I E 12 C5 12 C5^2 20 C3 15 C2 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ..... ....... ......... ........... T............ T1 3.00000 1.61803 -0.61803 0.00000 -1.00000 TTT TTT ..... ....... ......... TTT........ .TTT......... T2 3.00000 -0.61803 1.61803 0.00000 -1.00000 ... ... ..... ....TTT ......... ...TTT..... ............. G 4.00000 -1.00000 -1.00000 1.00000 0.00000 ... ... ..... TTTT... TTTT..... ........... ....TTTT..... H 5.00000 0.00000 0.00000 -1.00000 1.00000 ... ... TTTTT ....... ....TTTTT ......TTTTT ........TTTTT Irrational character values: 1.618033988750 = 2*cos(2*π/10) = 2*cos(π/5) = (√5+1)/2 -0.618033988750 = 2*cos(6*π/10) = 2*cos(3*π/5) = −(√5−1)/2 Symmetry of Rotations and Cartesian products A i+m T1 R+p+h+i+j+l+m T2 f+h+j+k+l+m G f+g+i+j+k+2l+m H d+g+h+i+j+2k+l+2m Notes: α The order of the I point group is 60, and the order of the principal axis (C_{5}) is 5. The group has 5 irreducible representations. β The I point group is isomorphic to the Alternating Group Alt(5). It is the smallest non-Abelian simple group, and also the smallest non-solvable group. δ The I point group is generated by two symmetry elements, which can be chosen as any two distinct C_{5} axes. Alternatively, the group can be generated from any C_{5} with any nonorthogonal C_{3} or C_{2}, or from some combination of C_{2} and/or C_{3}. In the latter case, however, the angle between the two generator elements must be carefully chosen, as some combinations may yield T which is a subgroup of I. ε The lowest nonvanishing multipole moment in I is 64 (tetrahexacontapole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η The point group is chiral, as it does not contain any mirroring operation. θ 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. ι 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. κ 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 |
T_{h} | O_{h} | I_{h} |
T_{d} |
This Character Table for the I point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.