Bilbao Crystallographic Server arrow Representations


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

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
GM1+
Ag
GM1+
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
GM4+
Bg
GM2+
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
GM5+
2E1g
GM3+
1
1
-(1+i3)/2
-(1-i3)/2
-(1-i3)/2
-(1+i3)/2
1
1
-(1+i3)/2
-(1-i3)/2
-(1-i3)/2
-(1+i3)/2
GM5-
2E1u
GM3-
1
1
-(1+i3)/2
-(1-i3)/2
-(1-i3)/2
-(1+i3)/2
-1
-1
(1+i3)/2
(1-i3)/2
(1-i3)/2
(1+i3)/2
GM2+
2E2g
GM4+
1
-1
-(1+i3)/2
-(1-i3)/2
(1-i3)/2
(1+i3)/2
1
-1
-(1+i3)/2
-(1-i3)/2
(1-i3)/2
(1+i3)/2
GM2-
2E2u
GM4-
1
-1
-(1+i3)/2
-(1-i3)/2
(1-i3)/2
(1+i3)/2
-1
1
(1+i3)/2
(1-i3)/2
-(1-i3)/2
-(1+i3)/2
GM6+
1E1g
GM5+
1
1
-(1-i3)/2
-(1+i3)/2
-(1+i3)/2
-(1-i3)/2
1
1
-(1-i3)/2
-(1+i3)/2
-(1+i3)/2
-(1-i3)/2
GM6-
1E1u
GM5-
1
1
-(1-i3)/2
-(1+i3)/2
-(1+i3)/2
-(1-i3)/2
-1
-1
(1-i3)/2
(1+i3)/2
(1+i3)/2
(1-i3)/2
GM3+
1E2g
GM6+
1
-1
-(1-i3)/2
-(1+i3)/2
(1+i3)/2
(1-i3)/2
1
-1
-(1-i3)/2
-(1+i3)/2
(1+i3)/2
(1-i3)/2
GM3-
1E2u
GM6-
1
-1
-(1-i3)/2
-(1+i3)/2
(1+i3)/2
(1-i3)/2
-1
1
(1-i3)/2
(1+i3)/2
-(1+i3)/2
-(1-i3)/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: 2001
C3: 3-001
C4: 3+001
C5: 6-001
C6: 6+001
C7: -1
C8: m001
C9: -3-001
C10: -3+001
C11: -6-001
C12: -6+001

List of pairs of conjugated irreducible representations

(GM3+,GM5+)
(GM3-,GM5-)
(GM4+,GM6+)
(GM4-,GM6-)
Matrices of the representations of the group

N
General position
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)
1
(
1 0 0
0 1 0
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
)
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
3
(
-1 1 0
-1 0 0
0 0 1
)
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
4
(
-1 0 0
0 -1 0
0 0 1
)
2001
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
5
(
0 1 0
-1 1 0
0 0 1
)
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
6
(
1 -1 0
1 0 0
0 0 1
)
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
7
(
-1 0 0
0 -1 0
0 0 -1
)
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
8
(
0 1 0
-1 1 0
0 0 -1
)
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
9
(
1 -1 0
1 0 0
0 0 -1
)
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
10
(
1 0 0
0 1 0
0 0 -1
)
m001
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
11
(
0 -1 0
1 -1 0
0 0 -1
)
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
12
(
-1 1 0
-1 0 0
0 0 -1
)
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
k-Subgroupsmag
Bilbao Crystallographic Server
http://www.cryst.ehu.es
For comments, please mail to
administrador.bcs@ehu.eus