Ordine |
Gruppo canonico |
Gruppi di simmetria molecolare |
Altri gruppi
|
1 |
G11 = Z1 |
|
|
2 |
G21 = Z2 = Sym2 |
Ci
|
E |
i
|
|
|
Ag |
1 |
1
|
Rx, Ry, Rz
|
x2, y2, z2, xy, xz, yz
|
Au |
1 |
-1
|
x, y, z
|
|
Cs
|
E |
σh
|
|
|
A' |
1 |
1
|
x, y, Rz
|
x2, y2, z2, xy
|
A'' |
1 |
-1
|
z, Rx, Ry
|
xz, yz
|
C2
|
E |
c2 |
|
|
A |
1 |
1 |
Rz, z |
x2, y2, z2, xy
|
B |
1 |
-1 |
Rx, Ry, x, y |
xz, yz
|
|
Z3× ; Z4× ; Z6×
|
3 |
G31 = Z3 = Alt3 |
C3
|
E |
C3 |
C32 |
|
|
A |
1 |
1 |
1 |
Rz, z |
x2 + y2
|
E
|
1
1
|
ω
ω*
|
ω*
ω
|
(Rx, Ry),
(x, y)
|
(x2 - y2, xy),
(xz, yz)
|
ω = e2πi/3
|
---
|
4 |
G41 = Z4 |
C4
|
E |
C4 |
C2 |
C43 |
|
|
A |
1 |
1 |
1 |
1 |
Rz, z |
x2 + y2, z2
|
B |
1 |
−1 |
1 |
−1 |
|
x2 − y2, xy
|
E |
1
1
|
i
−i
|
−1
−1
|
−i
i
|
(Rx, Ry),
(x, y)
|
(xz, yz)
|
S4
|
E |
S4 |
C2 |
S43 |
|
|
A |
1 |
1 |
1 |
1 |
Rz |
x2+y2, z2
|
B |
1 |
−1 |
1 |
−1 |
z |
x2−y2, xy
|
E |
1
1
|
i
−i
|
−1
−1
|
−i
i
|
(Rx, Ry),
(x, y)
|
(xz, yz)
|
|
Z5× ; Z10×
|
4 |
G42 = Dih2 = Z2 × Z2 |
D2
|
E |
C2(z) |
C2(x) |
C2(y) |
|
|
A |
1 |
1 |
1 |
1 |
|
x2, y2, z2
|
B1 |
1 |
1 |
−1 |
−1 |
Rz, z |
xy
|
B2 |
1 |
−1 |
−1 |
1 |
Ry, y |
xz
|
B3 |
1 |
−1 |
1 |
−1 |
Rx, x |
yz
|
C2v
|
E |
C2 |
σv |
σv' |
|
|
A1 |
1 |
1 |
1 |
1 |
z |
x2, y2, z2
|
A2 |
1 |
1 |
−1 |
−1 |
Rz |
xy
|
B1 |
1 |
−1 |
1 |
−1 |
Ry, x |
xz
|
B2 |
1 |
−1 |
−1 |
1 |
Rx, y |
yz
|
C2h
|
E |
C2 |
i |
σh |
|
|
Ag |
1 |
1 |
1 |
1 |
Rz |
x2, y2, z2, xy
|
Au |
1 |
1 |
−1 |
−1 |
z |
|
Bg |
1 |
−1 |
1 |
−1 |
Rx, Ry |
xz, yz
|
Bu |
1 |
−1 |
−1 |
1 |
x, y |
|
|
Z8× ; Z12×
|
5 |
G51 = Z5 |
C5
|
E |
C5 |
C52 |
C53 |
C54 |
|
|
A |
1 |
1 |
1 |
1 |
1 |
Rz, z |
x2 + y2, z2
|
E1 |
1
1
|
η
η*
|
η2
η2*
|
η2*
η2
|
η*
η
|
(Rx, Ry),
(x, y)
|
(xz, yz)
|
E2
|
1
1
|
η2
η2*
|
η*
η
|
η
η*
|
η2*
η2
|
|
(x2 - y2, xy)
|
η = e2πi/5
|
---
|
6 |
G61 = Sym3 = Dih3 |
D3
|
E |
2 C3 |
3 C2' |
|
|
A1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2 |
1 |
1 |
−1 |
Rz, z |
|
E |
2 |
−1 |
0 |
(Rx, Ry), (x, y) |
(x2 − y2, xy), (xz, yz)
|
C3v
|
E |
2 C3 |
3 σv |
|
|
A1 |
1 |
1 |
1 |
z |
x2 + y2, z2
|
A2 |
1 |
1 |
−1 |
Rz |
|
E |
2 |
−1 |
0 |
(Rx, Ry), (x, y) |
(x2 − y2, xy), (xz, yz)
|
|
---
|
6 |
G62 = Z6 = Z3×Z2 |
C6
|
E |
C6 |
C3 |
C2 |
C32 |
C65 |
|
|
A |
1 |
1 |
1 |
1 |
1 |
1 |
Rz, z |
x2 + y2, z2
|
B |
1 |
−1 |
1 |
−1 |
1 |
−1 |
|
|
E1
|
1
1
|
ζ
ζ*
|
−ζ*
−ζ
|
−1
−1
|
−ζ
−ζ*
|
ζ*
−ζ
|
(Rx, Ry),
(x, y)
|
(xz, yz)
|
E2
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
|
(x2 − y2, xy)
|
S6
|
E |
S6 |
C3 |
i |
C32 |
S65 |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
