A $15 \times 15$ matrix $M$ with complex entries has characteristic polynomial equal to $(x-1)^7(x-2)^8$. Find all possible minimal polynomials for $M$ such that the characteristic and minimal polynomials together determine the Jordan canonical form of $M$ up to the ordering of the blocks. Give the Jordan canonical form for each of these.
I am not sure when I know that the characteristic poly and minimal poly determine the JCF. I must be able to narrow it down somewhat since the minimal poly is of the form $(x-1)^a (x-2)^b$ for $1\leq a\leq 7$ and $1\leq b \leq 8$ because it seems like the problem would be too long if I would have to consider all the possibilities for the minimal poly.
It will be completely determined by the characteristic poly and the minimal poly when $m(x)=(x-1)^m(x-2)^n$ where m=1, 6, or 7 and n=1, 7, or 8