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Irreducible representations of the Double Point Group 62m (No. 26)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
GM1
A1'
GM1
1
1
1
1
1
1
1
1
1
GM3
A1''
GM2
1
1
-1
-1
1
-1
1
1
-1
GM4
A2''
GM3
1
1
-1
-1
-1
1
1
1
-1
GM2
A2'
GM4
1
1
1
1
-1
-1
1
1
1
GM6
E'
GM5
2
-1
2
-1
0
0
2
-1
-1
GM5
E''
GM6
2
-1
-2
1
0
0
2
-1
1
GM9
E3
GM7
2
-2
0
0
0
0
-2
2
0
GM8
E2
GM8
2
1
0
-√3
0
0
-2
-1
√3
GM7
E1
GM9
2
1
0
√3
0
0
-2
-1
-√3
(1): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press, based on Mulliken RS (1933) Phys. Rev. 43, 279-302.
(3): Notation of the irreps according to A. P. Cracknell, B. L. Davies, S. C. Miller and W. F. Love (1979) Kronecher Product Tables, 1, General Introduction and Tables of Irreducible Representations of Space groups. New York: IFI/Plenum, for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 3+001, 3-001
C3: m001, dm001
C4: -6-001, -6+001
C5: 2110, 2100, 2010, d2110, d2100, d2010
C6: m1-10, m120, m210, dm1-10, dm120, dm210
C7: d1
C8: d3+001, d3-001
C9: d-6-001, d-6+001

Matrices of the representations of the group

The number in parentheses after the label of the irrep indicates the "reality" of the irrep: (1) for real, (-1) for pseudoreal and (0) for complex representations.

N
Matrix presentation
Seitz Symbol
GM1(1)
GM2(1)
GM3(1)
GM4(1)
GM5(1)
GM6(1)
GM7(-1)
GM8(-1)
GM9(-1)
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
1
1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
2
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 (1+i√3)/2  0
  0 (1-i√3)/2
)
3+001
1
1
1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 -1  0
  0 -1
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
3
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 (1-i√3)/2  0
  0 (1+i√3)/2
)
3-001
1
1
1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 -1  0
  0 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
4
(
  1  0  0
  0  1  0
  0  0 -1
)
(
 -i  0
  0  i
)
m001
1
-1
-1
1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
5
(
  0 -1  0
  1 -1  0
  0  0 -1
)
(
 (√3-i)/2  0
  0 (√3+i)/2
)
6-001
1
-1
-1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
  i  0
  0 -i
)
(
 e-i5π/6  0
  0  ei5π/6
)
(
 e-iπ/6  0
  0  eiπ/6
)
6
(
 -1  1  0
 -1  0  0
  0  0 -1
)
(
 (√3+i)/2  0
  0 (√3-i)/2
)
6+001
1
-1
-1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 -i  0
  0  i
)
(
  ei5π/6  0
  0 e-i5π/6
)
(
  eiπ/6  0
  0 e-iπ/6
)
7
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0 -(1+i√3)/2
  (1-i√3)/2  0
)
2110
1
1
-1
-1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
8
(
  1 -1  0
  0 -1  0
  0  0 -1
)
(
  0 -1
  1  0
)
2100
1
1
-1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  1
 -1  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
(
  0 ei2π/3
  eiπ/3  0
)
9
(
 -1  0  0
 -1  1  0
  0  0 -1
)
(
  0 -(1-i√3)/2
  (1+i√3)/2  0
)
2010
1
1
-1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 -1
  1  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
(
  0  eiπ/3
 ei2π/3  0
)
10
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0 -(√3-i)/2
  (√3+i)/2  0
)
m110
1
-1
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
11
(
  1 -1  0
  0 -1  0
  0  0  1
)
(
  0 -i
 -i  0
)
m120
1
-1
1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
 0 i
 i 0
)
(
  0  e-iπ/6
 e-i5π/6  0
)
(
  0 e-i5π/6
  e-iπ/6  0
)
12
(
 -1  0  0
 -1  1  0
  0  0  1
)
(
  0  (√3+i)/2
 -(√3-i)/2  0
)
m210
1
-1
1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
 0 i
 i 0
)
(
  0 e-i5π/6
  e-iπ/6  0
)
(
  0  e-iπ/6
 e-i5π/6  0
)
13
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
1
1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
14
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 -(1+i√3)/2  0
  0 -(1-i√3)/2
)
d3+001
1
1
1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 1 0
 0 1
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
15
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 -(1-i√3)/2  0
  0 -(1+i√3)/2
)
d3-001
1
1
1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 1 0
 0 1
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
16
(
  1  0  0
  0  1  0
  0  0 -1
)
(
  i  0
  0 -i
)
dm001
1
-1
-1
1
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
17
(
  0 -1  0
  1 -1  0
  0  0 -1
)
(
 -(√3-i)/2  0
  0 -(√3+i)/2
)
d6-001
1
-1
-1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 -i  0
  0  i
)
(
  eiπ/6  0
  0 e-iπ/6
)
(
  ei5π/6  0
  0 e-i5π/6
)
18
(
 -1  1  0
 -1  0  0
  0  0 -1
)
(
 -(√3+i)/2  0
  0 -(√3-i)/2
)
d6+001
1
-1
-1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
  i  0
  0 -i
)
(
 e-iπ/6  0
  0  eiπ/6
)
(
 e-i5π/6  0
  0  ei5π/6
)
19
(
  0  1  0
  1  0  0
  0  0 -1
)
(
  0  (1+i√3)/2
 -(1-i√3)/2  0
)
d2110
1
1
-1
-1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
20
(
  1 -1  0
  0 -1  0
  0  0 -1
)
(
  0  1
 -1  0
)
d2100
1
1
-1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 -1
  1  0
)
(
  0  eiπ/3
 ei2π/3  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
21
(
 -1  0  0
 -1  1  0
  0  0 -1
)
(
  0  (1-i√3)/2
 -(1+i√3)/2  0
)
d2010
1
1
-1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  1
 -1  0
)
(
  0 ei2π/3
  eiπ/3  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
22
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0  (√3-i)/2
 -(√3+i)/2  0
)
dm110
1
-1
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
23
(
  1 -1  0
  0 -1  0
  0  0  1
)
(
 0 i
 i 0
)
dm120
1
-1
1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
  0 -i
 -i  0
)
(
  0 ei5π/6
  eiπ/6  0
)
(
  0  eiπ/6
 ei5π/6  0
)
24
(
 -1  0  0
 -1  1  0
  0  0  1
)
(
  0 -(√3+i)/2
  (√3-i)/2  0
)
dm210
1
-1
1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 -i
 -i  0
)
(
  0  eiπ/6
 ei5π/6  0
)
(
  0 ei5π/6
  eiπ/6  0
)
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