Consider a square where there are points on the four corners and the four midpoints. Findusing 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:

So

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.