Rz |
x2 + y2, z2
|
Eg
|
1
1
|
ζ*
ζ
|
ζ
ζ*
|
1
1
|
ζ*
ζ
|
ζ
ζ*
|
(Rx, Ry) |
(x2 − y2, xy), (xz, yz)
|
Au
|
1 |
−1 |
1 |
−1 |
1 |
−1 |
z |
|
Eu
|
1
1
|
−ζ*
−ζ
|
ζ
ζ*
|
−1
−1
|
ζ*
ζ
|
−ζ
−ζ*
|
(x, y)
|
|
C3h
|
E |
C3 |
C32 |
σh |
S3 |
S35 |
|
|
A' |
1 |
1 |
1 |
1 |
1 |
1 |
Rz |
x2 + y2, z2
|
E'
|
1
1
|
ω
ω*
|
ω*
ω
|
1
1
|
ω
ω*
|
ω*
ω
|
(x, y)
|
(x2 − y2, xy)
|
A'' |
1 |
1 |
1 |
−1 |
−1 |
−1 |
z |
|
E'' |
1
1
|
ω
ω*
|
ω*
ω
|
−1
−1
|
−ω
−ω*
|
−ω*
−ω
|
(Rx, Ry)
|
(xz, yz)
|
ω = e2πi/3
ζ = e2πi/6
|
Z7× ; Z9× ; Z14× ; Z18×
|
7
|
G71 = Z7
|
---
|
---
|
8 |
G81 = Z8 |
C8
|
E |
C8 |
C4 |
C83 |
C2 |
C85 |
C43 |
C87 |
|
!
|
A
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
Rz, z
|
x2 + y2, z2
|
B
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
|
|
E1
|
1
1
|
λ
λ*
|
i
−i
|
−λ*
−λ
|
−1
−1
|
−λ
−λ*
|
−i
i
|
λ*
λ
|
(Rx, Ry),
(x, y)
|
(xz, yz)
|
E2
|
1
1
|
i
−i
|
−1
−1
|
−i
i
|
1
1
|
i
−i
|
−1
−1
|
−i
i
|
|
(x2 − y2, xy)
|
E3
|
1
1
|
−λ
−λ*
|
i
−i
|
λ*
λ
|
−1
−1
|
λ
λ*
|
−i
i
|
−λ*
−λ
|
|
|
S8
|
E |
S8 |
C4 |
S83 |
i |
S85 |
C42 |
S87 |
|
|
A
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
Rz
|
x2 + y2, z2
|
B
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
z
|
|
E1
|
1
1
|
λ
λ*
|
i
−i
|
−λ*
−λ
|
−1
−1
|
−λ
−λ*
|
−i
i
|
λ*
λ
|
(x, y)
|
(xz, yz)
|
E2
|
1
1
|
i
−i
|
−1
−1
|
−i
i
|
1
1
|
i
−i
|
−1
−1
|
−i
i
|
|
(x2 − y2, xy)
|
E3
|
1
1
|
−λ*
−λ
|
−i
i
|
λ
λ*
|
−1
−1
|
λ*
λ
|
i
−i
|
−λ
−λ*
|
(Rx, Ry)
|
(xz, yz)
|
λ = e2πi/8 = (1+i)/√2
|
---
|
8 |
G82 = Z2 × Z4 |
C4h
|
E |
C4 |
C2 |
C43 |
i |
S43 |
σh |
S4 |
|
|
Ag
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
Rz
|
x2 + y2, z2
|
Bg
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
|
x2 − y2, xy
|
Eg
|
1
1
|
i
−i
|
−1
−1
|
−i
i
|
1
1
|
i
−i
|
−1
−1
|
−i
i
|
(Rx, Ry)
|
(xz, yz)
|
Au
|
1
|
1
|
1
|
1
|
−1
|
−1
|
−1
|
−1
|
z
|
|
Bu
|
1
|
−1
|
1
|
−1
|
−1
|
1
|
−1
|
1
|
|
|
Eu
|
1
1
|
i
−i
|
−1
−1
|
−i
i
|
−1
−1
|
−i
i
|
1
1
|
i
−i
|
(x, y)
|
|
|
Z15× ; Z16× ; Z20× ; Z30×
|
8 |
G83 = Dih4 |
D4
|
E |
2 C4 |
C2 |
2 C2' |
2 C2" |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
−1 |
−1 |
Rz, z |
|
B1 |
1 |
−1 |
1 |
1 |
−1 |
|
x2 − y2
|
B2 |
1 |
−1 |
1 |
−1 |
1 |
|
xy
|
E |
2 |
0 |
−2 |
0 |
0 |
(Rx, Ry), (x, y) |
(xz, yz)
|
C4v
|
E |
2 C4 |
C2 |
2 σv |
2 σd |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
z |
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1 |
1 |
−1 |
1 |
1 |
−1 |
|
x2 − y2
|
B2 |
1 |
−1 |
1 |
−1 |
1 |
|
xy
|
E |
2 |
0 |
−2 |
0 |
0 |
(Rx, Ry), (x, y) |
(xz, yz)
|
D2d
