The iterated monodromy group of a post-critically finite complex polynomial of degree d ≥ 2 acts naturally on the complete d-ary rooted tree T of preimages of a generic point. They have been used by Bartholdi and Nekrashevych to resolve the well-known “twisted rabbit” problem of J. Hubbard. They also have some interesting group-theoretic properties. For example, the Basilica group is the first known example separating the classes of amenable groups and sub-exponentially amenable groups. This talk is to introduce a way to compute the iterated monodromy group with the Basilica group as an example. We will give a proof of “the Basilica group has exponential growth” if time permits.