Groups of order 3.2^n

This article discusses the groups of order $$3 \cdot 2^n$$, where $$n$$ varies over nonnegative integers. Note that any such group has a 3-Sylow subgroup which is cyclic group:Z3, and a 2-Sylow subgroup, which is of order $$2^n$$. Further, because order has only two prime factors implies solvable, any such group is a solvable group.

Composition length
As noted above, all groups of order $$3 \cdot 2^n$$ are solvable groups. Thus, the composition factors are all simple abelian groups, and hence cyclic groups of prime order. In particular, there are $$n$$ composition factors isomorphic to cyclic group:Z2 and one composition factor isomorphic to cyclic group:Z3. The composition length is thus $$n + 1$$.

Fitting length
Since all groups of this order are solvable, the Fitting lengths are well defined and meaningful. Also note that the Fitting length is bounded from above by the derived length.

Note that Fitting length 1 means that the group is a nilpotent group (specifically, a finite nilpotent group) which in this case means it is a direct product of its 2-Sylow subgroup and 3-Sylow subgroup, the latter being cyclic group:Z3. The number of such groups equals the number of groups of order $$2^n$$.