Ih E 12 C5 12 C5^2 20 C3 15 C2 i 12 S10 12 S10^3 20 S6 15 s <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ..... ....... ......... ........... T............ T1g 3.00000 1.61803 -0.61803 0.00000 -1.00000 3.00000 -0.61803 1.61803 0.00000 -1.00000 TTT ... ..... ....... ......... ........... .TTT......... T2g 3.00000 -0.61803 1.61803 0.00000 -1.00000 3.00000 1.61803 -0.61803 0.00000 -1.00000 ... ... ..... ....... ......... ........... ............. Gg 4.00000 -1.00000 -1.00000 1.00000 0.00000 4.00000 -1.00000 -1.00000 1.00000 0.00000 ... ... ..... ....... TTTT..... ........... ....TTTT..... Hg 5.00000 0.00000 0.00000 -1.00000 1.00000 5.00000 0.00000 0.00000 -1.00000 1.00000 ... ... TTTTT ....... ....TTTTT ........... ........TTTTT Au 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. T1u 3.00000 1.61803 -0.61803 0.00000 -1.00000 -3.00000 0.61803 -1.61803 0.00000 1.00000 ... TTT ..... ....... ......... TTT........ ............. T2u 3.00000 -0.61803 1.61803 0.00000 -1.00000 -3.00000 -1.61803 0.61803 0.00000 1.00000 ... ... ..... ....TTT ......... ...TTT..... ............. Gu 4.00000 -1.00000 -1.00000 1.00000 0.00000 -4.00000 1.00000 1.00000 -1.00000 0.00000 ... ... ..... TTTT... ......... ........... ............. Hu 5.00000 0.00000 0.00000 -1.00000 1.00000 -5.00000 0.00000 0.00000 1.00000 -1.00000 ... ... ..... ....... ......... ......TTTTT ............. Irrational character values: 1.618033988750 = 2*cos(2*π/10) = 2*cos(π/5) = (√5+1)/2 0.618033988750 = 2*cos(4*π/10) = 2*cos(2*π/5) = (√5−1)/2 Symmetry of Rotations and Cartesian products Ag i+m T1g R+i+m T2g k+m Gg g+i+k+m Hg d+g+i+2k+2m T1u p+h+j+l T2u f+h+j+l Gu f+j+2l Hu h+j+l Notes: α The order of the Ih point group is 120, and the order of the principal axis (S10) is 10. The group has 10 irreducible representations. γ The Ih point group is generated by two symmetry elements, which can be chosen as any two distinct S10 axes. There are more possible pairs, provided at least one has order 5 or 10, at least one has negative parity, and the elements are neither coplanar nor orthogonal. Because Ih contains the tetrahedral groups T and Th as subgroups, choosing elements of lower orders as generators may lead to smaller groups. Yet it is possible to generate Ih from two low-order elements if the elements and their angle are chosen carefully. δ The lowest nonvanishing multipole moment in Ih is 64 (tetrahexacontapole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ This point group has several symmetry elements of order 3 or higher which are not coaxial. Therefore, it has at least three-dimensional irreducible representations. The symmetry described by this group may be called “isometric”, as the three Cartesian directions are degenerate. η This point group corresponds to icosahedral symmetry, because it is isometric and contains five-fold axes. I was not able to calculate symmetry-adapted forms for Cartesian products in icosahedral symmetry. θ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values are related to cosine values of n*π/5 and have simple algebraic expressions.
| T | O | I |
| Th | Oh | Ih |
| Td | ||
This Character Table for the Ih point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.