Permutation Groups · Hard problem

Let $G = \mathbb{Z}_2 \times \mathbb{Z}_6$ (order $12$, written additively, with $\mathbb{Z}_2 = \{0,1\}$ and $\mathbb{Z}_6=\{0,\ldots,5\}$). In the regular representation of $G$, how many elements $g \in G$ give a permutation $\lambda_g$ of order exactly $3$?