|
E |
2 S4 |
C2 |
2 C2' |
2 σd |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
|
x2, y2, z2
|
A2 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1 |
1 |
−1 |
1 |
1 |
−1 |
|
x2 − y2
|
B2 |
1 |
−1 |
1 |
−1 |
1 |
z |
xy
|
E |
2 |
0 |
−2 |
0 |
0 |
(Rx, Ry), (x, y) |
(xz, yz)
|
|
---
|
8 |
G84 = Dic2 = Q8 |
---
|
---
|
8 |
G85 = Z23 |
D2h
|
E |
C2 |
C2(x) |
C2(y) |
i |
σ(xy) |
σ(xz) |
σ(yz) |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2, y2, z2
|
B1g |
1 |
1 |
−1 |
−1 |
1 |
1 |
−1 |
−1 |
Rz |
xy
|
B2g |
1 |
−1 |
−1 |
1 |
1 |
−1 |
1 |
−1 |
Ry |
xz
|
B3g |
1 |
−1 |
1 |
−1 |
1 |
−1 |
−1 |
1 |
Rx |
yz
|
Au |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
|
|
B1u |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
1 |
1 |
z |
|
B2u |
1 |
−1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
y |
|
B3u |
1 |
−1 |
1 |
−1 |
−1 |
1 |
1 |
−1 |
x |
|
|
Z24×
|
9 |
G91 = Z9 |
---
|
---
|
9 |
G92 = Z32 |
---
|
---
|
10 |
G101 = Dih5 |
D5
|
E |
2 C5 |
2 C52 |
5 C2 |
|
|
A1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
−1 |
Rz, z |
|
E1 |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
(Rx, Ry), (x, y) |
(xz, yz)
|
E2 |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
|
(x2 − y2, xy)
|
C5v
|
E |
2 C5 |
2 C52 |
5 σv |
|
|
A1 |
1 |
1 |
1 |
1 |
z |
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
−1 |
Rz |
|
E1 |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
0 |
(Rx, Ry), (x, y) |
(xz, yz)
|
E2 |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
|
(x2 − y2, xy)
|
|
---
|
10 |
G102 = Z10 = Z5 × Z2 |
C5h
|
E |
C5 |
C52 |
C53 |
C54 |
σh |
S5 |
S57 |
S53 |
S59 |
|
|
A' |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
Rz |
x2 + y2, z2
|
E1'
|
1
1
|
η
η*
|
η2
η2*
|
η2*
η2
|
η*
η
|
1
1
|
η
η*
|
η2
η2*
|
η2*
η2
|
η*
η
|
(x, y)
|
|
E2'
|
1
1
|
η2
η2*
|
η*
η
|
η
η*
|
η2*
η2
|
1
1
|
η2
η2*
|
η*
η
|
η
η*
|
η2*
η2
|
|
(x2 - y2, xy)
|
A'' |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
z |
|
E1''
|
1
1
|
η
η*
|
η2
η2*
|
η2*
η2
|
η*
η
|
−1
−1
|
−η
−η*
|
−η2
−η2*
|
−η2*
−η2
|
−η*
−η
|
(Rx, Ry)
|
(xz, yz)
|
E2''
|
1
1
|
η2
η2*
|
η*
η
|
η
η*
|
η2*
η2
|
−1
−1
|
−η2
−η2*
|
−η*
−η
|
−η
−η*
|
−η2*
−η2
|
|
|
S10
|
E |
C5 |
C52 |
C53 |
C54 |
σh |
S5 |
S57 |
S53 |
S59 |
|
|
A' |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
Rz |
x2 + y2, z2
|
E1'
|
1
1
|
η
η*
|
η2
η2*
|
η2*
η2
|
η*
η
|
1
1
|
η
η*
|
η2
η2*
|
η2*
η2
|
η*
η
|
(x, y)
|
|
E2'
|
1
1
|
η2
η2*
|
η*
η
|
η
η*
|
η2*
η2
|
1
1
|
η2
η2*
|
η*
η
|
η
η*
|
η2*
η2
|
|
(x2 - y2, xy)
