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Reducible or irreducible? A reducible representation can be of arbitrary high dimension. An irreducible representation always has dimension $\leq\sqrt{\left|G\right|}$, if the base field is algebraically closed and of characteristic $0$ (in fact, in this case, the sum of the squares of the dimensions of all irreducible representations is $\left|G\right|$). However, if the base field is not necessarily algebraically closed, but still of characteristic $0$, then we can only say that the dimension of any irreducible representation is $\leq \left|G\right|$. (Wait a minute for the proof.)