Point Group 1: C1=1
E
A1 1 all
Point Group 2: Ci=S2=-1
E i
Ag 1 1 Rx, Ry, Rz, x2, y2, z2, xy, xz, yz
Au 1 -1 x, y, z
Point Group 3: Cs=C1h
E sigmah
A' 1 1 x, y, Rz, x2, y2, z2, xy
A'' 1 -1 z, Rz, Ry, yz, xz
Point Group 4: C2=2
E C2
A 1 1 z, Rz, x2, y2, z2, xy
B 1 -1 x, y, Rx, Ry, yz, xz
Point Group 5: C2v=mm2
E C2 sigmav sigmav'
A1 1 1 1 1 z, x2, y2, z2
A2 1 1 -1 -1 Rz, xy
B1 1 -1 1 -1 x, Ry, xz
B2 1 -1 -1 1 y, Rx, yz
Point Group 6: C2h=2/m
E C2 i sigmah
Ag 1 1 1 1 Rz, x2, y2, z2, xy
Bg 1 -1 1 -1 Rx, Ry, xz, yz
Au 1 1 -1 -1 z
Bu 1 -1 -1 1 x, y
Point Group 7: D2=222
E C4 C2 C43
A 1 1 1 1 x2, y2, z2
B1 1 1 -1 -1 z, Rz, xy
B1 1 -1 1 -1 y, Ry, xz
B2 1 -1 -1 1 x, Rx, yz
Point Group 8: D2h=2/mmm
E C2 C2' C2'' i sigma(xy) sigma'(yz) sigma''(xz)
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
Point Group 9: D2d=-42m
E 2S4 C2 2C2' 2sigmad
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 (x,y), (Rx,Ry), (xz,yz)
Point Group 10: C3=3
E C3
A 1 1 z, Rz, x2+y2, z2
E {1,1} {e,e*} (x,y), (Rx,Ry), (x2-y2,xy), (xz,yz)
Point Group 11: C3v=3m
E 2C3 3sigmav
A1 1 1 1 x2+y2, z2
A2 1 1 -1 Rz
E 2 -1 0 (x,y), (Rx, Ry), (x2-y2,xy),(yz,zx)
Point Group 12: C3h=-6
E C3 C32 sigmah
A' 1 1 1 1 Rz, x2+y2, z2
E' {1,1} {e,e*} {e*,e} {1,1} (x,y), (x2-y2,xy)
A'' 1 1 1 -1 Z
E'' {1,1} {e,e*} {e*,e} {-1,-1} (Rx,Ry), (xz, yz)
Point Group 13: S6=-3
E C3 C32 i
Ag 1 1 1 1 Rz, x2+y2, z2
Eg {1,1} {e,e*} {e*,e} {1,1} (Rx,Ry), (x2-y2,xy), (xz,yz)
Au 1 1 1 -1 Z
Eu {1,1} {e,e*} {e*,e} {-1,-1} (x,y)
Point Group 14: D3=32
E 2C3 3C2
A1 1 1 1 x2+y2, z2
A2 1 1 -1 Rz
E 2 -1 0 (x,y), (Rx,Ry),(x2-y2,xy),(yz,zx)
Point Group 15: D3d=-3m
E 2C3 3C2 i 2S6 3sigmad
A1g 1 1 1 1 1 1 x2+y2, z2
A2g 1 1 -1 1 1 -1 Rz
Eg 2 -1 0 2 -1 0 (Rx,Ry), (x2-y2,xy), (xz,yz)
A1u 1 1 1 -1 -1 -1
A2u 1 1 -1 -1 -1 1 z
Eu 2 -1 0 -2 1 0 (x,y)
Point Group 16: D3h=-6m2
E 2C3 3C2 sigmah 2S3 3sigmav
A1' 1 1 1 1 1 1 x2+y2, z2
A2' 1 1 -1 1 1 -1 Rz
E' 2 -1 0 2 -1 0 (x,y),(x2-y2,xy)
A1' ' 1 1 1 -1 -1 -1
A2' ' 1 1 -1 -1 -1 1 z
E' ' 2 -1 0 -2 1 0 (Rx,Ry), (yz,zx)
Point Group 17: C4=4
E C4 C2
A 1 1 1 z, Rz, x2+y2, z2
B 1 -1 1 x2-y2, xy
E {1,1} {i,-i} {-1,1} (x,y), (Rx,Ry), (xz,yz)
Point Group 18: C4h=4/m
E C4 C42 C43 i S43
Ag 1 1 1 1 1 1 Rz,x2+y2,z2
Bg 1 -1 1 -1 1 -1 x2-y2,xy
Eg {1,1} {i,-i} {-1,-1} {-i,i} {1,1} {i,-i} (Rx,Ry),(yz,zx)
Au 1 1 1 1 -1 -1 Z
Bu 1 -1 1 -1 -1 1
Eu {1,1} {i,-i} {-1,-1} {-i,i} {-1,-1} {-i,i} (x,y)
Point Group 19: C4v=4mm
E 2C4 C2 2sigmav 2sigmad
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 (x,y), (Rx,Ry), (xz, yz)
Point Group 20: D4=422
E 2C4 C2 2C2' 2C2''
A1 1 1 1 1 1 x2+y2, z2
A2 1 1 1 -1 -1 z, Rz
B1 1 -1 1 1 -1 x2-y2
B2 1 -1 1 -1 1 xy
