There is a unique homomorphism $\displaystyle \theta: \mathbb{Z}_{6}\rightarrow S_{3} $ such that $\displaystyle \theta ([1]) =(1 \ 2 \ 3).$ Determine $\displaystyle \theta ([k])$ for each $\displaystyle [k] \in \mathbb{Z}_{6}$. Which elements are in $\displaystyle Ker \theta $?

I don't understand how that single condition can be used to determine the rest of the $\displaystyle \theta ([k])$. It doesn't seem like enough information.