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 $S_4/H$?