To be more specific, I'm looking on $S6$ right now. I want to show that the subgroup generated by: $\langle a=(1234),b=(3456) \rangle$ has order $16$, meaning the multiple of it's generators (EDIT: this is wrong see comments).
This is clearly correct if the generators commute, as we just need to choose 2 numbers from 0 to 3, for the powers of the generators.
But what happens when the generators don't commute, like in the example? I haven't been able to find a clear rule for this case. What confuses me is that when they don't commute, $a^i*b^i$ and $b^i*a^i$ could be different for some $i$, resulting in a larger subgroup.