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We give an explicit decomposition of $\hbox{Ind}(1)_{B_n}^{S_{2n}}$,following Barbasch and Vogan [1]. We define two natural generalizations of$B_n$, and extend the proof in [1] to recursively compute these decompositions.Although the decompositions do not appear to follow a simple pattern, we proveenough of their structure to show that they are almost never multiplicity-free.

The representation theory of generalized hyperoctahedral groups