Finite \(p\)-groups of nilpotency class 3 with two conjugacy class sizes (Q2182051)

A finite group G is said to be of conjugate type \((1 = m_1,m_2,\dots,m_r)\), \(m_i < m_{i+1}\), if \(m_i\)'s are precisely the different sizes of conjugacy classes of \(G\). By a theorem of P. Hall, any two isoclinic groups are of the same conjugate type. The authors prove that if \(p\) is an odd prime and \(n>1\), then there exist finite \(p\)-groups of nilpotency class 3 and conjugate type \((1,p^n)\) if and only if \(n\) is even. Moreover, for a given even positive integer \(n\), such a group is unique up to isoclinism.