|
A'' |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1
|
z |
|
E1''
|
1
1
|
η
η*
|
η2
η2*
|
η2*
η2
|
η*
η
|
−1
−1
|
−η
-η*
|
−η2
−η2*
|
−η2*
−η2
|
−η*
−η
|
(Rx, Ry)
|
(xz, yz)
|
E2''
|
1
1
|
η2
η2*
|
η*
η
|
η
η*
|
η2*
η2
|
−1
−1
|
−η2
−η2*
|
−η*
−η
|
−η
−η*
|
−η2*
−η2
|
|
|
η = e2πi/5
|
C10 ;
Z10× ; Z22×
|
11
|
G111 = Z11
|
---
|
---
|
12
|
G121 = Dic3 = Q12
|
---
|
---
|
12
|
G122 = Z12 = Z4 × Z3
|
---
|
---
|
12 |
G123 = Alt4 |
T
|
E |
4 C3 |
4 C32 |
3 C2 |
|
|
A
|
1
|
1
|
1
|
1
|
|
x2 + y2 + z2
|
E
|
1
1
|
ω
ω*
|
ω*
ω
|
1
1
|
|
(2 z2 − x2 − y2,
x2 − y2)
|
T
|
3
|
0
|
0
|
−1
|
(Rx, Ry, Rz),
(x, y, z)
|
(xy, xz, yz)
|
ω = e2πi/3
|
|
12
|
G124 = Dih6 = Dih3 × Z2 |
D6
|
E |
2 C6 |
2 C3 |
C2 |
3 C2' |
3 C2" |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
1 |
−1 |
−1 |
Rz, z |
|
B1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
|
|
B2 |
1 |
−1 |
1 |
−1 |
−1 |
1 |
|
|
E1 |
2 |
1 |
−1 |
−2 |
0 |
0 |
(Rx, Ry), (x, y) |
(xz, yz)
|
E2 |
2 |
−1 |
−1 |
2 |
0 |
0 |
|
(x2 − y2, xy)
|
C6v
|
E |
2 C6 |
2 C3 |
C2 |
3 σv |
3 σd |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
1 |
z |
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
|
|
B2 |
1 |
−1 |
1 |
−1 |
−1 |
1 |
|
|
E1 |
2 |
1 |
−1 |
−2 |
0 |
0 |
(Rx, Ry), (x, y) |
(xz, yz)
|
E2 |
2 |
−1 |
−1 |
2 |
0 |
0 |
|
(x2 − y2, xy)
|
D3h
|
E |
2 C3 |
3 C2 ' |
σh |
2 S3 |
3 σv |
|
|
A1' |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A1'' |
1 |
1 |
1 |
−1 |
−1 |
−1 |
|
|
A2' |
1 |
1 |
−1 |
1 |
1 |
−1 |
Rz |
|
A2'' |
1 |
1 |
−1 |
−1 |
−1 |
1 |
z |
|
E' |
2 |
−1 |
0 |
2 |
−1 |
0 |
(x, y) |
(x2 − y2, xy)
|
E'' |
2 |
−1 |
0 |
−2 |
1 |
0 |
(Rx, Ry) |
(xz, yz)
|
D3d
|
E |
2 C3 |
3 C2 ' |
i |
2 S6 |
3 σd |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2g |
1 |
1 |
−1 |
1 |
1 |
−1 |
Rz |
|
A1u |
1 |
1 |
1 |
−1 |
−1 |
−1 |
|
|
A2u |
1 |
1 |
−1 |
−1 |
−1 |
1 |
z |
|
Eg |
2 |
−1 |
0 |
2 |
−1 |
0 |
(Rx, Ry) |
(x2 − y2, xy), (xz, yz)
|
Eu |
2 |
−1 |
0 |
−2 |
1 |
0 |
(x, y) |
|
|
|
12
|
G125 = Z6 × Z2 = Z3 × Z22 = Z3 × Dih2 |
C6h
|
E |
C6 |
C3 |
C2 |
C32 |
C65 |
i |
S35 |
S65 |
σh |
S6 |
S3 |
|
|
Ag
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
Rz
|
x2 + y2, z2
|
Bg
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
|
|
E1g
|
1
1
|
ζ
ζ*
|
−ζ*
−ζ
|
−1
−1
|
−ζ
−ζ*
|
ζ*
ζ
|
1
1
|
ζ
ζ*
|
−ζ*
−ζ
|
−1
−1
|
−ζ
−ζ*
|
ζ*
ζ
|
(Rx, Ry)
|
(xz, yz)
|
E2g
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
|
(x2 − y2, xy)
|
Au
|
1
|
1
|
1
|
1
|
1
|
1
|
−1
|
−1
|
−1
|
−1
|
−1
|
−1
|
z
|
|
Bu
|
1
|
−1
|
1
|
−1
