▁U . g ▁aff ords ▁the ▁character ▁ μ ( g ) ▁and ▁. ▁The ▁respective ▁equal ities ▁follow ▁because ▁ χ ▁is ▁a ▁class - function ▁of ▁G ▁and ▁N ▁ ▁is ▁a ▁normal ▁subgroup . ▁The ▁integer ▁e ▁appearing ▁in ▁the ▁statement ▁of ▁the ▁theorem ▁is ▁this ▁common ▁multip licity . ▁ ▁Cor ollary ▁of ▁Cl iff ord ' s ▁theorem ▁ ▁A ▁cor ollary ▁of ▁Cl iff ord ' s ▁theorem , ▁which ▁is ▁often ▁explo ited , ▁is ▁that ▁the ▁ir reducible ▁character ▁ χ ▁appearing ▁in ▁the ▁theorem ▁is ▁induced ▁from ▁an ▁ir reducible ▁character ▁of ▁the ▁in ert ial ▁subgroup ▁I G ( μ ). ▁If , ▁for ▁example , ▁the ▁ir reducible ▁character ▁ χ ▁is ▁primitive ▁( that ▁is , ▁ χ ▁is ▁not ▁induced ▁from ▁any ▁proper ▁subgroup ▁of ▁G ), ▁then ▁G ▁= ▁I G ( μ ) ▁and ▁ χ N ▁= ▁e μ . ▁A ▁case ▁where ▁this ▁property ▁of ▁primitive ▁characters ▁is ▁used ▁particularly ▁frequently ▁is ▁when ▁N ▁is ▁Ab elian ▁and ▁ χ ▁is ▁faithful ▁( that ▁is , ▁its ▁kernel ▁contains ▁just ▁the ▁identity ▁element ). ▁In ▁that ▁case , ▁ μ ▁is ▁linear , ▁N ▁is ▁represented ▁by ▁scalar ▁matrices ▁in ▁any ▁representation ▁aff ording ▁character ▁ χ ▁and ▁N ▁is ▁thus ▁contained ▁in ▁the ▁center ▁of ▁G . ▁For ▁example , ▁if ▁G ▁is ▁the ▁symmetric ▁group ▁S 4 , ▁then ▁G ▁has ▁a ▁faithful ▁complex ▁ir reducible ▁character ▁ χ ▁of ▁degree ▁ 3 .