What is the cycle index for the Dihedral Group acting on a hexagon?

Question

What is the cycle index for the Dihedral Group acting on a hexagon?

So I believe the answer is: $$ \frac{1}{12}(x_1^6 + 2x_6^1 + 2x_3^3 + 4x_2^3 + 3x_2^1x_2^2) $$ but I'm not sure if I may have mixed up something. These things seem quite tedious!

Answer 1

I think you're not quite right, so let me try and go through the elements one at a time. If I have this correct $x_n^m$ means there are $m$ cycles of order $n$. Let me know if you think one of these is incorrect, and I'll try and explain my reasoning.

id=$x_1^6$

$f$ contributes to $x_2^2x_1^2$

$r,r^{-1}$ both equal $x_6^1$

$r^2,r^{-2}$ both equal $x_3^2$

$r^3,r^3f,rf,r^{-1}f$ contribute to $x_2^3$

$r^2f,r^{-2}f$ both equal $x_2^2x_2^1$

so in total $x_1^6+4x_2^3+2x_6^1+2x_3^2+3x_2^2x_1^2$