  We consider compact matrix quantum groups whose fundamental corepresentation
matrix has entries which are partial isometries with central support. We show
that such quantum groups have a simple representation as semi-direct product
quantum groups of a group dual quantum group by an action of a permutation
group. This general result allows us to completely classify easy quantum groups
with the above property by certain reflection groups. We give four applications
of our result. First, there are uncountably many easy quantum groups. Second,
there are non-easy quantum groups between the free orthogonal quantum group and
the permutation group. Third, we study operator algebraic properties of the
hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for
easy quantum groups in the scope of this article.