|
1
|
−1
|
−1
|
1
|
−1
|
1
|
−1
|
1
|
|
|
E1u
|
1
1
|
ζ
ζ*
|
−ζ*
−ζ
|
−1
−1
|
−ζ
−ζ*
|
ζ*
ζ
|
−1
−1
|
−ζ
−ζ*
|
ζ*
ζ
|
1
1
|
ζ
ζ*
|
−ζ*
−ζ
|
(x, y)
|
|
E2u
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
1
1
|
−ζ*
−ζ
|
−ζ
−ζ*
|
−1
−1
|
ζ*
ζ
|
ζ
ζ*
|
−1
−1
|
ζ*
ζ
|
ζ
ζ*
|
|
|
ζ = e2πi/6
|
|
13
|
G131 = Z13
|
---
|
---
|
14
|
G141 = Dih7
|
---
|
---
|
14
|
G142 = Z14 = Z7 × Z2
|
---
|
---
|
15
|
G151 = Z15 = Z5 × Z3
|
---
|
---
|
16
|
G165 = Z8 × Z2 |
---
|
C8h
|
16 |
G167 = Dih8
|
D4d
|
E |
2 S8 |
2 C4 |
2 S83 |
C2 |
4 C2' |
4 σd |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1 |
1 |
−1 |
1 |
−1 |
1 |
1 |
−1 |
|
|
B2 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
z |
|
E1 |
2 |
√2 |
0 |
−√2 |
−2 |
0 |
0 |
(x, y) |
|
E2 |
2 |
0 |
−2 |
0 |
2 |
0 |
0 |
|
(x2 − y2, xy)
|
E3 |
2 |
−√2 |
0 |
√2 |
−2 |
0 |
0 |
(Rx, Ry) |
(xz, yz)
|
|
D8 ; C8v
|
16
|
G1611 = Dih4 × Z2 |
D4h
|
E |
2 C4 |
C2 |
2 C2' |
2 C2"
|
i |
2 S4 |
σh |
2 σv |
2 σd |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2g |
1 |
1 |
1 |
−1 |
−1 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1g |
1 |
−1 |
1 |
1 |
−1 |
1 |
−1 |
1 |
1 |
−1 |
|
x2 − y2
|
B2g |
1 |
−1 |
1 |
−1 |
1 |
1 |
−1 |
1 |
−1 |
1 |
|
xy
|
Eg |
2 |
0 |
−2 |
0 |
0 |
2 |
0 |
−2 |
0 |
0 |
(Rx, Ry) |
(xz, yz)
|
A1u |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
|
|
A2u |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
1 |
1 |
z |
|
B1u |
1 |
−1 |
1 |
1 |
−1 |
−1 |
1 |
−1 |
−1 |
1 |
|
|
B2u |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
|
|
Eu |
2 |
0 |
−2 |
0 |
0 |
−2 |
0 |
2 |
0 |
0 |
(x, y) |
|
|
|
20
|
G205 = Z10 × Z2 = Z5 × Z22 = Z5 × Dih2 |
---
|
C10h
|
20
|
G204 = Dih10 = Dih5 × Z2
|
D5h
|
E |
2 C5 |
2 C52 |
5 C2 |
σh |
2 S5 |
2 S53 |
5 σv |
|
|
A1' |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2' |
1 |
1 |
1 |
−1 |
1 |
1 |
1 |
−1 |
Rz |
|
E1' |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
0 |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
0 |
(x, y) |
|
E2' |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
|
(x2 − y2, xy)
|
A1'' |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
|
|
A2'' |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
1 |
z |
|
E1'' |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
0 |
−2 |
−2 cos 2π/5 |
−2 cos 4π/5 |
0 |
(Rx, Ry) |
(xz, yz)
|
E2'' |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
−2 |
−2 cos 4π/5 |
−2 cos 2π/5 |
0 |
|
|
D5d
|
E |
2 C5 |
2 C52 |
5 C2 |
i |
2 S10 |
2 S103 |
5 σd |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2g |
