## The 32 Crystal Classes

Crystal system

Crystal Class

International
symbol

Schoenflies
symbol

Space groups

2-fold axes

3-fold axes

4-fold axes

6-fold axes

mirror planes

inversion

Examples

Number of symmetry elements

Generating matrices

rank 1 tensors1

rank 2 symmetric tensors2

rank 2 asymmetric tensors

rank 3 tensors3

rank 4 tensors

Group elements

Triclinic
a ≠ b ≠ c
α ≠ β ≠ γ

triclinic-pedial

1

C1

1: P1

-

-

-

-

-

n

1

triclinic-pinacoidal

1

S2 = Ci

2: P1

-

-

-

-

-

y

2

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

monoclinic-sphenoidal

2

C2

3: P2, 4: P21, 5: C2

1

-

-

-

-

n

2

monoclinic-domatic

m

C1h = Cs

6: Pm, 7: Pc, 8: Cm, 9: Cc

-

-

-

-

1

n

2

monoclinic-prismatic

$\frac{2}{m}$

C2h

10: P2/m, 11: P21/m, 12: C2/m, 13: P2/c, 14: P21/c, 15: C2/c

1

-

-

-

1

y

4

Crystal system

Crystal Class

International
symbol

Schoenflies
symbol

Space groups

2-fold axes

3-fold axes

4-fold axes

6-fold axes

mirror planes

inversion

Examples

Number of symmetry elements

Generating matrices

rank 1 tensors1

rank 2 symmetric tensors2

rank 2 asymmetric tensors

rank 3 tensors3

rank 4 tensors

Group elements

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

orthorhombic-disphenoidal

222

V = D2

16: P222, 17: P2221, 18: P21212, 19: P212121, 20: C2221, 21: C222, 22: F222, 23: I222, 24: I212121

3

-

-

-

-

n

4

orthorhombic-pyramidal

mm2

C2v

25: Pmm2, 26: Pmc21, 27: Pcc2, 28: Pma2, 29: Pca21, 30: Pnc2, 31: Pmn21, 32: Pba2, 33: Pna21, 34: Pnn2 35: Cmm2, 36: Cmc21, 37: Ccc2, 38: Amm2, 39: Aem2, 40: Ama2, 41: Aea2, 42: Fmm2, 43: Fdd2, 44: Imm2, 45: Iba2, 46: Ima2

1

-

-

-

2

n

4

orthorhombic-dipyramidal

mmm

Vh = D2h

47: Pmmm, 48: Pnnn, 49: Pccm, 50: Pban, 51: Pmma, 52: Pnna, 53: Pmna, 54: Pcca, 55: Pbam, 56: Pccn, 57: Pbcm, 58: Pnnm, 59: Pmmn, 60: Pbcn, 61: Pbca, 62: Pnma 63: Cmcm, 64: Cmce, 65: Cmmm, 66: Cccm, 67: Cmme, 68: Ccce, 69: Fmmm, 70: Fddd, 71: Immm, 72: Ibam, 73: Ibca, 74: Imma

3

-

-

-

3

y

47: YBa2Cu3O7-x
62: Fe3C
64: αGa
70: αS

8

Crystal system

Crystal Class

International
symbol

Schoenflies
symbol

Space groups

2-fold axes

3-fold axes

4-fold axes

6-fold axes

mirror planes

inversion

Examples

Number of symmetry elements

Generating matrices

rank 1 tensors1

rank 2 symmetric tensors2

rank 2 asymmetric tensors

rank 3 tensors3

rank 4 tensors

Group elements

tetragonal-pyramidal

4

C4

75: P4, 76: P41, 77: P42, 78: P43, 79: I4, 80: I41

-

-

1

-

-

n

4

tetragonal-disphenoidal

4

S4

81: P4, 82: I4

1

-

-

-

-

n

4

tetragonal-dipyramidal

$\frac{4}{m}$

C4h

83: P4/m, 84: P42/m, 85: P4/n, 86: P42/n, 87: I4/m, 88: I41/a

-

-

1

-

1

y

8

tetragonal-trapezoidal

422

D4

89: P422, 90: P4212, 91: P4122, 92: P41212, 93: P4222, 94: P42212, 95: P4322, 96: P43212, 97: I422, 98: I4122

4

-

1

-

-

n

8

ditetragonal-pyramidal

4mm

C4v

99: P4mm, 100: P4bm, 101: P42cm, 102: P42nm, 103: P4cc, 104: P4nc, 105: P42mc, 106: P42bc, 107: I4mm, 108: I4cm, 109: I41md, 110: I41cd

-

-

-

-

4

n

99: PZT PbZrxTi1-xO3 x<0.52

8

tetragonal-scalenoidal

42m

Vd = D2d

111: P42m, 112: P42c, 113: P421m, 114: P421c, 115: P4m2, 116: P4c2, 117: P4b2, 118: P4n2, 119: I4m2, 120: I4c2, 121: I42m, 122: I42d

