International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.2, pp. 56-62

## Section 1.2.6. Tables

T. Janssena*

aInstitute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

### 1.2.6. Tables

| top | pdf |

In the following, a short description of the tables is given in order to facilitate consultation without reading the introductory theoretical Sections 1.2.2 to 1.2.5.

Table 1.2.6.1. Finite point groups in three dimensions. The point groups are grouped by isomorphism class. There are four infinite families and six other isomorphism classes. (Notation: for the cyclic group of order n, for the dihedral group of order 2n, T, O and I the tetrahedral, octahedral and icosahedral groups, respectively). Point groups of the first class are subgroups of SO(3), those of the second class contain −E, and those of the third class are not subgroups of SO(3), but do not contain −E either. The families and are also isomorphism classes of two-dimensional finite point groups.

 Table 1.2.6.1| top | pdf | Finite point groups in three dimensions
Isomorphism classFirst class with determinants Second class with −EThird class without −EOrder
n   (n even, 2)
m ()
n22 (n even)   (n even)
n2 (n odd, )   (n even)
(n odd)
(n odd)
(n even)
(n even, )
()
(n odd, )
T 23
O 432
I 532

Table 1.2.6.2. Among the infinite number of finite three-dimensional point groups, 32 are crystallographic.

 Table 1.2.6.2| top | pdf | Crystallographic point groups in three dimensions
Isomorphism classFirst classSecond class with −EThird class without −EOrder
1     1
2 m 2
3     3
4   4
222 4
6 6
32   6
8
422   , 8
8
622 , 12
T 23     12
12
16
O 432   24
24
24
48

Table 1.2.6.3. Character table for the cyclic groups . The generator is denoted by . The number of elements in the conjugacy classes () is one for each class. The order is the smallest nonnegative power p for which . The n irreducible representations are denoted by .

 Table 1.2.6.3| top | pdf | Irreducible representations for cyclic groups
 , l.c.d. = largest common divisor.

11111
Order1nn/l.c.d.(n, 2)n/l.c.d.(n, 3)n
1 1 1 1 1
1
1
1

Table 1.2.6.4. Character tables for the dihedral groups of order . is the number of elements in the conjugacy class . The irreducible representations are denoted by .

 Table 1.2.6.4| top | pdf | Irreducible representations for dihedral groups
 (a) n odd. , l.c.d. = largest common divisor.

112n
Order1n/l.c.d.()2
1 1 1
1 1 −1
2 0
 (b) n even. , l.c.d. = largest common divisor.

112
Order12n/l.c.d.22
1 1 1 1 1
1 1 1 −1 −1
1 1 −1
1 −1 1
2 2 2cos() 0 0

Table 1.2.6.5. The character tables for the 32 three-dimensional crystallographic point groups. The groups are grouped by isomorphism class (there are 18 isomorphism classes).

 Table 1.2.6.5| top | pdf | Irreducible representations and character tables for the 32 crystallographic point groups in three dimensions
 (a)
 n 1 Order 1 1
 (b)
 n 1 1 Order 1 2 1 1 1 −1
 z z
 (c) [].
 n 1 1 1 Order 1 3 3 1 1 1 1 1
 Matrices of the real two-dimensional representation:

 z
 (d)
n1
Order1
1
1
1
1
 Matrices of the real two-dimensional representation:

 z z
 (e) [].
n1
Order1
1
1
1
1
1
1
 Matrices of the real representations:

 z : : z : : z : :
 (f)
n1
Order1
1
1
1
1
 x y z z y x z z
 (g)
n
Order
 Matrices of the two-dimensional representation:

 z z
 (h)
n1
Order1
1
1
1
1
2
 Matrices of the two-dimensional representation:

 422 z z z
 (i)
n1
Order1
1
1
1
1
2
2
 Matrices of the two-dimensional representations:

 z z z z
 (j) T [].
T
n144
Order133
1 1 1
1
1
3 0 0
 Real representations of dimension :

 T
 (k) O
O
n1
Order1
1
1
2
3
3
 Higher-dimensional representations:

 O
 Other point groups which are of second class and contain . See Table 1.2.6.6(a).
GroupIsomorphism classRotation subgroup
4
6
222
422
622
23
432

For each isomorphism class, the character table is given, including the symbol for the isomorphism class, the number n of elements per conjugacy class and the order of the elements in each such class. The conjugation classes are specified by representative elements expressed in terms of the generators . The irreps are denoted by , where i takes as many values as there are conjugation classes. In each isomorphism class for each point group, given by its international symbol and its Schoenflies symbol, identification is made between the generators of the abstract group () and the generating orthogonal transformations. Notation: is a rotation of along the x axis, is a reflection from a plane perpendicular to the x axis, is a rotation over along the z axis multiplied by and is a reflection from a plane through the unique axis.

