Let $K$ be a local field and $G_K$ be its absolute Galois group. A character is a group homomorphism $\phi: G_K \to \mathbb{C}^*$ with finite image, i.e. $|\phi(G_K)| < \infty$. It is unramified if $\phi(I_K) = 1$ where $I_K$ denotes the inertia subgroup of $G_K$.
In the proof of Lemma 2 of this paper, I found the following sentence:
We have two cases: If F/K is not totally ramified, pick an unramified character $\phi$ of $G_K$ of order $n$ with $\phi(\operatorname{Frob}_K) = \chi(\operatorname{Frob}_K)^{-1}$. Otherwise, pick any unramified character $\phi$ of $G_K$ of order $n$.
Question: What is the definition of the order of an unramified character?
I thought first it could have something to do with a power of $\phi$ (like with the order of group elements), but this does not make sense since $\phi$ is not an endomorphism. Could you explain what this means? Thank you!