E 2 0 -2 0 0 (x,y), (Rx,Ry), (xz,yz)
Point Group 21: D4h=4/mmm
E 2C4 C2 2C2' 2C2'' i 2S4 sigmah 2sigmav 2sigmad
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 (x,y), (x2-y2,xy)
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)
Point Group 22: D4d
E 2S8 2C4 2S83 2C2 4C2' 4sigmad
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)
Point Group 23: S4=-4
E S4 C2
A 1 1 1 Rz, x2+y2, z2
B 1 -1 1 z, x2-y2, xy
E {1,1} {i,-i} {-1,-1} (x,y), (Rx,Ry), (xz,yz)
Point Group 24: C6=6
E C6 C62 C63
A 1 1 1 1 z, Rz,x2+y2, z2
B 1 -1 1 -1
E1 {1,1} {e, e*} {-e*, -e} {-1, -1} (x,y), (Rx,Ry), (xz, yz)
E2 {1,1} {-e*, -e} {-e, -e*} {1,1} (x2-y2,xy)
Point Group 25: C6h=6/m
E C6 C62 C63 C64 C65 i S35
Ag 1 1 1 1 1 1 1 1 Rz, x2+y2, z2
Bg 1 -1 1 -1 1 -1 1 -1
E1g {1,1} {e,e*} {-e*,-e} {-1,-1} {-e,-e*} {e*,e} {1,1} {e,e*} (Rx,Ry),(yz,zx)
E2g {1,1} {-e*,-e} {-e,-e*} {1,1} {-e*,-e} {-e,-e*} {1,1} {-e*,-e} (x2-y2,xy)
Au 1 1 1 1 1 1 -1 -1 z
Bu 1 -1 1 -1 1 -1 -1 1
E1u {1,1} {e,e*} {-e*,-e} {-1,-1} {-e,-e*} {e*,e} {-1,-1} {-e,-e*} (x,y)
E2u v {-e*,-e} {-e,-e*} {1,1} {-e*,-e} {-e,-e*} {-1,-1} {e*,e}
Point Group 26: C6v=6mm
E 2C6 2C3 C2 3sigmah 3sigmad
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 (x,y), (Rx,Ry), (xz, yz)
E2 2 -1 -1 2 0 0 (x2-y2,xy)
Point Group 27: D6=622
E 2C6 2C62 C63 3C2' 3C2''
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 (x,y), (Rx,Ry), (xz, yz)
E2 2 -1 -1 2 0 0 (x2-y2,xy)
Point Group 28: D6h=6/mmm
E 2C6 2C3 C2 3C2' 2C2'' i 2S2 2S6 sigmah 3sigmad 3sigmav
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
Point Group 29: T=23
E 4C3 4C32
A 1 1 1 x2+y2+z2
E {1,1} {e,e*} {e*,e} (x2-y2,2z2-x2-y2)
T 3 0 0 (Rx,Ry,Rz),(x,y,z),(xy,xz,yz)
Point Group 30: Th=m3
E 4C3 4C32 3C2 i 4S6
Ag 1 1 1 1 1 1 x2+y2+z2
Eg {1,1} {e,e*} {e*,e} {1,1} {1,1} {e,e*} (x2-y2,2z2-x2-y2)
Tg 3 0 0 1 0 0 (Rx,Ry,Rz),(xy,xz,yz)
Au 1 1 1 1 -1 -1
Eu {1,1} {e,e*} {e*,e} {-e,-e*} {-1,-1} {-e,-e*}
Tu 3 0 0 0 -1 0 (x,y,z)
Point Group 31: Td=-43m
E 8C3 3C2 6S43 6sigmad
A1 1 1 1 1 1 x2+y2+z2
A2 1 1 1 -1 -1
E 2 -1 2 0 0 (x2-y2,2z2-x2-y2)
T1 3 0 -1 1 -1 (Rx,Ry,Rz)
T2 3 0 -1 -1 1 (x,y,z), (xy,xz,yz)
Point Group 32: O=432
E 8C3 3C2 6C4 6C2
A1 1 1 1 1 1 x2+y2+z2
A2 1 1 1 -1 -1
E 2 -1 2 0 0 (x2-y2,2z2-x2-y2)
T1 3 0 -1 1 -1 (Rx,Ry,Rz)
T2 3 0 -1 -1 1 (xy,xz,yz)
Point Group 33: Oh=m-3m
E 8C3 3C2 6C2' 6C4 i 8S6 3sigmah 6S4 6sigmad
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 2 0 0 2 -1 2 0 0 (x2-y2,2z2-x2-y2)
T1g 3 0 -1 1 -1 3 0 -1 1 -1 (Rx,Ry,Rz)
T2g 3 0 -1 -1 1 3 0 -1 -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 2 0 0 -2 1 -2 0 0
T1g 3 0 -1 1 -1 -3 0 1 -1 1 (x,y,z)
T2g 3 0 -1 -1 1 -3 0 1 1 -1