In S4, what is the subgroup generated by the cycle (123)?

Question

Let $S_4$ be the symmetric group of degree $4$ and $H$ the subgroup of $S_4$ generated by $(1\ 2\ 3)$. I want to list out the members of $H$.

I know they are the powers of (123), but I get (132) when I raised (123) to power of 2 which seems to be the same thing as (123). Does it mean the subgroup has only one member (123)?

Also what is the quotient group $S_4/H$?

Answer 1

No: $(1\ \ 2\ \ 3)\neq(1\ \ 3\ \ 2)$, since $(1\ \ 2\ \ 3)$ maps $1$ into $2$, whereas $(1\ \ 3\ \ 2)$ maps $1$ into $3$.

You can also check that $(1\ \ 2\ \ 3)^3=e$. Therefore,$$H\left(=\bigl\langle(1\ \ 2\ \ 3)\bigr\rangle\right)=\bigl\{(1\ \ 2\ \ 3),(1\ \ 3\ \ 2),e\bigr\}.$$