Bilbao Crystallographic Server arrow Representations


Irreducible representations of the Double Point Group 6/m (No. 23)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
GM4+
Bg
GM2+
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
GM4-
Bu
GM2-
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
-1
-1
-1
1
1
1
-1
-1
-1
1
1
1
GM5+
2E1g
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
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-
2E1u
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
-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
GM2+
2E2g
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
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
GM2-
2E2u
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
-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+
1E1g
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
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-
1E1u
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
-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+
1E2g
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
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-
1E2u
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
-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+
2E1g
GM7
1
-1
-1
-i
i
-i
-1
1
1
i
-i
i
1
-1
-1
-i
i
-i
-1
1
1
i
-i
i
GM11+
1E1g
GM8
1
-1
-1
i
-i
i
-1
1
1
-i
i
-i
1
-1
-1
i
-i
i
-1
1
1
-i
i
-i
GM9+
2E2g
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
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+
1E3g
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
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+
2E3g
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
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+
1E2g
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
(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
GM12-
2E1u
GM13
1
-1
-1
-i
i
-i
-1
1
1
i
-i
i
-1
1
1
i
-i
i
1
-1
-1
-i
i
-i
GM11-
1E1u
GM14
1
-1
-1
i
-i
i
-1
1
1
-i
i
-i
-1
1
1
-i
i
-i
1
-1
-1
i
-i
i
GM9-
2E2u
GM15
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
-(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-
1E3u
GM16
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
-(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-
2E3u
GM17
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
-(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-
1E2u
GM18
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
-(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: 2001
C5: 6-001
C6: 6+001
C7d1
C8d3+001
C9d3-001
C10d2001
C11d6-001
C12d6+001
C13: -1
C14: -3+001
C15: -3-001
C16: m001
C17: -6-001
C18: -6+001
C19d-1
C20d-3+001
C21d-3-001
C22dm001
C23d-6-001
C24d-6+001

List of pairs of conjugated irreducible representations

(*GM3+,*GM5+)
(*GM3-,*GM5-)
(*GM4+,*GM6+)
(*GM4-,*GM6-)
(*GM7,*GM8)
(*GM9,*GM12)
(*GM10,*GM11)
(*GM13,*GM14)
(*GM15,*GM18)
(*GM16,*GM17)
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)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(0)
GM3-(0)
GM4+(0)
GM4-(0)
GM5+(0)
GM5-(0)
GM6+(0)
GM6-(0)
GM7(0)
GM8(0)
GM9(0)
GM10(0)
GM11(0)
GM12(0)
GM13(0)
GM14(0)
GM15(0)
GM16(0)
GM17(0)
GM18(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
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
1
1
ei2π/3
ei2π/3
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
e-i2π/3
e-i2π/3
-1
-1
e-iπ/3
e-iπ/3
eiπ/3
eiπ/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
1
1
e-i2π/3
e-i2π/3
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
ei2π/3
ei2π/3
-1
-1
eiπ/3
eiπ/3
e-iπ/3
e-iπ/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
)
2001
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
-i
i
-i
i
-i
i
-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
-1
-1
ei2π/3
ei2π/3
e-iπ/3
e-iπ/3
e-i2π/3
e-i2π/3
eiπ/3
eiπ/3
i
-i
e-i5π/6
eiπ/6
e-iπ/6
ei5π/6
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
-1
-1
e-i2π/3
e-i2π/3
eiπ/3
eiπ/3
ei2π/3
ei2π/3
e-iπ/3
e-iπ/3
-i
i
ei5π/6
e-iπ/6
eiπ/6
e-i5π/6
-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
)
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
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
)
3+001
1
-1
1
-1
ei2π/3
e-iπ/3
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
e-i2π/3
eiπ/3
-1
-1
e-iπ/3
e-iπ/3
eiπ/3
eiπ/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
)
3-001
1
-1
1
-1
e-i2π/3
eiπ/3
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
ei2π/3
e-iπ/3
-1
-1
eiπ/3
eiπ/3
e-iπ/3
e-iπ/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
)
m001
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
-i
i
-i
i
-i
i
i
-i
i
-i
i
-i
11
(
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
e-iπ/3
e-iπ/3
ei2π/3
e-i2π/3
eiπ/3
eiπ/3
e-i2π/3
i
-i
e-i5π/6
eiπ/6
e-iπ/6
ei5π/6
-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
)
6+001
1
-1
-1
1
e-i2π/3
eiπ/3
eiπ/3
e-i2π/3
ei2π/3
e-iπ/3
e-iπ/3
ei2π/3
-i
i
ei5π/6
e-iπ/6
eiπ/6
e-i5π/6
i
-i
e-iπ/6
ei5π/6
e-i5π/6
eiπ/6
13
(
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
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
14
(
0 -1 0
1 -1 0
0 0 1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
1
1
1
ei2π/3
ei2π/3
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
e-i2π/3
e-i2π/3
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
15
(
-1 1 0
-1 0 0
0 0 1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
1
1
1
e-i2π/3
e-i2π/3
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
ei2π/3
ei2π/3
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
16
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
i
-i
i
-i
i
-i
i
-i
i
-i
i
-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
ei2π/3
e-iπ/3
e-iπ/3
e-i2π/3
e-i2π/3
eiπ/3
eiπ/3
-i
i
eiπ/6
e-i5π/6
ei5π/6
e-iπ/6
-i
i
eiπ/6
e-i5π/6
ei5π/6
e-iπ/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
e-i2π/3
eiπ/3
eiπ/3
ei2π/3
ei2π/3
e-iπ/3
e-iπ/3
i
-i
e-iπ/6
ei5π/6
e-i5π/6
eiπ/6
i
-i
e-iπ/6
ei5π/6
e-i5π/6
eiπ/6
19
(
-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
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
20
(
0 1 0
-1 1 0
0 0 -1
)
(
-(1+i3)/2 0
0 -(1-i3)/2
)
d3+001
1
-1
1
-1
ei2π/3
e-iπ/3
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
e-i2π/3
eiπ/3
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
-1
-1
e-iπ/3
e-iπ/3
eiπ/3
eiπ/3
21
(
1 -1 0
1 0 0
0 0 -1
)
(
-(1-i3)/2 0
0 -(1+i3)/2
)
d3-001
1
-1
1
-1
e-i2π/3
eiπ/3
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
ei2π/3
e-iπ/3
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
-1
-1
eiπ/3
eiπ/3
e-iπ/3
e-iπ/3
22
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
i
-i
i
-i
i
-i
-i
i
-i
i
-i
i
23
(
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
e-iπ/3
e-iπ/3
ei2π/3
e-i2π/3
eiπ/3
eiπ/3
e-i2π/3
-i
i
eiπ/6
e-i5π/6
ei5π/6
e-iπ/6
i
-i
e-i5π/6
eiπ/6
e-iπ/6
ei5π/6
24
(
-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
eiπ/3
eiπ/3
e-i2π/3
ei2π/3
e-iπ/3
e-iπ/3
ei2π/3
i
-i
e-iπ/6
ei5π/6
e-i5π/6
eiπ/6
-i
i
ei5π/6
e-iπ/6
eiπ/6
e-i5π/6
k-Subgroupsmag
Bilbao Crystallographic Server
http://www.cryst.ehu.es
For comments, please mail to
administrador.bcs@ehu.eus