In the slides on commutators in groups, the author states one fact (p.10-11), whose reference is not mentioned there.
There is a group of order $216$ in which derived subgroup is non-abelian of order $24$ and the set of commutators is not a subgroup.
I wanted to see the structure of such group(s). Can anyone help to get structure (presentation) of such group. Also, is such group of order $216$ unique?
There are appear to be three groups with those properties, namely $\mathtt{SmallGroup}(216,i)$, with $i=39,41,42$.
They have very similar structures with $[G,G] \cong Q_8 \times C_3$, $G/[G,G] \cong C_3 \times C_3$. They all have $22$ commutators.
They all have nonabelian Sylow $3$-subgroups of order $27$, of exponent 3 in one case and exponent 9 for the other 2. This was just a quick calculation and I have not found any structural difference between those last two groups.