194
A Character Tables
The Dihedral Groups D n (n = 2, 3, 4, 5, 6)
D 2
ˆ
E
ˆ
C
z
2
ˆ
C
y
2
ˆ
C x
2
A
1
111
x 2 ,y 2 ,z 2
B 1
11
−1
−1
z, R z
xy
B 2
1
−11
−1
y,R y
xz
B 3
1
−1
−11
x,R x
yz
D 3
ˆ
E
2 ˆ
C 3
3 ˆ
C 2
A 1
111
x 2 + y 2 ,z 2
A 2
11
−1
z, R z
E
2
−10
(x, y)(R x ,R y )
(xz, yz)(x 2 − y 2 ,xy)
D 4
ˆ
E
2 ˆ
C 4
ˆ
C 2 (= ˆ
C 2
4 )
2 ˆ
C ′
2
2 ˆ
C ′′
2
A 1
11
1
11
x 2 + y 2 ,z 2
A 2
11
1
−1
−1
z, R z
B 1
1
−11
1
−1
x 2 − y 2
B 2
1
−11
−11
xy
E
20
−20
0
(x, y)(R x ,R y )( x z , y z )
D 5
ˆ
E 2 ˆ
C 5
2 ˆ
C 2
5
5 ˆ
C ′
2
A 1
11
1
1
x 2 + y 2 ,z 2
A 2
11
1
−1 z, R z
E 1
2
2 cos(2π/5) 2 cos(4π/5)
0 (x, y)(R x ,R y )( x z , y z )
E 2
2
2 cos(4π/5) 2 cos(2π/5)
0
(x 2 − y 2 ,xy)
D 6
ˆ
E
2 ˆ
C 6
2 ˆ
C 3
ˆ
C 2
3 ˆ
C ′
2
3 ˆ
C ′′
2
A 1
111
111
x 2 + y 2 ,z 2
A 2
111
1
−1
−1
z, R z
B 1
1
−11
−11
−1
x(x 2 − 3y 2 )
B 2
1
−11
−1
−11
y(3x 2 − y 2 )
E 1
21
−1
−200
(x, y)(R x ,R y )( x z , y z )
E 2
2
−1
−12
0
0
(x 2 − y 2 ,xy)