Commutator subgroup of the affine group of the complex line
= Commutator subgroup of the affine group of the complex line
For the <Affine group of the complex line>,
$$
[G,G]=\{f_{1,b}:b\in\mathbb C\}\cong(\mathbb C,+),
\qquad G/[G,G]\cong\mathbb C^\times.
$$
Indeed, the multiplier map $f_{a,b}\mapsto a$ has the displayed translation group as its kernel and abelian image, while
$$
[f_{a,1},f_{1,b}]=f_{1,b(1-a^{-1})}
$$
produces every translation.