Denote $G$ and $H$ are two finite groups. Let $r(G)$ be the order of the set of conjugacy class of G. We know $$r(G\times H)=r(G)\times r(H).$$My problem is that there is a semi-direct product $G\rtimes H$ such that $G\rtimes H$ cannot be decompose the form $G\times H$,then if $$r(G\rtimes H)<r(G\times H)?$$