Groups which do not have four irreducible characters of degrees divisible by a prime $p$

Given a finite group $G$, we say that $G$ has property $cP_n$ if for every prime integer $p$, $G$ has at most $n-1$ irreducible characters whose degrees are multiples of $p$. ##In this paper, we classify all finite groups that have property $cP_4$. ##We show that the groups satisfying property $cP_4$ are exactly the finite groups with at most three nonlinear irreducible characters, one solvable group of order $168$, $SL_2(3)$, $Alt_5$, $Sym_5$, $PSL_2(7)$ and $Alt_6$.