Using representation theory, we obtain a necessary and sufficient condition for a discrete--time Markov chain on a finite state space $E$ to be representable as $\Psi_n \Psi_{n-1} \cdots \Psi_1 z$, $n \ge 0$, for any $z \in E$, where the $\Psi_i$ are independent, identically distributed random permutations taking values in some given transitive group of permutations on $E$. The condition is particularly simple when the group is $2$-transitive on $E$.