I'm working with $\mathbb S_4$, and I have a subgroup of $\mathbb S_4$ called $G$.
$G$ is generated by $a=(12)(34)$ and $b=(123)$, which I've actually found to be $A_4$ by multiplying elements by $a$ and $b$ until I can't find any newer elements (actually , is there a simpler way to do this as well?)
Then I have a subgroup of $G$ called $H$, and $H$ is generated by $a=(12)(34)$ and $c=(13)(24)$. I know I can just test whether each conjugate ($ghg^{-1}$) is an element of $H$, but this is long and tedious.
Is there a trick to show this without having to calculate all the conjugates?