Let \(G\) be a finite group. An \(r\)-tuple \(\{n_1,n_2,\ldots,n_r\}\) is said to be the conjugacy type vector of \(G\) if \(n_1>n_2>\cdots>n_r\) are all the numbers that occur as sizes of conjugacy classes of \(G\). Define \(\{n_1,\ldots,n_r\}\times\{m_1,\ldots,m_s\}\) as the set \(\{n_im_j\mid 1\leq i\leq r\), \(1\leq j\leq s\}\). The authors prove that \(G\) is nilpotent if the conjugacy type vector of \(G\) has the form \(\{p_1,1\}\times\cdots\times\{p_r,1\}\) where \(p_1,\ldots,p_r\) are distinct primes. They also study the structure of \(G\) in the case when, for some prime \(q\) dividing the order of \(G\), the size of every conjugacy class of \(q\)-elements of \(G\) is a power of a prime. In particular, such a group is \(q\)-soluble of \(q\)-length 1.