m 2/m \Gamma ^{9} _{h} 72s \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) : 4 /\tilde 6 ⋅ m 8 ^{\circ{:}2} [*{ ⋅ } 4 { ⋅ } 4 {:} 2 ]{:} 3 [ 2 _{1{*}{ ⋅ }2{ ⋅ }2]{:}6} 230 Iad I 41/a 2/d \Gamma ^{10} _{h} 99a \left ( \tfrac{a + b + c}{ 2 }/a:a:a\right ) : 4 _{1//\tilde} 6 ⋅ \tfrac{ 1 }{ 2 }\widetilde{abc} 8 ^{\circ/4} (* 4 _{14{:}2){:}3} (* 2 _{12{:}2{:}2){:}6} External links International Union of Crystallography Point Groups and Bravais Lattices Full list of 230 crystallographic space groups Conway et al. on fibrifold notation

Notes and References

1. Book: Bradley, C. J. . Cracknell . A. P. . The mathematical theory of symmetry in solids: representation theory for point groups and space groups . Clarendon Press . Oxford New York . 2010 . 978-0-19-958258-7 . 859155300 . 127–134.