Let \(\sigma \in S_n\) be a permutation and let \([\sigma]\) be the class of all cyclic permutations of \(\sigma\). For \(\pi \in S_n\), \(\pi =(b_1,b_2,...,b_n)\), denote by \(\pi^R\) the permutation \((b_n,...,b_1)\in S_n\). Also denote by \(\) the set \([\sigma]\cup \{\tau^R;\quad \tau \in [\sigma]\}.\) In this paper the authors study the function \(F(n)=\max \min I(\sigma)\), where \(I(\sigma)\) is the number of inversions in \(\sigma\), the max is over \(\pi \in S_n\) and the min over \(\sigma\in[\pi].\) The main result is the following theorem: \[ 0.304^-n^2+0(n)=\frac{8-\pi}{16}\cdot n^2-\frac{3n}{2}\leq F(n)\leq \frac{n^2}{3}-\frac{3n-1}{6}=0.333^+n^2+0(n). \] The measures of complexity considered are the number of inversions and the diameter of the permutation.