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We define the position of an irreducible complex character of a finite groupas an alternative to the degree. We then use this to define three classes ofgroups: PR-groups, IPR-groups and weak IPR-groups. We show that IPR-groups andweak IPR-groups are solvable and satisfy the Taketa inequality (ie, that thederived length of the group is at most the number of degrees of irreduciblecomplex characters of the group), and we show that any M-group is a weakIPR-group. We also show that even though PR-groups need not be solvable, theycannot be perfect.

Positions of characters in finite groups and the Taketa inequality