The notation for the irreducible representations can be given as , but other systems have been used as well. Indicated below are the relations between and a system that uses a characterization according to the dimension of the representation and (for groups of the second kind) the sign of the representative of . This nomenclature is often used by spectroscopists.The other notation for which the relation with the present notation is indicated is that of Kopský, and is used in the accompanying software.

The three functions x, y and z transform according to the vector representation of the point group, which is generally reducible. The reduction into irreducible components of this three-dimensional vector representation is indicated.

The six bilinear functions , , , , , transform according to the symmetrized product of the vector representation. The basis functions of the irreducible components are indicated. Because the basis functions are real, one should consider the physically irreducible representations.

Table 1.2.6.6. The point groups of the second class containing are obtained from those of the first class by taking the direct product with the group generated by . From the point groups, one obtains nonmagnetic point groups by the direct product with the group generated by the time reversal . The relation between the characters of a point group and its direct products with groups generated by , and are given in Tables 1.2.6.6(a), (b) and (c), respectively.

 Table 1.2.6.6| top | pdf | Direct products with and
 (a) With .
 cf. 4 cf. 6 cf. 222 cf. 422 cf. 622 cf. 23 cf. 432
 (b) With .
 cf. 1 cf. 2 cf. m cf. 222 cf. cf. 4 cf. cf. cf. 422 cf. cf. 3 cf. 32 cf. cf. cf. cf. 6 cf. cf. 622 cf. cf. 23 cf. 432 cf.
 (c) With and .
 cf. 1 cf. 2 cf. 4 cf. 6 cf. 222 cf. 422 cf. cf. 622 cf. 23 cf. 432

Table 1.2.6.7. The representations of a point group are also representations of their double groups. In addition, there are extra representations which give projective representations of the point groups. For several cases, these are associated with an ordinary representation. As extra representations, those irreducible representations of the double point groups that give rise to projective representations of the point groups with a factor system that is not associated with the trivial one are given. These do not correspond to ordinary representations of the single group.

 Table 1.2.6.7| top | pdf | Extra representations of double point groups
 222d E −E A B AB 2 −2 422d E −E A2 −A B AB 2 −2 2 −2 622d E −E −A2 B A3 −A5 A3B 2 −2 −1 2 −2 −1 2 −2 −2 23d E −E −A −A2 B 2 −2 −1 −1 2 −2 2 −2 432d E −E −B A2 −A AB 2 −2 −1 2 −2 −1 4 −4 −1

Table 1.2.6.8. If one chooses for each element of a point group one of the two corresponding elements, the latter form a projective representation of the point group. If one selects for the rotation the element where is the rotation angle and the rotation axis, and for the element where and are the rotation angle and axis of the rotation , the matrices form a projective representation: The factor system is the spin factor system. It is determined via the generators and defining relations of the point group K. Then and the factors fix uniquely the class of the factor system . These factors are given in the table.

 Table 1.2.6.8| top | pdf | Projective spin representations of the 32 crystallographic point groups
Point groupRelations giving Double groupExtra representations
1 No
No
2, m No
,
222, Yes

,
4, 4d No
,
422, , 422d Yes
As above, plus , ,
3 3d No

32, 32d No
,
6, 6d No
,
622, , 622d Yes
As above, plus , ,
23 23d Yes
As above, plus , ,
432, 432d Yes
As above, plus , ,

Because is represented by the unit matrix in spin space, the double groups of two isomorphic point groups obtained from each other by replacing the elements by are the same.

The projective representations with factor system may sometimes be associated with one with a trivial factor system. If this is the case, there are actually no extra representations of the double group. If there are extra representations, these are irreducible representations of the double group: see Table 1.2.6.7.

Table 1.2.6.9. For the 32 three-dimensional crystallographic point groups, the character of the vector representation and the number of times the identity representation occurs in a number of tensor products of this vector representation are given. This is identical to the number of free parameters in a tensor of the corresponding type. For the direct products , the character is equal to that of K on the rotation subgroup, and its opposite [] for the coset .

 Table 1.2.6.9| top | pdf | Number of free parameters of some tensors
GroupIsomorphism classCharacter of the vector representationMultiplicity identity representation in
1 3 9 6 27 18 21
3, −3 9 6 0 0 21
2 3, −1 5 4 13 8 13
m 3, 1 5 4 14 10 13
5 4 0 0 13
222 3, −1, −1, −1 3 3 6 3 9
3, 1, 1, −1 3 3 7 5 9
3 3 0 0 9
3 3, 0, 0 3 2 9 6 9
3 2 0 0 9
32 3, 0, −1 2 2 4 2 6
3, 0, 1 2 2 5 4 6
2 2 0 0 6
6 3, 2, 0, −1, 0, 2 3 2 7 4 5
3, 2, 0, 1, 0, −2 3 2 2 2 5
3 2 0 0 5
622 3, 2, 0, −1, −1, −1 2 2 3 1 5
3, 2, 0, −1, 1, 1 2 2 4 3 5
3, −2, 0, 1, −1, 1 2 2 1 1 5
2 2 0 0 5
4 3, 1, −1, 1 3 2 7 4 7
3, −1, −1, −1 3 2 6 4 7
3 2 0 0 7
422 3, 1, −1, −1, −1 2 2 3 1 6
3, 1, −1, 1, 1 2 2 4 3 6
3, −1, −1, −1, 1 2 2 3 2 6
2 2 0 0 6
23 T 3, 0, 0, −1 1 1 2 1 3
1 1 0 0 3
432 O 3, 0, −1, 1, −1 1 1 1 0 3
O 3, 0, −1, −1, 1 1 1 1 1 3
1 1 0 0 3

