What's the smallest non-$p$ group that isn't a semidirect product?

group-theory finite-groups semidirect-product

Solution 1:

What about $\mathtt{SmallGroup}(48,28)$, which is a group that is similar to ${\rm GL}(2,3)$ but has no non-central elements of order $2$? It is one of the two Schur covering groups $2 \cdot S_4$ of $S_4$ (the other one being ${\rm GL}(2,3)$).

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