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Irreducible representations of the Double Point Group 42m (No. 14)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
GM1
A1
GM1
1
1
1
1
1
1
1
GM3
B1
GM2
1
1
-1
1
-1
1
-1
GM4
B2
GM3
1
1
-1
-1
1
1
-1
GM2
A2
GM4
1
1
1
-1
-1
1
1
GM5
E
GM5
2
-2
0
0
0
2
0
GM7
E2
GM6
2
0
-2
0
0
-2
2
GM6
E1
GM7
2
0
2
0
0
-2
-2
(1): Notation of the irreps according to Koster GF, Dimmok JO, Wheeler RG and Statz H, (1963) Properties of the thirty-two point groups, M.I.T. Press, Cambridge, Mass.
(2): Notation of the irreps according to Mulliken RS (1933) Phys. Rev. 43, 279-302.
(3): Notation of the irreps according to C. J. Bradley, A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 2001d2001
C3: -4+001, -4-001
C4: 2010, 2100d2010d2100
C5: m110, m1-10dm110dm1-10
C6d1
C7d-4+001d-4-001

Matrices of the representations of the group

The number in parenthesis 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)
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
)
2
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2001
1
1
1
1
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
3
(
0 1 0
-1 0 0
0 0 -1
)
(
(1-i)2/2 0
0 (1+i)2/2
)
4+001
1
-1
-1
1
(
0 -1
1 0
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
4
(
0 -1 0
1 0 0
0 0 -1
)
(
(1+i)2/2 0
0 (1-i)2/2
)
4-001
1
-1
-1
1
(
0 1
-1 0
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
5
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2010
1
1
-1
-1
(
0 1
1 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
6
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2100
1
1
-1
-1
(
0 -1
-1 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
7
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
m110
1
-1
1
-1
(
1 0
0 -1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
8
(
0 1 0
1 0 0
0 0 1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
m110
1
-1
1
-1
(
-1 0
0 1
)
(
0 i
i 0
)
(
0 i
i 0
)
9
(
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
)
10
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
11
(
0 1 0
-1 0 0
0 0 -1
)
(
-(1-i)2/2 0
0 -(1+i)2/2
)
d4+001
1
-1
-1
1
(
0 -1
1 0
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
12
(
0 -1 0
1 0 0
0 0 -1
)
(
-(1+i)2/2 0
0 -(1-i)2/2
)
d4-001
1
-1
-1
1
(
0 1
-1 0
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
13
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2010
1
1
-1
-1
(
0 1
1 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
14
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2100
1
1
-1
-1
(
0 -1
-1 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
15
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
dm110
1
-1
1
-1
(
1 0
0 -1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
16
(
0 1 0
1 0 0
0 0 1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
dm110
1
-1
1
-1
(
-1 0
0 1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
k-Subgroupsmag
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