2.2: Understanding Character Tables of Symmetry Groups
- Page ID
- 83479
Nonaxial Groups
These groups are characterized by a lack of a proper rotation axis,
\(C_1\) | E |
A | 1 |
\(C_s\) | E | σh | ||
A' | 1 | 1 | x, y, Rz | x2, y2, z2, xy |
A" | 1 | -1 | z, Rx, Ry | yz, xz |
\(C_i\) | E | i | ||
Au | 1 | 1 | Rx, Ry, Rz | x2, y2, z2, xy, yz, zx |
Ag | 1 | -1 | x,y,z |
Cyclic \(C_n\) Groups
These groups are characterized by an n-fold proper rotation axis \(C_n\).
C2 | E | C2 | ||
A | 1 | 1 | z, Rz | x2, y2, z2, xy |
B | 1 | -1 | x, y, Rx, Ry | yz,xz |
C3 | E | C3 | C32 | ε=exp (2πi/3) | |||||||
A | 1 | 1 | 1 | z, Rz | x2+y2, z2 | ||||||
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(x,y) (Rx,Ry) | (x2-y2, xy), (xz, yz) |
C4 | E | C4 | C2 | C43 | ||||||||||
A | 1 | 1 | 1 | 1 | z, Rz | x2+y2, z2 | ||||||||
B | 1 | -1 | 1 | -1 | x2-y2, xy | |||||||||
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(x,y) (Rx,Ry) | (xz, yz) |
C5 | E | C5 | C52 | C53 | C54 | ||||||||||||
A | 1 | 1 | 1 | 1 | 1 | Z, Rz | x2+y2, z2 | ||||||||||
E1 |
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(x, y)(Rx,Ry) | (xz, yz) | ||||||||||
E2 |
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(x2-y2, xy) |
C6 | E | C6 | C3 | C2 | C32 | C65 | ε=exp (2πi/6) | |||||||||||||
A | 1 | 1 | 1 | 1 | 1 | 1 | z, Rz | x2+y2, z2 | ||||||||||||
B | 1 | -1 | 1 | -1 | 1 | -1 | ||||||||||||||
E1 |
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(Rx,Ry) (x,y) | (xz, yz) | ||||||||||||
E2 |
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(x2-y2, xy) |
C7 | E | C7 | C72 | C73 | C74 | C75 | C76 | ε=exp (2πi/7) | |||||||||||||||
A | 1 | 1 | 1 | 1 | 1 | 1 | 1 | z, Rz | x2+y2, z2 | ||||||||||||||
E1 |
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(Rx,Ry) (x,y) | (xz, yz) | ||||||||||||||
E2 |
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E3 |
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C8 | E | C8 | C4 | C83 | C2 | C85 | C43 | C87 | ε=exp (2πi/8) | |||||||||||||||||
A | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | z, Rz | x2+y2, z2 | ||||||||||||||||
B | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | ||||||||||||||||||
E1 |
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(Rx,Ry) (x,y) | (xz, yz) | ||||||||||||||||
E2 |
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E3 |
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Reflection \(C_{nh}\) Groups
These groups are characterized by an n-fold proper rotation axis \(C_n\) and a mirror plane \(\sigma_h\) normal to \(C_n\).
