Crystal system

International
symbol

Schoenflies
symbol

Space groups

Examples

Number of symmetry elements

Generating matrices
By multiplying the generating matrices with themselves and with each other, all of the elements of the group can be generated.

Independent components of rank 1 tensors¹

Independent components of symmetric rank 2 tensors²

Independent components of asymmetric rank 2 tensors

Independent components of asymmetric rank 3 tensors³

Independent components of symmetric rank 4 tensors

Independent components of asymmetric rank 4 tensors

Group elements

Triclinic
a ≠ b ≠ c
α ≠ β ≠ γ

1

C₁

1

1

1

S₂ = C_i

2

2

Monoclinic
a ≠ b ≠ c
α ≠ 90°, β = γ = 90°

2

C₂

3-5

2

m

C_1h = C_s

6-9

2

2/m

C_2h

10-15

4

Orthorhombic
a ≠ b ≠ c
α = β = γ = 90°

222

V = D₂

16-24

4

mm2

C_2v

25-46

4

mmm

V_h = D_2h

47-74

47: YBa₂Cu₃O_7-x
62: Fe₃C
64: αGa
70: αS

8

Tetragonal

4

C₄

75-80

4

4

S₄

81-82

4

4/m

C_4h

83-88

8

422

D₄

89-98

8

4mm

C_4v

99-110

99: PZT PbZr_xTi_1-xO₃ x<0.52

8

42m

V_d = D_2d

111-122

8

4/mmm

D_4h

123-142

139: face centered tetragonal, In
141: βSn

16

Trigonal

3

C₃

143-146

3

3

S₆ = C_6i

147-148

6

32

D₃

149-155

154: α-Quartz

6

3m

C_3v

156-161

161: ferroelectric LiNbO₃

6

3m

D₃d

162-167

167: calcite, paraelectric LiNbO₃, sapphire (α-Al₂O₃)

12

Hexagonal

6

C₆

168-173

6

6

C_3h = S₃

174

6

6/m

C_6h

175-176

12

622

D₆

177-182

180: β-Quartz

12

6mm

C_6v

183-186

186: Wurtzite, ZnS

12

6m2

D_3h

187-190

12

6/mmm

D_6h

191-194

194: hcp, Mg, Be, Sc, α-Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Gd, Tb, Dy, Ho, Er, Tm, Lu, Hf, Re, Os, Tl, graphite, MoS₂, ice Ih

24

Cubic

23

T

195-199

12

m3

T_h

200-206

24

432

O

207-214

24

43m

T_d

215-220

216: Zincblende, ZnS, GaAs, GaP, InAs, SiC

24

m3m

O_h

221-230

221: CsCl, cubic perovskite
225: fcc, Al, Cu, Ni, Ag, Pt, Au, Pb, γ-Fe, NaCl
227: diamond, C, Si, Ge, α-Sn, spinel
229: bcc, Na, K, Cr, α-Fe, β-Ti, Nb, Mo, Ta

48