It turns out that there is an explicit characterization of $\mathbb{Z}_n^{ imes}$ that depends on the factorization of $n$; in your case this becomes $\mathbb{Z}_{21}^{ imes} \cong \mathbb{Z}_3^{ imes} imes \mathbb{Z}_7^{ imes}$, so once you know this theorem, showing whether this group is cyclic or not becomes pretty easy.