Group codes of dimension 2 and 3 are abelian

Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dimF⁡(C)≤3 then C is an abelian group code. Since there exist non-abelian group codes of dimension 4 when charF>2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite field is 4.