Description of $S_4/H$ as a set where $H$ is order $8$ subgroup of $S_4$

Question

Let $H$ be subgroup of $S_4$ of order $8$. It is conjugate of $\langle (1,2,3,4),(2,4)\rangle $. I would like to specify $S_4/H$ as a set. $S_4/H$ is order $3$, so the set is like $\{ id,\sigma H,\tau H\}$. I would like to specify $\sigma $ and $\tau$.

I would like to know the process of finding $\sigma, \tau$.

Thank you for your help.

Answer 1

First, lets recall that in this setting, $id$ is just H.

Now take an element $\sigma \in S_4-H$. Since this element is not in H, you get that $\sigma H\ne H$.

Take $\tau \in S_4 - (H \cup \sigma H)$ and you get $\tau H \ne \sigma H \land \tau H \ne H$.

So $S_4/H = \{H, \sigma H, \tau H\}$.