Character table for the I point group

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 (C5) 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 C5 axes.
       Alternatively, the group can be generated from any C5 with any nonorthogonal C3 or C2, or from some combination of C2 and/or C3. 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.