Table 1.2.6.10. The irreducible projective representations of the 32 three-dimensional crystallographic point groups that have a factor system that is not associated to a trivial one. In three (and two) dimensions all factor systems are of order two.

 Table 1.2.6.10| top | pdf | Irreducible projective representations of the 32 crystallographic point groups
 (a)
ElementsEABAB
2 0 0 0
 (b)
ElementsEA2A3BA2BABA3B
2 0 0 0 0 0
2 0 0 0 0 0
 (c)
ElementsEBA2BA4BA3ABA3BA5B
2 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0
 (d) T [].
 Elements Elements
 (e) O
 Elements Elements Elements Elements
 (f)
ElementsEAA3BABA2BA3B
2 0 0 0 0 0 0
2 0 0 0 0 0 0
 (g)
ElementsEAA2A3A4A5BABA2BA3BA4BA5B
2 0 2 0 2 0 0 0 0 0 0 0
2 0 2 0 2 0 0 0 0 0 0 0
2 0 2 0 2 0 0 0 0 0 0 0
 (h)
ElementsE
2
2
Elements E
2
2
Elements E
2
2
Elements E
2
2
Elements E
2
2
Elements E
2
2
Elements E
2
2

Table 1.2.6.11. The special points in the Brillouin zones. Strata of irreducible representations of the space groups are characterized by the wavevector of such a point and a (possibly projective) irreducible representation of the point group . The latter is the intersection of the symmetry group of (the group of for the holohedral point group) and the point group of the space group. For each Bravais class the special points for the holohedry are given. These are given by their coordinates with respect to a basis of the reciprocal lattice of the conventional cell. These points correspond to Wyckoff positions in the corresponding dual lattice. The symbols for these Wyckoff positions and their site symmetry are given. A well known notation for the special points is that of Kovalev, as used in his book on representations of space groups. Correspondence with the notation in Kovalev (1987) is given.

 Table 1.2.6.11| top | pdf | Special points in the Brillouin zones in three dimensions
 (a) Triclinic
Kovalev
a
b
c
d
e
f
g
h
 (b) Monoclinic P
Kovalev
a
b
c
d
e
f
g
h
i 2
j 2
k 2
l 2
m m
n m
 (c) Monoclinic A
Kovalev
a
b
c
d
e
f
g 2
h 2
i m
 (d) Orthorhombic P
Kovalev
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u m
v
w
x
y
z
 (e) Orthorhombic C
Kovalev
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
 (f) Orthorhombic I
Kovalev
a
b
c

d

e

f
g
h
i
j
k
l
m
n
o
 (g) Orthorhombic F
Kovalev
a
b
c
d
e
f
g
h
i 2
j
k

l
m
n
 (h) Tetragonal P
Kovalev
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r m
s
t
 (i) Tetragonal I
Kovalev
a
b
c
d
e
f
g
h
i
j
k
l
m m
m
n
 (j) Trigonal R (rhombohedral axes)
Kovalev
a
b
c
d
e
f
g
h m
 (k) Hexagonal P
Kovalev
a
b
c
d
e
f
g
h
i
j
k
l
m
n m
o m
p m
q m
 (l) Cubic P
Kovalev
a
b
c
d
e
f
g
h
i
j
k
l
m
 (m) Cubic F
Kovalev
a
b
c
d
e
f
g
h
i
j
k m
 (n) Cubic I
Kovalev
a
b
c
d
e
f
g
h
i
j
k m
m

Table 1.2.6.12. The three-dimensional crystallographic magnetic and nonmagnetic point groups of type I (trivial magnetic, no antichronous elements), type II (nonmagnetic, containing time reversal as an element) and type III (nontrivial magnetic, without time reversal itself, but with antichronous elements).

 Table 1.2.6.12| top | pdf | Magnetic point groups
Type IType IIType III
1
2
m
, , ,
222
,
, ,
4
, ,
422 ,
,
, ,
, , , ,
3
32
, ,
6
, ,
622 ,
,
, ,
, , , ,
23
432
, ,

### References

Kovalev, O. V. (1987). Representations of the Crystallographic Space Groups. New York: Gordon and Breach.