  A generalization of Dowling lattices was recently introduced by Bibby and
Gadish, in a work on orbit configuration spaces. The authors left open the
question as to whether these posets are shellable. In this paper we prove
EL-shellability and use it to determine the homotopy type. Our result
generalizes shellability of Dowling lattices and of posets of layers of abelian
arrangements defined by root systems. We also show that subposets corresponding
to invariant subarrangements are not shellable in general.
