I'm working on my dissertation about semidirect products. I'm ending with a proof of Schur's Theorem and using factor sets to construct the quaternion. For my final discussion I'm looking for a group that cannot be constructed using factor sets. So I believe i'm looking for a group that has no abelian normal subgroups. Ideally not a simple group as they have no normal subgroups at all so aren't a true limitation of my work.
Thanks In advance