I was thinking of building on top of known non Abelian groups, like $S_3$, and taking a direct product with $\Bbb Z_n$'s but those groups' order would be a multiple of order of $S_3$.
- So, is there is a clever way to do it for any order, like use an Abelian group of order close to, say $39$, and make it non-Abelian?
- Here, Finding presentation of group of order 39 they give a general representation but can we come up with an actual example without using Sylow theorems? Appreciate your response.
There are composite orders (e.g. 15 or 765, or prime squares) such that all groups of that order will be abelian, and there is no all-purpose construction, but here are a few constructions of nonabelian groups that cover lots of orders:
If the order is even and $>4$, one can construct a dihedral group.
If the order involves a prime power $p^k$ such that another prime divisor (that could be $p$ as well, so this in particular covers all orders that are multiples of a prime cubed) that divides the order of $GL_k(p)$, one can form a nonabelian semidirect product.