Exercise from Artin's Algebra.
Describe the centralizer $Z (\sigma) $ of the permutation $\sigma = (153)(246)$ in the symmetric group $S_{7} $, and compute the orders of $Z (\sigma) $ and of $C (\sigma)$.
My progress:
Since $P= (1/\sqrt {2}) \begin {pmatrix} 1 & i \\ 1 & -i \end {pmatrix}$
has the property that $PAP^{*}$ is diagonal, and $PP^{*}=I $. For this sort of problem I think I am suppose to find a basis for eigenvalues vectors for A and normalize them so they give a unitary change of basis matrix. How would this be done?