Compute the left cosets of $\langle (12)\rangle \times \langle[1]\rangle$ in $S_3\times \Bbb Z_3$

Question

Compute the left cosets of $\langle (12)\rangle\times\langle[1]\rangle$ in $S_3\times \Bbb Z_3$

Ok, so I know $\langle (12)\rangle$ is in $S_3$ and $\langle[1]\rangle$ is in $\Bbb Z_3$

Answer 1

Am I multiplying $(12)$ by $[1]$?

No.

Note that $\langle (12)\rangle=\{(1)(2)(3), (12)\}$, whereas $\langle [1]_3\rangle=\{[0]_3, [1]_3, [2]_3\}$. Thus there are six elements in $H=\langle (12)\rangle\times\langle[1]_3\rangle$.

The left cosets of $H$ in your group are of the form

$$(\sigma, [a]_3)H=\{(\tau, [b]_3)\in S_3\times \Bbb Z_3 \mid (\sigma\tau^{-1}, [a-b]_3)\in H\},$$

noting that $(12)^{-1}=(12)$.