What is the number of similarity classes of 6×6 matrices over $\mathbb{C}$ with the minimal polynomial $(x-1)(x-2)^2$ and characteristic polynomial $(x-1)^2(x-2)^4$.
Similar matrices have same jordan normal form upto permutation of the jordan blocks. In the above problem there two jordan normal forms possible and therefore there are two similarity classes. I am not sure if my argument is correct. Help me.
