Permutation representation (reviewed)

A permutation representation of a finite group $G$ is a homomorphism $\rho : G \to S_n$ for some $n$. The integer $n$ is the degree of $\rho$, and a representation is faithful if $\rho$ is injective. The representation is transitive if the set $\{1, \dots, n\}$ cannot be broken up into smaller orbits under the action of $G$, i.e. for all $1 \le i \le j \le n$ there is some $g \in G$ with $\rho(g)(i) = j$.