Let $\chi$ be 2-dimensional complex character of the group $ S_3 $. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$.
There is hint in my book: " Use the Maschke's theorem and the properties of the commutator subgroup"
I try. Assume that $\chi$ is reducible character of the group $ S_3. $ Using the Maschke's theorem we have $\chi=\chi_1+\chi_2,$ as $\chi_1, \chi_2$ are $1$-dimensional complex character of the group $ S_3 $. But $$\chi_1(123)= \chi_2(123)=-1$$
I can not prove " if $\chi$ is irreducible character then $\chi((123))=-1$"
I know that $S_3 '=A_3.$ But I can not use it.
I am sorry for my English. Thanks a lot in advance for any help!