I am working on some calculations for a conjecture that I have that involves looking at direct and semi direct product of finite groups. So far the only groups that I have been able to use has been of the form $L=F\rtimes G$ where $F$ is normal in $L$ and $G$ is abelian. Can anyone point me in the direction of a finite group $L$ that has the form $L=F\rtimes G$ where both $F$ and $G$ are nonabelian?
Thanks for any help.