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Irreducible corepresentations of the Magnetic Point Group -6m'2' (N. 26.5.99)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
GM1
A'
GM1
1
1
1
1
1
1
1
1
1
1
1
1
GM4
A''
GM2
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
GM2
2E'
GM3
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
GM5
2E''
GM4
1
-(1-i3)/2
-(1+i3)/2
-1
(1-i3)/2
(1+i3)/2
1
-(1-i3)/2
-(1+i3)/2
-1
(1-i3)/2
(1+i3)/2
GM3
1E'
GM5
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
GM6
1E''
GM6
1
-(1+i3)/2
-(1-i3)/2
-1
(1+i3)/2
(1-i3)/2
1
-(1+i3)/2
-(1-i3)/2
-1
(1+i3)/2
(1-i3)/2
GM12
2E1
GM7
1
-1
-1
-i
i
-i
-1
1
1
i
-i
i
GM11
1E1
GM8
1
-1
-1
i
-i
i
-1
1
1
-i
i
-i
GM9
2E2
GM9
1
(1-i3)/2
(1+i3)/2
-i
-(3+i)/2
-(3-i)/2
-1
-(1-i3)/2
-(1+i3)/2
i
(3+i)/2
(3-i)/2
GM7
1E3
GM10
1
(1-i3)/2
(1+i3)/2
i
(3+i)/2
(3-i)/2
-1
-(1-i3)/2
-(1+i3)/2
-i
-(3+i)/2
-(3-i)/2
GM8
2E3
GM11
1
(1+i3)/2
(1-i3)/2
-i
(3-i)/2
(3+i)/2
-1
-(1+i3)/2
-(1-i3)/2
i
-(3-i)/2
-(3+i)/2
GM10
1E2
GM12
1
(1+i3)/2
(1-i3)/2
i
-(3-i)/2
-(3+i)/2
-1
-(1+i3)/2
-(1-i3)/2
-i
(3-i)/2
(3+i)/2
The notation used in this table is an extension to corepresentations of the following notations used for irreducible representations:
(1): Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): 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): 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 unitary symmetry operations in the conjugacy classes

C1: 1
C2: 3+001
C3: 3-001
C4: m001
C56-001
C66+001
C7d1
C8d3+001
C9d3-001
C10dm001
C11d6-001
C12d6+001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6GM7GM8GM9GM10GM11GM12
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
1
1
1
1
1
1
1
1
2
(
0 -1 0
1 -1 0
0 0 1
)
(
(1+i3)/2 0
0 (1-i3)/2
)
3+001
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
-1
-1
e-iπ/3
e-iπ/3
eiπ/3
eiπ/3
3
(
-1 1 0
-1 0 0
0 0 1
)
(
(1-i3)/2 0
0 (1+i3)/2
)
3-001
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
-1
-1
eiπ/3
eiπ/3
e-iπ/3
e-iπ/3
4
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
-1
1
-1
1
-1
-i
i
-i
i
-i
i
5
(
0 -1 0
1 -1 0
0 0 -1
)
(
(3-i)/2 0
0 (3+i)/2
)
6-001
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
i
-i
e-i5π/6
eiπ/6
e-iπ/6
ei5π/6
6
(
-1 1 0
-1 0 0
0 0 -1
)
(
(3+i)/2 0
0 (3-i)/2
)
6+001
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
-i
i
ei5π/6
e-iπ/6
eiπ/6
e-i5π/6
7
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
8
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
9
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
10
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
1
-1
1
-1
i
-i
i
-i
i
-i
11
(
0 -1 0
1 -1 0
0 0 -1
)
(
-(3-i)/2 0
0 -(3+i)/2
)
d6-001
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
-i
i
eiπ/6
e-i5π/6
ei5π/6
e-iπ/6
12
(
-1 1 0
-1 0 0
0 0 -1
)
(
-(3+i)/2 0
0 -(3-i)/2
)
d6+001
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
i
-i
e-iπ/6
ei5π/6
e-i5π/6
eiπ/6
13
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i3)/2
(1-i3)/2 0
)
m'110
1
1
1
1
1
1
1
1
1
1
1
1
14
(
-1 1 0
0 1 0
0 0 1
)
(
0 -1
1 0
)
m'100
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
-1
-1
eiπ/3
eiπ/3
e-iπ/3
e-iπ/3
15
(
1 0 0
1 -1 0
0 0 1
)
(
0 -(1-i3)/2
(1+i3)/2 0
)
m'010
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
16
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(3-i)/2
(3+i)/2 0
)
2'1-10
1
-1
1
-1
1
-1
-i
i
-i
i
-i
i
17
(
-1 1 0
0 1 0
0 0 -1
)
(
0 -i
-i 0
)
2'120
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
-i
i
ei5π/6
e-iπ/6
eiπ/6
e-i5π/6
18
(
1 0 0
1 -1 0
0 0 -1
)
(
0 (3+i)/2
-(3-i)/2 0
)
2'210
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
-i
i
eiπ/6
e-i5π/6
ei5π/6
e-iπ/6
19
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i3)/2
-(1-i3)/2 0
)
dm'110
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
20
(
-1 1 0
0 1 0
0 0 1
)
(
0 1
-1 0
)
dm'100
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
21
(
1 0 0
1 -1 0
0 0 1
)
(
0 (1-i3)/2
-(1+i3)/2 0
)
dm'010
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
-1
-1
e-iπ/3
e-iπ/3
eiπ/3
eiπ/3
22
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (3-i)/2
-(3+i)/2 0
)
d2'1-10
1
-1
1
-1
1
-1
i
-i
i
-i
i
-i
23
(
-1 1 0
0 1 0
0 0 -1
)
(
0 i
i 0
)
d2'120
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
i
-i
e-iπ/6
ei5π/6
e-i5π/6
eiπ/6
24
(
1 0 0
1 -1 0
0 0 -1
)
(
0 -(3+i)/2
(3-i)/2 0
)
d2'210
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
i
-i
e-i5π/6
eiπ/6
e-iπ/6
ei5π/6
k-Subgroupsmag
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