Definition:Permutation Representation

Definition
Let $G$ be a group.

Let $X$ be a set.

Let $\operatorname{Sym} \left({X}\right)$ be the group of permutations on $X$.

A permutation representation of $G$ is a group homomorphism from $G$ to $\operatorname{Sym}(X)$.

Also see

 * Correspondence Between Group Actions and Permutation Representations