Is there an irreducible linear representation of $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ of dimension 2?

Question

More explicitly, let $G$ be given by the additive group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

Question: Can there be an irreducible linear representation $\rho: G \to \operatorname{GL}(\mathbb{C}^2)$?

I am trying to construct such an example, but as time flies, I suspect that there might be none. Could you please verify my presumption and explain the reason? Thank you!

Answer 1

Irreducible representations of abelian groups are one-dimensional.