What does the subgroup generated by a specific 3-cycle look like?

Question

I am struggling to simply understand how you would work out which elements are 'generated' by a single 3-cycle, see the following example

$ H = \langle (4, 5, 6) \rangle \leqslant S_6 $

Is it just $ H =\{ e, (4,5,6), (4,6,5) \} $ ?

Since $$(4,5,6)(4,5,6)=(4,6,5) $$and $$ (4,5,6)(4,5,6)(4,5,6)=e $$

Or am I missing other ways of generating elements?

Answer 1

Yes, all three cycles have order three; that is, if $\sigma$ is a three cycle, then $\sigma^3=e$.

The underlying set of the group is indeed

$$\{e, (456), (456)^2\}=\{e, (456), (465)\}.$$