\(C_{2h}\) | E | C2 | i | σh | ||
Ag | 1 | 1 | 1 | 1 | Rz | x2, y2, z2 |
Bg | 1 | -1 | 1 | -1 | Rx, Ry | xz, yz |
Au | 1 | 1 | -1 | -1 | z | |
Bu | 1 | -1 | -1 | 1 | x,y |
\(C_{3h}\) | E | C3 | C32 | σh | S3 | S35 | ε=exp (2πi/3) | |||||||||||||
A' | 1 | 1 | 1 | 1 | 1 | 1 | Rz | x2+y2, z2 | ||||||||||||
E' |
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(x,y) | (x2-y2, xy) | ||||||||||||
A" | 1 | 1 | 1 | -1 | -1 | -1 | z | |||||||||||||
E" |
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(Rx, Ry) | (xz, yz) |
\(C_{4h}\) | 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 |
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(Rx, Ry) | (xz, yz) | ||||||||||||||||
Au | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | z | |||||||||||||||||
Bu | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | ||||||||||||||||||
Eu |
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\(C_{5h}\) | E | C5 | C52 | C53 | C54 | σh | S5 | S57 | S53 | S59 | ε=exp (2πi/5) | |||||||||||||||||||||
A' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Rz | x2+y2, z2 | ||||||||||||||||||||
E1' |
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E2' |
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A" | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | z | |||||||||||||||||||||
E1" |
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(Rx, Ry) | (xz, yz) | ||||||||||||||||||||
E2" |
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\(C_{6h}\) | E | C6 | C3 | C2 | C32 | C65 | i | S35 | S65 | σh | S6 | S3 | ε=exp (2πi/6) | |||||||||||||||||||||||||
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 |
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(Rx, Ry) | (xz, yz) | ||||||||||||||||||||||||
E2g |
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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 |
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E2u |
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Pyramidal \(C_{nv}\) Groups
These groups are characterized by an n-fold proper rotation axis \(C_n\) and n mirror planes \(σ_v\) which contain \(C_n\)
\(C_{2v}\) | E | C2 | σV | σh' | ||
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 |
\(C_{3v}\) | E | 2C3 | 3σv | ||
A1 | 1 | 1 | 1 | z | x2+y2, z2 |
A2 | 1 | 1 | -1 | Rz | |
E | 2 | -1 | 0 | (Rx, Ry), (x,y) | (xz, yz) (x2-y2, xy) |
\(C_{4v}\) | E | 2C4 | C2 | 2σv | 2σd | ||
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) |
\(C_{5v}\) | E | 2C5 | 2C52 | 5σv | ||
A1 | 1 | 1 | 1 | 1 | z | x2+y2, z2 |
A2 | 1 | 1 | 1 | -1 | Rz | |
E1 | 2 | 2cos 72 | 2cos 144 | 0 | (Rx, Ry)(x,y) | (xz, yz) |
E2 | 2 | 2cos144 | 2cos 72 | 0 | (x2-y2, xy) |
\(C_{6v}\) | E | 2C6 | 2C3 | C2 | 3σv | 3σd | ||
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) |
C∞v | E | 2C∞ | ... | ∞σv | ||
A1 | 1 | 1 | ... | 1 | z | x2+y2, z2 |
A2 | 1 | 1 | ... | -1 | Rz | |
E1 | 2 | 2cos θ | ... | 0 | (x,y);(Rx, Ry) | (xz, yz) |
E2 | 2 | 2cos 2θ | ... | 0 | (x2-y2, xy) | |
E3 | 2 | 2cos 3θ | ... | 0 | ||
... | ... | ... | ... | ... |
Dihedral \(D_n\) Groups
\(D_2\) | E | C2(z) | C2(y) | C2(x) | ||
A | 1 | 1 | 1 | 1 | x2, y2, z2 | |
B1 | 1 | 1 | -1 | -1 | z, Rz | xy |
B2 | 1 | -1 | 1 | -1 | y, Ry | zx |
B3 | 1 | -1 | -1 | 1 | x, Rx | yz |
\(D_3\) | E | 2C3 | 3C2 | ||
A1 | 1 | 1 | 1 | x2+y2, z2 | |
A2 | 1 | 1 | -1 | z, Rz | |
E | 2 | -1 | 0 | (Rx, Ry)(x,y) | (x2-y2, xy) (xz, yz) |
\(D_4\) | E | 2C4 | C2(C42) | 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 | (Rx, Ry)(x,y) | (xz, yz) |
\(D_5\) | E | 2C5 | 2C52 | 5C2 | ||
A1 | 1 | 1 | 1 | 1 | x2+y2, z2 | |
A2 | 1 | 1 | 1 | -1 | z, Rz | |
E1 | 2 | 2cos72 | 2cos144 | (Rx, Ry)(x,y) | (xz, yz) | |
E2 | 2 | 2cos144 | 2cos72 | (x2-y2, xy) |
\(D_6\) | E | 2C6 | 2C3 | C2 | 2C2' | 3C2" | ||
A1 | 1 | 1 | 1 | 1 | 1 | 1 | x2+y2, z2 | |
A2 | 1 | 1 | 1 | 1 | -1 | -1 | z, 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) |
Prismatic \(D_{nh}\) Groups
These groups are characterized by
- an n-fold proper rotation axis \(C_n\)
- n 2-fold proper rotation axes \(C_2\) normal to \(C_n\)
- a mirror plane \(\sigma_h\) normal to \(C_n\) and containing the \(C_2\) axes.
