I'm asked in this exercise to find all permutations that commute with $\omega$=(1 9 7 10 12 2 5)(4 11)(3 6 8) in $S_{12}$.
What I have so far: We could write $x$(1 9 7 10 12 2 5)(3 6 8)=(1 9 7 10 12 2 5)(3 6 8)$x$, which means (1 9 7 10 12 2 5)(4 11)(3 6 8)=$x^{-1}$(1 9 7 10 12 2 5)(4 11)(3 6 8)$x$=(1x 9x 7x 10x 12x 2x 5x)(4x 11x)(3x 6x 8x). This means $x$ commutes with $\omega$ iff $x$ 'transfers' one of the distinct cycles that construct $\omega$ unto itself.
There are $7\cdot 2\cdot 3$ ways to 'present' $\omega$ disregarding the order of multiplication of the cycle (should I disregard it?), each of which creates a distinct commuting permutation if constructed by the algorithm: 1x$\rightarrow$(1 9 7 10 12 2 5), 9x$\rightarrow$(1 9 7 10 12 2 5), et cetera. So all in all we end up with $7\cdot 2\cdot 3$ permutations.
I guess what I should ask is: (a) does this sound correct? (b) since the question wants us to find all permutations and not count them, perhaps there is a more general 'form' for the commuting permutations?