Groups with Small Centralizers

Denote by $w(n)$ the number of factors in the representation of a positive integer $n$ in the form of a product of primes. For a subgroup $>H$ of a finite group $G$, we set $w(H)=w(|H|)$ and $v(G)=\max\{w(C(g))\mid g\in G\setminus Z(G)\}$. In the present paper, the complete description of centerfree groups satisfying the condition $v(G)= 4$ is presented.