Image and kernel of the action of the group $D_{12}$ on the $D_{12}/H$
Solution 1:
Hint: For all $\sigma, \tau\in S_n$ with $\tau=(t_{a_1}\dots t_{a_j})\dots(t_{z_1}\dots t_{z_k})$ being the cyclic decomposition of $\tau$, we have
$$\sigma\tau\sigma^{-1}=(\sigma(t_{a_1})\dots \sigma(t_{a_j}))\dots(\sigma(t_{z_1})\dots\sigma(t_{z_k})),$$
where $\sigma(x)$ is $\sigma$ evaluated at $x$.