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Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr\mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. Weprove that the irreducible representations for this class of groups are indexedby a certain type of rooted trees. This provides a Bratteli diagram for thegeneralized iterated wreath product, a simple recursion formula for the numberof irreducible representations, and a strategy to calculate the dimension ofeach irreducible representation. We calculate explicitly fast Fouriertransforms (FFT) for this class of groups, giving literature's fastest FFTupper bound estimate.

Generalized iterated wreath products of cyclic groups and rooted trees correspondence