# Cycle index for rotation

• Jun 3rd 2013, 07:00 PM
Yuuki
Cycle index for rotation
Consider a square where there are points on the four corners and the four midpoints. Find $\hat{C}(r,b)$ using the cycle index for the group D4.

The identity has 8 cycles of length 1; the three rotations have 2 cycles of length 4; the reflections have 2 cycles of length 1 and 3 cycles of length 2.
Hence, the cycle index is:
$(z_{1}^{8}+3z_{4}^{2}+4z_{1}^{2}z_{2}^{3})/8$
So $\hat{C}(r,b)=((r+b)^8+3(r^4+b^4)^2+4(r+b)^2(r^2+b^ 2)^3)/8$
But when I expanded this using Wolfram Alpha, I got fractions for some of the coefficients.
I can't seem to find what I did wrong, so I need help finding out.