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For a character  χ  of a finite group  G , the numberis called the co-degree of  χ . Letbe an integer anddenote the set of irreducible characters whose kernels do not contain. In this paper, we show that if  G  is solvable andfor every prime divisor  p  ofand every, then the derived length of  G  is at most. Then, we classify the finite non-solvable groups with non-trivial Fitting subgroups such that the co-degrees of their irreducible characters whose kernels do not contain the Fitting subgroups are cube-free.

Groups with some cube-free irreducible character co-degrees