On conjugacy classes of groups of Squarefree order

The problem of finding the largest finite group with a certain class number (number of conjugacy classes), $k(G)$, has been investigated by a number of researchers since the early 1900's and has been solved by computer for $k(G) \leq 9$. (For the restriction to simple groups for $k(G) \leq 12$.) One has also tried to find a general upper bound on $|G|$ in terms of $k(G)$. The best known upper bound in the general case is in the order of magnitude $|G| \leq k(G)^{2^{k(G)-1}}$. In this paper we consider the restriction of this longstanding problem to groups of squarefree order. We derive an explicit formula for the class number $k(G)$ of any group of squarefree order and we also obtain an estimate $|G| \leq k(G)^3$ in this case. Combining the two results we get an efficient way to compute the largest squarefree order a group with a certain class number can have. We also provide an implementation of this algorithm to compute the maximal squarefree $|G|$ for arbitrary $k(G)$ and display the results obtained for $k(G) \leq 100$.