That seems a reasonable starting point dragon.
I do not know if you did it yourself but it is a better start than i had when I tried to do it with no hints.
I had no idea where to start.
However:
I am not convinced about your continued logic.
Just for one of these configurations I think you have to start by placing one number. I chose to place one of the A which is a multiple of 3
So I think it would be 2! * 3! * 3! = 24 ways
But you have
C B A C B A C B A YES THAT WILL WORK 24 ways
B C A B C A B C A THAT WILL WORK TOO 24 ways
I can't see any other way to do it. But maybe you can.
So I only get 48 ways but I could easily be wrong.