Alternating group $A_5$ has subgroup of order $6$ (group theory)

Question

In my lectures, I've read that $A_5$ $($the alternating group of even length cycles in $S_5$$)$, has a subgroup of order $6$, and the example is: the group generated by $\langle (12) (34), (123)\rangle$.

I don't even understand what group we are generating? Wasn't a cyclic group only generated by one element? Why would this group be of order $6$?

Answer 1

https://groupprops.subwiki.org/wiki/Twisted_S3_in_A5

Read the above link, there is a subgroup of $A_5$ which is quite similar to $S_3$