Characters of projective representations of the infinite generalized symmetric group

By the infinite generalized symmetric group we mean the group $B_m=\mathfrak{S}_\infty\ltimes\mathbb{Z}_m^\infty$, where $\mathbb{Z}_m^\infty$ is the group of all sequences $\{z_k\}_{k=1}^\infty$ in $\mathbb{Z}_m$ containing only finitely many non-zero elements $z_k$ and $\mathfrak{S}_\infty$ is the group of all finitely supported permutations of the positive integers. A complete description of the projective factor representations of $B_m$ of finite type is obtained.
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