The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$.

- $\Sigma$ is
*regular*if it acts transitively and freely on $\Omega$, i.e., for any two $i,j\in \Omega$ there is a unique $\sigma\in\Sigma$ with $\sigma(i)=j$. - $\Sigma$ is
*multiplicity-free*if its permutation character (the character of the linear representation of $\Sigma$ by permutation matrices) is the sum of*distinct*irreducible characters.

Examples are the permutation groups generated by a single cyclic permutation.