Classify, up to similarity, real matrices of dimension $6\times6$ with minimal polynomial $(t-1)^2(t+1)(t-2)$ Classify, up to similarity, real matrices of dimension $6\times6$ with minimal polynomial $(t-1)^2(t+1)(t-2)$
My attempt:
If the minimal polynomial is $(t-1)^2(t+1)(t-2)$ then we only have the following invariant factors:
$(t-1)^2(t+1)(t-2)=t^4-3t^3+t^2+3t-2, (t-1)^2=t^2-2t+1$
So, every matrix of dimension $6\times6$  with minimal polynomial $(t-1)^2(t+1)(t-2)$ is similar to
$$A=\begin{bmatrix}
0 & 0 & 0 & 2 & 0 & 0 \\
1 & 0 & 0 & -3 & 0 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 \\
0 & 0 & 1 & 3 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & 2 \\
\end{bmatrix}$$.
Is this reasoning correct?
Thanks in advance.

Hint
The matrices having the desired minimal polynomial, are similar to the ones having for only non zero elements the elements $2,-1,1$ or $B=\begin{pmatrix}
1 & 1\\
0 & 1\end{pmatrix}$ on the diagonal, with at least one $2$, one $-1$ and one $B$ on the diagonal.