3

-

-

-

2

n

8

ditetragonal-dipyramidal

$\frac{4}{m}mm$

D4h

123: P4/mmm, 124: P4/mcc, 125: P4/nbm, 126: P4/nnc, 127: P4/mbm, 128: P4/mnc, 129: P4/nmm, 130: P4/ncc, 131: P42/mmc, 132: P42/mcm, 133: P42/nbc, 134: P42/nnm, 135: P42/mbc, 136: P42/mnm, 137: P42/nmc, 138: P42/ncm, 139: I4/mmm, 140: I4/mcm, 141: I41/amd, 142: I41/acd

4

-

1

-

5

y

139: face centered tetragonal, In
141: βSn

16

Crystal system

Crystal Class

International
symbol

Schoenflies
symbol

Space groups

2-fold axes

3-fold axes

4-fold axes

6-fold axes

mirror planes

inversion

Examples

Number of symmetry elements

Generating matrices

rank 1 tensors1

rank 2 symmetric tensors2

rank 2 asymmetric tensors

rank 3 tensors3

rank 4 tensors

Group elements

trigonal-pyramidal

3

C3

143: P3, 144: P31, 145: P32, 146: R3

-

1

-

-

-

n

3

rhombohedral

3

S6 = C6i

147: P3, 148: R3

-

1

-

-

-

y

6

trigonal-trapezoidal

32

D3

149: P312, 150: P321, 151: P3112, 152: P3121, 153: P3212, 154: P3221, 155: R32

3

1

-

-

-

n

154: α-Quartz

6

ditrigonal-pyramidal

3m

C3v

156: P3m1, 157: P31m, 158: P3c1, 159: P31c, 160: R3m, 161: R3c

-

1

-

-

3

n

161: ferroelectric LiNbO3

6

ditrigonal-scalahedral

3m

D3d

162: P31m, 163: P31c, 164: P3m1, 165: P3c1, 166: R3m, 167: R3c

3

1

-

-

3

y

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

12

Crystal system

Crystal Class

International
symbol

Schoenflies
symbol

Space groups

2-fold axes

3-fold axes

4-fold axes

6-fold axes

mirror planes

inversion

Examples

Number of symmetry elements

Generating matrices

rank 1 tensors1

rank 2 symmetric tensors2

rank 2 asymmetric tensors

rank 3 tensors3

rank 4 tensors

Group elements

hexagonal-pyramidal

6

C6

168: P6, 169: P61, 170: P65, 171: P62, 172: P64, 173: P63

-

-

-

1

-

n

6

trigonal-dipyramidal

6

C3h = S3

174: P6

-

1

-

-

1

n

6

hexagonal-dipyramidal

$\frac{6}{m}$

C6h

175: P6/m, 176: P63/m

-

-

-

1

1

y

12

hexagonal-trapezoidal

622

D6

177: P622, 178: P6122, 179: P6522, 180: P6222, 181: P6422, 182: P6322

6

-

-

1

-

n

180: β-Quartz

12

dihexagonal-pyramidals

6mm

C6v

183: P6mm, 184: P6cc, 185: P63cm, 186: P63mc

-

-

-

1

6

n

186: Wurtzite,

12

ditrigonal-dipyramidal

6m2

D3h

187: P6m2, 188: P6c2, 189: P62m, 190: P62c

3

1

-s

-

4

n

12

dihexagonal-dipyramidal

$\frac{6}{m}mm$

D6h

191: P6/mmm, 192: P6/mcc, 193: P63/mcm, 194: P63/mmc

6

-

-

1

7

y

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

Crystal system

Crystal Class

International
symbol

Schoenflies
symbol

Space groups

2-fold axes

3-fold axes

4-fold axes

6-fold axes

mirror planes

inversion

Examples

Number of symmetry elements

Generating matrices

rank 1 tensors1

rank 2 symmetric tensors2

rank 2 asymmetric tensors

rank 3 tensors3

rank 4 tensors

Group elements

tetrahedral

23

T

195: P23, 196: F23, 197: I23, 198: P213, 199: I213

3

4

-

-

-

n

12

diploidal

m3

Th

200: Pm3, 201: Pn3, 202: Fm3, 203: Fd3, 204: Im3, 205: Pa3, 206: Ia3

3

4

-

-

3

y

24

gyroidal

432

O

207: P432, 208: P4232, 209: F432, 210: F4132, 211: I432, 212: P4332, 213: P4132, 214: I4132

6

4

3

-

-

n

24

hextetrahedral

43m

Td

215: P43m, 216: F43m, 217: I43m, 218: P43n, 219: F43c, 220: I43d

3

4

-

-

6

n

216: Zincblende, GaAs, GaP, InAs, SiC

24

hexoctahedral

m3m

Oh

221: Pm3m, 222: Pn3n, 223: Pm3n, 224: Pn3m 225: Fm3m, 226: Fm3c, 227: Fd3m, 228: Fd3c 229: Im3m, 230: Ia3d

6

4

3

-

9

y

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

2 Rank 2 tensors are matrices. Examples of properties described by rank 2 tensors are electrical conductivity, thermal conductivity, dielectic constant, magnetic susceptibility, and thermal expansion. Only the upper elements of the matrices are given because gij = gji.