A class of special \(p\)-groups (Q1816492)

By means of the multiplication table of a finite group \(G\) of order \(n\), for every odd prime \(p\) a special \(p\)-group \(P\) is constructed, with the following properties: \(|P|=p^{3n}\), \(\exp(P)=p\), \(|P'|=p^n\), \(P=AB\), where \(|A|=|B|=p^{2n}\) and \(A'= B'=1\). Some connections between the structures of \(G\) and \(P\) are proved. In particular, \(p^{[G:G']}\) is the number of abelian subgroups \(C\) of order \(p^{2n}\) such that \(AC=P\). If \(p\) does not divide \(n\), for every quotient of \(G\) a direct factorization of \(P\) is determined. Last, noncentral elements of \(P\) with large centralizers are characterized in terms of the multiplication table of \(G\).