\(D_{2h}\) | E | C2(z) | C2(y) | C2(x) | 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 |
\(D_{3h}\) | E | 2C3 | 3C2 | σh | 2S3 | 3σv | ||
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) | (xz, yz) |
\(D_{4h}\) | E | 2C4 | C2 | 2C2' | 2C2" | i | 2S4 | σh | 2σv | 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) |
\(D_{5h}\) | E | 2C5 | 2C52 | 5C2 | σh | 2S5 | 2S53 | 5σv | ||
A1' | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | x2+y2, z2 | |
A2' | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | Rz | |
E1' | 2 | 2cos72 | 2cos144 | 0 | 2 | 2cos72 | 2cos144 | (x,y) | ||
E2' | 2 | 2cos144 | 2cos72 | 0 | 2 | 2cos144 | 2cos72 | (x2-y2, xy) | ||
A1" | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ||
A2" | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | z | |
E1" | 2 | 2cos72 | 2cos144 | 0 | -2 | -2cos72 | -2cos144 | 0 | (Rx, Ry) | (xz, yz) |
E2" | 2 | 2cos144 | 2cos72 | 0 | -2 | -2cos144 | -2cos72 | 0 |
D6h | E | 2C6 | 2C3 | C2 | 3C2' | 3C2" | i | 2S3 | 2S6 | σh | 3σd | 3σv | ||
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 |
D∞h | E | 2C∞ | ... | ∞σv | i | 2S∞ | ... | ∞ C2 | ||
Sg+ | 1 | 1 | ... | 1 | 1 | 1 | ... | 1 | x2+y2, z2 | |
Sg- | 1 | 1 | ... | -1 | 1 | 1 | ... | -1 | Rz | |
πg | 2 | 2cos | ... | 0 | 2 | -2cos | ... | 0 | (Rx, Ry) | (xz, yz) |
Dg | 2 | 2cos2 | ... | 0 | 2 | 2cos2 | ... | 0 | (x2-y2, xy) | |
... | ... | ... | ... | ...... | ... | ... | ... | ... | ||
Su+ | 1 | 1 | ... | 1 | -1 | -1 | ... | -1 | z | |
Su- | 1 | 1 | ... | -1 | -1 | -1 | ... | 1 | ||
πu | 2 | 2cos | ... | 0 | -2 | 2cos | ... | 0 | (x, y) | |
Du | 2 | 2cos2 | ... | 0 | -2 | -2cos | ... | 0 | ||
... | ... | ... | ... | ... | ... | ... | ... | ... |
Antiprismatic \(D_{nd}\) Groups
These groups are characterized by
- an n-fold proper rotation axis Cn
- n 2-fold proper rotation axes C2 normal to Cn
- n mirror planes σd which contain Cn.