1 |
1 |
1 |
−1 |
1 |
1 |
1 |
−1 |
Rz |
|
E1g |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
0 |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
(Rx, Ry) |
(xz, yz)
|
E2g |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
0 |
|
(x2 − y2, xy)
|
A1u |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
|
|
A2u |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
1 |
z |
|
E1u |
2 |
2 cos 2π/5 |
2 cos 4π/5 |
0 |
−2 |
−2 cos 4π/5 |
−2 cos 2π/5 |
0 |
(x, y) |
|
E2u |
2 |
2 cos 4π/5 |
2 cos 2π/5 |
0 |
−2 |
−2 cos 2π/5 |
−2 cos 4π/5 |
0 |
|
|
|
D10 ; C10v
|
24 |
G2412 = Sym4 |
Td
|
E |
8 C3 |
3 C2 |
6 S4 |
6 σd |
|
|
A1
|
1
|
1
|
1
|
1
|
1
|
|
x2 + y2 + z2
|
A2
|
1
|
1
|
1
|
−1
|
−1
|
|
|
E
|
2
|
−1
|
2
|
0
|
0
|
|
(2 z2 − x2 − y2, x2 − y2)
|
T1
|
3
|
0
|
−1
|
1
|
−1
|
(Rx, Ry, Rz)
|
|
T2
|
3
|
0
|
−1
|
−1
|
1
|
(x, y, z)
|
(xy, xz, yz)
|
O
|
E
|
6 C4 |
3 C2(=C42) |
8 C3 |
6 C2 ' |
|
|
A1
|
1
|
1
|
1
|
1
|
1
|
|
x2 + y2 + z2
|
A2
|
1
|
−1
|
1
|
1
|
−1
|
|
|
E
|
2
|
0
|
2
|
−1
|
0
|
|
(2 z2 − x2 − y2,
x2 − y2)
|
T1
|
3
|
1
|
−1
|
0
|
−1
|
(Rx, Ry, Rz),
(x, y, z)
|
|
T2
|
3
|
−1
|
−1
|
0
|
1
|
|
(xy, xz, yz)
|
|
|
24
|
G2413 = Alt4 × Z2 |
Th
|
E |
4 C3 |
4 C32 |
3 C2 |
i |
4 S6 |
4 S65 |
3 σh |
|
|
Ag
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
|
x2 + y2 + z2
|
Au
|
1
|
1
|
1
|
1
|
−1
|
−1
|
−1
|
−1
|
|
|
Eg
|
1
1
|
ω
ω*
|
ω*
ω
|
1
1
|
1
1
|
ω
ω*
|
ω*
ω
|
1
1
|
|
(2 z2 − x2 − y2,
x2 − y2)
|
Eu
|
1
1
|
ω
ω*
|
ω*
ω
|
1
1
|
−1
−1
|
−ω
−ω*
|
−ω*
−ω
|
−1
−1
|
|
|
Tg
|
3
|
0
|
0
|
−1
|
3
|
0
|
0
|
−1
|
(Rx, Ry, Rz)
|
(xy, xz, yz)
|
Tu
|
3
|
0
|
0
|
−1
|
−3
|
0
|
0
|
1
|
(x, y, z)
|
|
ω=e2πi/3
|
|
24
|
G2414 = Dih6 × Z2 = Dih3 × Z22 |
D6h
|
E |
2 C6 |
2 C3 |
C2 |
3 C2' |
3 C2"
|
i |
2 S3 |
2 S6 |
σh |
3 σd |
3 σv |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2g |
1 |
1 |
1 |
1 |
−1 |
−1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1g |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
|
|
B2g |
1 |
−1 |
1 |
−1 |
−1 |
1 |
1 |
−1 |
1 |
−1 |
−1 |
1 |
|
|
E1g |
2 |
1 |
−1 |
−2 |
0 |
0 |
2 |
1 |
−1 |
−2 |
0 |
0 |
(Rx, Ry) |
(xz, yz)
|
E2g |
2 |
−1 |
−1 |
2 |
0 |
0 |
2 |
−1 |
−1 |
2 |
0 |
0 |
|
(x2 − y2, xy)
|
A1u |
1 |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
−1 |
|
|
A2u |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
−1 |
1 |
1 |
z |
|
B1u |
1 |
−1 |
1 |
−1 |
1 |
−1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
|
|
B2u |
1 |
−1 |
1 |
−1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
