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 | |