Utente:Giacomo Lanza/Sandbox

Lista dei gruppi finiti

La seguente tabella riporta:

• tutti i gruppi canonici di ordine ≤ 15;
• tutti i gruppi simmetrici e alternanti su ≤ 5 elementi;
• tutti i gruppi di simmetria che possono descrivere la simmetria di molecole o cristalli;
• altri gruppi isomorfi ai precedenti.

L'elenco dei gruppi di ordine ≥ 16 è necessariamente incompleto.

La prima colonna riporta l'ordine (numero di elementi) dei gruppi.

La seconda colonna elenca i gruppi canonici (gruppi astratti, unici a meno di isomorfismi).

• G_oⁱ = numerazione sistematica della Small Groups library: o = ordine del gruppo, i = numero progressivo.
• Z_n = Z/nZ = gruppo ciclico di ordine n.
• Sym_n = gruppo simmetrico di n elementi, avente ordine n! .
• Alt_n = gruppo alternante di n elementi, avente ordine n!/2 (per n≥2).
• Dih_n = gruppo diedrale a simmetria n-aria, avente ordine 2n.
• Dic_n = gruppo diciclico a simmetria n-aria, avente ordine 4n

Viene riportato anche il diagramma ciclico.

La terza colonna descrive i gruppi di simmetria in 3D che appaiono che possono essere rappresentati in simmetria molecolare o cristallina.

• C_n = simmetria rotazionale; S_2n = simmetria di rotoriflessione; C_nh = simmetria di riflessione; C_nv = simmetria piramidale;
• D_n = simmetria diedrale; D_nh = simmetria antiprismatica; D_nd = simmetria prismatica;
• T = simmetria tetraedrica chirale; T_d = simmetria tetraedrica completa; T_h = simmetria piritoedrica;
• O = simmetria ottaedrica chirale; O_h = simmetria ottaedrica completa;
• I = simmetria icosaedrica chirale; I_h = simmetria icosaedrica completa.

Per ogni gruppo vengono mostrate anche le rappresentazioni irriducibili mediante i simboli di Mulliken:

• A / B = rappresentazione monodimensionale, simmetrica / antisimmetrica rispetto all'asse principale c_n;
• E / T / G / H = rappresentazione bi/tri/quadri/quintidimensionale;
• ...₁ / ...₂ , ...₃ = rappresentazione simmetrica / antisimmetrica rispetto a un asse secondario c₂';
• ..._g / ..._u = rappresentazione simmetrica / antisimmetrica rispetto al centro di inversione i;
• ...' / ...'' = rappresentazione simmetrica / antisimmetrica rispetto al piano di simmetria orizzontale σ_h.

Infine vengono assegnate anche le classi di simmetria delle coordinate:

• x, y, z = coordinate lineari (--> orbitali p);
• R_x, R_y, R_z = rotazioni;
• x², y², z², xy, xz, yz = combinazioni quadratiche (--> orbitali d).

Nella quarta colonna vengono elencati altri gruppi isomorfi ai gruppi astratti riportati; per esempio gruppi di simmetria non rappresentabili in strutture molecolari o cristalline, o gruppi moltiplicativi:

• Z_m^× = Sistema ridotto di residui modulo m, avente ordine φ(m)

Ordine	Gruppo canonico	Gruppi di simmetria molecolare	Altri gruppi
1	G11 = Z1	E C1 A 1
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 ${\displaystyle \sin {\frac {(2\ell +1)\Phi }{2}} \over \sin {\frac {\Phi }{2}}}$
∞∞	O(3)	Kh

Collegamenti

List of character tables for chemically important 3D point groups

List of small groups

Point groups in three dimensions

Nontotient

Collegamenti esterni

Point Group Symmetry Character Tables

Character Tables for Point Groups used in Chemistry

Group names