1 |
−1 |
|
|
E1u |
2 |
1 |
−1 |
−2 |
0 |
0 |
−2 |
−1 |
1 |
2 |
0 |
0 |
(x, y) |
|
E2u |
2 |
−1 |
−1 |
2 |
0 |
0 |
−2 |
1 |
1 |
−2 |
0 |
0 |
|
|
D6d
|
E |
2 S12 |
2 C6 |
2 S4 |
2 C3 |
2 S125 |
C2 |
6 C2' |
6 σd |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
1 |
−1 |
|
|
B2 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
−1 |
1 |
z |
|
E1 |
2 |
√3 |
1 |
0 |
−1 |
−√3 |
−2 |
0 |
0 |
(x, y) |
|
E2 |
2 |
1 |
−1 |
−2 |
−1 |
1 |
2 |
0 |
0 |
|
(x2 − y2, xy)
|
E3 |
2 |
0 |
−2 |
0 |
2 |
0 |
−2 |
0 |
0 |
|
|
E4 |
2 |
−1 |
−1 |
2 |
−1 |
−1 |
2 |
0 |
0 |
|
|
E5 |
2 |
−√3 |
1 |
0 |
−1 |
√3 |
−2 |
0 |
0 |
(Rx, Ry) |
(xz, yz)
|
|
|
32
|
Dih8 × Z2 |
D8h
|
E |
2 C8 |
2 C83 |
2 C4 |
C2 |
4 C2' |
4 C2"
|
i |
2 S83 |
2 S8 |
2 S4 |
σh |
4 σd |
4 σv |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2, z2
|
A2g |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
Rz |
|
B1g |
1 |
−1 |
−1 |
1 |
1 |
1 |
−1 |
1 |
−1 |
−1 |
1 |
1 |
1 |
−1 |
|
|
B2g |
1 |
−1 |
−1 |
1 |
1 |
−1 |
1 |
1 |
−1 |
−1 |
1 |
1 |
−1 |
1 |
|
|
E1g |
2 |
√2 |
−√2 |
0 |
−2 |
0 |
0 |
2 |
√2 |
−√2 |
0 |
−2 |
0 |
0 |
(Rx, Ry) |
(xz, yz)
|
E2g |
2 |
0 |
0 |
−2 |
2 |
0 |
0 |
2 |
0 |
0 |
−2 |
2 |
0 |
0 |
|
(x2 − y2, xy)
|
E3g |
2 |
−√2 |
√2 |
0 |
−2 |
0 |
0 |
2 |
−√2 |
√2 |
0 |
−2 |
0 |
0 |
|
|
A1u |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
−1 |
−1 |
|
|
A2u |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
−1 |
−1 |
1 |
1 |
z |
|
B1u |
1 |
−1 |
−1 |
1 |
1 |
1 |
−1 |
−1 |
1 |
1 |
−1 |
−1 |
−1 |
1 |
|
|
B2u |
1 |
−1 |
−1 |
1 |
1 |
−1 |
1 |
−1 |
1 |
1 |
−1 |
−1 |
1 |
−1 |
|
|
E1u |
2 |
√2 |
−√2 |
0 |
−2 |
0 |
0 |
−2 |
−√2 |
√2 |
0 |
2 |
0 |
0 |
(x, y) |
|
E2u |
2 |
0 |
0 |
−2 |
2 |
0 |
0 |
−2 |
0 |
0 |
2 |
−2 |
0 |
0 |
|
|
E3u |
2 |
−√2 |
√2 |
0 |
−2 |
0 |
0 |
−2 |
√2 |
−√2 |
0 |
2 |
0 |
0 |
|
|
|
|
48 |
Sym4 × Z2 |
Oh
|
E |
8 C3 |
6 C2 ' |
6 C4 |
3 C2(=C42) |
i |
6 S4 |
8 S6 |
3 σh |
6 σd |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2 + z2
|
A2g |
1 |
1 |
−1 |
−1 |
1 |
1 |
−1 |
1 |
1 |
−1 |
|
|
Eg |
2 |
−1 |
0 |
0 |
2 |
2 |
0 |
−1 |
2 |
0 |
|
(2 z2 − x2 − y2,
x2 − y2)
|
T1g |
3 |
0 |
−1 |
1 |
−1 |
3 |
1 |
0 |
−1 |
−1 |
(Rx, Ry, Rz) |
|
T2g |
3 |
0 |
1 |
−1 |
−1 |
3 |
−1 |
0 |
−1 |
1 |
|
(xy, xz, yz)
|
A1u |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
|
|
A2u |
1 |
1 |
−1 |
−1 |
1 |
−1 |
1 |
−1 |
−1 |
1 |
|
|
Eu |
2 |
−1 |
0 |
0 |
2 |
−2 |
0 |
1 |
−2 |
0 |
|
|
T1u |
3 |
0 |
−1 |
1 |
−1 |
−3 |
−1 |
0 |
1 |
1 |
(x, y, z) |
|
T2u |
3 |
0 |
1 |
−1 |
−1 |
−3 |
1 |
0 |
1 |
