Consider a square where there are points on the four corners and the four midpoints. Find$\displaystyle \hat{C}(r,b)$using thecycle index for the group D_{4}.

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:

$\displaystyle (z_{1}^{8}+3z_{4}^{2}+4z_{1}^{2}z_{2}^{3})/8$

So $\displaystyle \hat{C}(r,b)=((r+b)^8+3(r^4+b^4)^2+4(r+b)^2(r^2+b^ 2)^3)/8$

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.

Thanks in advance.