Let $W$ be a complex reflection group and $B$ and $P$ be respectively the
braid group and the pure braid group of $W$.

The aim of this article is to give a nice description of the following short
exact sequence as an element of $H^2(W,P^\textrm{ab})$.

$$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] &
1}$$

To obtain this description, we need to construct a linear character of a
subgroup of the braid group. This is the heart of the first part of the
article. In this first part, we also determine completely the abelianization of
the preimage in $B$ of the subgroups of $W$ generated by reflections. Using
simple methods of low-dimensional cohomology, we give the explicit description
of the 2-cocycle associated to our exact sequence in the second part of the
article. The third part of the article presents table for the so-called "index
of ramification" of the various hyperplanes.