−1 |
|
|
|
|
60 |
Alt5 |
I
|
E |
12 C5 |
12 C52 |
20 C3 |
15 C2 |
|
|
A |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2 + z2
|
T1 |
3 |
2 cos π/5 = (1+√5)/2 |
2 cos 3π/5 = (1−√5)/2 |
0 |
−1 |
(Rx, Ry, Rz),
(x, y, z)||
|
T2 |
3 |
2 cos 3π/5 = (1−√5)/2 |
2 cos π/5 = (1+√5)/2 |
0 |
−1 |
|
|
G |
4 |
−1 |
−1 |
1 |
0 |
|
|
H |
5 |
0 |
0 |
−1 |
1 |
|
(2 z2 − x2 − y2,
x2 − y2, xy, xz, yz)
|
|
|
120 |
Alt5 × Z2 |
Ih
|
E |
12 C5 |
12 C52 |
20 C3 |
15 C2 |
i |
12 S10 |
12 S103 |
20 S6 |
15 σ |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
x2 + y2 + z2
|
T1g |
3 |
2 cos π/5 = (1+√5)/2 |
2 cos 3π/5 = (1−√5)/2 |
0 |
−1 |
3 |
2 cos 3π/5 = (1−√5)/2 |
2 cos π/5 = (1+√5)/2 |
0 |
−1 |
(Rx, Ry, Rz) |
|
T2g |
3 |
2 cos 3π/5 = (1−√5)/2 |
2 cos π/5 = (1+√5)/2 |
0 |
−1 |
3 |
2 cos π/5 = (1+√5)/2 |
2 cos 3π/5 = (1−√5)/2 |
0 |
−1 |
|
|
Gg |
4 |
−1 |
−1 |
1 |
0 |
4 |
−1 |
−1 |
1 |
0 |
|
|
Hg |
5 |
0 |
0 |
−1 |
1 |
5 |
0 |
0 |
−1 |
1 |
|
(2 z2 − x2 − y2,
x2 − y2, xy, xz, yz)
|
Au |
1 |
1 |
1 |
1 |
1 |
−1 |
−1 |
−1 |
−1 |
−1 |
|
|
T1u |
3 |
2 cos π/5 = (1+√5)/2 |
2 cos 3π/5 = (1−√5)/2 |
0 |
−1 |
−3 |
−2 cos 3π/5 = −(1−√5)/2 |
−2 cos π/5 = −(1+√5)/2 |
0 |
1 |
(x, y, z) |
|
T2u |
3 |
2 cos 3π/5 = (1−√5)/2 |
2 cos π/5 = (1+√5)/2 |
0 |
−1 |
−3 |
−2 cos π/5 = −(1+√5)/2 |
−2 cos 3π/5 = −(1−√5)/2 |
0 |
1 |
|
|
Gu |
4 |
−1 |
−1 |
1 |
0 |
−4 |
1 |
1 |
−1 |
0 |
|
|
Hu |
5 |
0 |
0 |
−1 |
1 |
−5 |
0 |
0 |
1 |
−1 |
|
|
|
|
120
|
Sym5
|
|
|
∞ |
O(2) |
C∞v
|
E |
2 C∞Φ |
... |
∞ σv |
|
|
A1=Σ+ |
1 |
1 |
... |
1 |
z |
x2 + y2, z2
|
A2=Σ− |
1 |
1 |
... |
−1 |
Rz |
|
E1=Π |
2 |
2 cos Φ |
... |
(x, y), (Rx, Ry) |
(xz, yz)
|
E2=Δ |
2 |
2 cos 2Φ |
... |
0 |
|
(x2 - y2, xy)
|
E3=Φ |
2 |
2 cos 3Φ |
... |
0 |
|
|
... |
... |
... |
... |
... |
|
|
|
|
∞ |
Z2×O(2) |
D∞h
|
E |
2 C∞Φ |
... |
∞ σv |
i |
2 S∞Φ |
... |
∞ C2 |
|
|
Σg+ |
1 |
1 |
... |
1 |
1 |
1 |
... |
1 |
|
x2 + y2, z2
|
Σg− |
1 |
1 |
... |
−1 |
1 |
1 |
... |
−1 |
Rz |
|
Πg |
2 |
2 cos Φ |
... |
0 |
2 |
−2 cos Φ |
... |
0 |
(Rx, Ry) |
(xz, yz)
|
Δg |
2 |
2 cos 2Φ |
... |
0 |
2 |
2 cos 2Φ |
... |
0 |
|
(x2 − y2, xy)
|
... |
... |
... |
... |
... |
... |
... |
... |
... |
|
|
Σu+ |
1 |
1 |
... |
1 |
−1 |
−1 |
... |
−1 |
z |
|
Σu− |
1 |
1 |
... |
−1 |
−1 |
−1 |
... |
1 |
|
|
Πu |
2 |
2 cos Φ |
... |
0 |
−2 |
2 cos Φ |
... |
0 |
(x, y) |
|
Δu |
2 |
2 cos 2Φ |
... |
0 |
−2 |
−2 cos 2Φ |
... |
0 |
|
|
... |
... |
... |
... |
... |
... |
... |
... |
... |
|
|
|
|
∞∞ |
SO(3) |
K
K
|
E |
∞ C∞Φ |
|
|
Σ |
1 |
1 |
|
|
Γl |
1 |
|
|
|
|
|
∞∞ |
O(3) |
Kh |
|