This is a follow-up to
Conceptual reason why Coxeter groups are never simple
A complex reflection group is a finite subgroup of $ U_n $ that is generated by pseudo reflections. A pseudo reflection is a matrix in $ U_n $ with $ n-1 $ dimensional $ \lambda=1 $ eigenspace. Real complex reflection groups are Coxeter groups.
I was looking at the complex reflection groups given here
https://en.wikipedia.org/wiki/Complex_reflection_group
and it seems that none of them are perfect (although the $ S_n $ and a number of the exceptional cases have index 2 perfect subgroups).
Is it true that complex reflection groups are never perfect? Is there a nice conceptual proof of this fact?
Complex reflections have the form $$\begin{pmatrix}\zeta &0&\cdots& 0\\ 0&1&\cdots &0 \\ \vdots &&\ddots & \\ 0&\cdots &0&1\end{pmatrix}$$ and so do not have determinant $1$. Therefore the subgroup is not a subgroup of $\mathrm{SL}_n(\mathbb{C})$, and the determinant map has a kernel that is not the whole group.