I'm trying to find the characteristic polynomial of the matrix which represents an arbitrary element $g$ of symmetric group $S_n$.
My idea is that if $g = c_1 c_2 ... c_k$ in cycle notation, where each $c_i$ is a cycle of $g$, then the characteristic polynomial of its matrix $M_g$ is given by $\det(M_g) = (x^{\lvert{c_1}\rvert}-1)(x^{\lvert{c_2}\rvert}-1)...(x^{\lvert{c_k}\rvert}-1) $.
Through Leibniz formula for determinants, I can prove that this is the case for elements $g$ which have only one cycle with $n$ elements, but I'm struggling to prove why I can multiply together the characteristic polynomials of matrices consisting of other smaller cycles.
Could someone give me a nod in the right direction?