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Irreducible representations of the Double Point Group 6 (No. 22)

Table of characters

(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
-(i+3)/2
(i-3)/2
-1
-(1-i3)/2
-(1+i3)/2
i
(i+3)/2
-(i-3)/2
GM7
1E3
GM10
1
(1-i3)/2
(1+i3)/2
i
(i+3)/2
-(i-3)/2
-1
-(1-i3)/2
-(1+i3)/2
-i
-(i+3)/2
(i-3)/2
GM8
2E3
GM11
1
(1+i3)/2
(1-i3)/2
-i
-(i-3)/2
(i+3)/2
-1
-(1+i3)/2
-(1-i3)/2
i
(i-3)/2
-(i+3)/2
GM10
1E2
GM12
1
(1+i3)/2
(1-i3)/2
i
(i-3)/2
-(i+3)/2
-1
-(1+i3)/2
-(1-i3)/2
-i
-(i-3)/2
(i+3)/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: 3+001
C3: 3-001
C4: m001
C5: -6-001
C6: -6+001
C7d1
C8d3+001
C9d3-001
C10dm001
C11d-6-001
C12d-6+001

List of pairs of conjugated irreducible representations

(*GM3,*GM5)
(*GM4,*GM6)
(*GM7,*GM8)
(*GM9,*GM12)
(*GM10,*GM11)
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(0)
GM4(0)
GM5(0)
GM6(0)
GM7(0)
GM8(0)
GM9(0)
GM10(0)
GM11(0)
GM12(0)
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
k-Subgroupsmag
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