D2d | E | 2S4 | C2 | 2C2' | 2σd | ||
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) |
D3d | E | 2C3 | 3C2 | i | 2S6 | 3σd | ||
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) |
D4d | E | 2S8 | 2C4 | 2S83 | C2 | 4C2' | 4σd | ||
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 | 1.414 | 0 | - 1.414 | -2 | 0 | 0 | (x, y) | |
E2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | (x2-y2, xy) | |
E3 | 2 | - 1.414 | 0 | 1.414 | -2 | 0 | 0 | (Rx, Ry) | (xz, yz) |
D5d | E | 2C5 | 2C52 | 5C2 | i | 2S103 | 2S10 | 5σd | ||
A1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | x2+y2, z2 | |
A2g | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | Rz | |
E1g | 2 | 2cos 72 | 2cos 144 | 0 | 2 | 2cos 72 | 2cos 144 | 0 | (Rx, Ry) | (xz, yz) |
E2g | 2 | 2cos 144 | 2cos 72 | 0 | 2 | 2cos 144 | 2cos 72 | 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 | 2cos 72 | 2cos 144 | 0 | -2 | -2cos 72 | -2cos 144 | 0 | (x, y) | |
E2u | 2 | 2cos 144 | 2cos 72 | 0 | -2 | -2cos 144 | -2cos 72 | 0 |
D6d | E | 2S12 | 2C6 | 2S4 | 2C3 | 2S125 | C2 | 6C2' | 6σd | ||
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 | 1.732 | 1 | 0 | -1 | -1.732 | -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 | -1.732 | 1 | 0 | -1 | 1.732 | -2 | 0 | 0 | (Rx, Ry) | (xz, yz) |
Improper Rotation \(S_n\) Groups
These groups are characterized by an n-fold improper rotation axis \(S_n\), where \(n\) is necessarily even
\(S_4\) | E | S4 | C2 | S43 | ||||||||||
A | 1 | 1 | 1 | 1 | Rz | x2+y2, z2 | ||||||||
B | 1 | -1 | 1 | -1 | z | x2-y2, xy | ||||||||
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S6 | E | C3 | C32 | i | S65 | S6 | ||||||||||||||
Ag | 1 | 1 | 1 | 1 | 1 | 1 | Rz | x2+y2, z2 | ||||||||||||
Eg |
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(Rx, Ry) | (x2-y2, xy)(xz, yz) | ||||||||||||
Au | 1 | 1 | 1 | -1 | -1 | -1 | z | |||||||||||||
Eu |
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S8 | E | S8 | C4 | S83 | C2 | S85 | C43 | S87 | ε=exp (2πi/8) | |||||||||||||||||
A | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Rz | x2+y2, z2 | ||||||||||||||||
B | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | z | |||||||||||||||||
E1 |
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E2 |
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E3 |
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Cubic Groups
These polyhedral groups are characterized by not having a \(C_5\) proper rotation axis.
\(T\) | E | 4C3 | 4C32 | 3C2 | ||||||||||
A | 1 | 1 | 1 | 1 | x2+y2+z2 | |||||||||
E |
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T | 3 | 0 | 0 | (Rx, Ry, Rz) (x, y, z) | (xz, yz, xy) |
Th | E | 4C3 | 4C32 | 3C2 | i | 4S6 | 4S65 | 3σh | ε=exp (2πi/3) | |||||||||||||||||
Ag | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | x2+y2+z2 | |||||||||||||||||
Eg |
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Tg | 3 | 0 | 0 | -1 | 1 | 0 | 0 | -1 | (Rx, Ry, Rz) | (xz, yz, xy) | ||||||||||||||||
Au | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ||||||||||||||||||
Eu |
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Tu | 3 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | (x, y, z) |
Td | E | 8C3 | 3C2 | 6S4 | 6σd | ||
A1 | 1 | 1 | 1 | 1 | 1 | x2+y2+z2 | |
A2 | 1 | 1 | 1 | -1 | -1 | ||
E | 2 | -1 | 2 | 0 | 0 | (2z2-x2-y2, x2-y2) | |
T1 | 3 | 0 | -1 | 1 | -1 | (Rx, Ry, Rz) | |
T2 | 3 | 0 | -1 | -1 | 1 | (x, y, z) | (xz, yz, xy) |
O | 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 | (2z2-x2-y2, x2-y2) | |
T1 | 3 | 0 | -1 | 1 | -1 | (Rx, Ry, Rz)(x, y, z) | |
T2 | 3 | 0 | -1 | -1 | 1 | (xz, yz, xy) |
Oh | E | 8C2 | 6C2 | 6C4 | 3C2(C42) | i | 6S4 | 8S6 | 3σh | 6σd | ||
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 | (2z2-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 | (xz, yz, xy) | |
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 |