Finding what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to

You already did much of the work when you calculated the product of the two elements.

$(135)(246)(12)(34)(56)=(145236)$, which clearly has order $6$. On the other hand, if this permutation is $\pi$, it’s easy to check that $\pi^3=(12)(34)(56)$ and $\pi^4=(135)(246)$, so $\pi$ generates the same subgroup.