A group G is called completely reducible if it is a direct product of simple groups. It is known that a Krull-Schmidt Remak type unicity for the decomposition in the direct product of simple groups hold. The proof s of this facts can be found for example in Chapter 3 of D. Robinson's book A Course in the Theory of Groups
My questions are the following:
Are any known characterizations for finite completely reducible groups given in the literature? Given a finite group how can one decide whether the group is completely reducible or not?, of course without searching for direct product decompositions if possible.
If one replaces the direct product decomposition with iterated crossed product decomposition is anything known about unicity in this case?