Let \(G\) denote a finite group and let \(\text{lcs}(G)\) denote the largest conjugacy class size of \(G\). If \(G_p\) is a Sylow \(p\)-subgroup of \(G\), the authors prove the following Theorem: Let \(G\) be an Abelian-by-nilpotent finite group. Then: \[ \text{lcs}(G)\geq\prod_{p\in\pi(G)}\text{lcs}(G_p). \] The restriction in the above theorem is essential; for every \(\varepsilon>0\) the authors construct an example \(G\) of derived length 3 which satisfies: \[ \text{lcs}(G)<\varepsilon(\prod_{p\in\pi(G)}\text{lcs}(G_p)). \] The proof of the theorem (which goes by induction on the number of prime divisors of \(|G|\)) and the construction of the examples are far from being routine. Taken together, these results illustrate very well the difficulties involved in estimating \(\text{lcs}(G)\).