I am doing an exercise that asks me to find what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to.
I am allowed to only use the groups $D_n,S_n,\mathbb{Z}_n$ and the direct sums ) where $S_n$ is the permutatin group of $n$ elements, $D_n$ is the dihedral group of order $2n$.
I have noted that the first element is of order $3$ and that the second one is of order $2$. I also noted that these elements commutes hence generate an abelian group. I can also say that this group is of order at least $6$ since $gcd(2,3)=1$.
How can I find what is Finding $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ ? If there was a good argument that say that this group is at most of order $6$ then I can clain that since the only groups of order $6$ are $S_3$ and $\mathbb{Z}_6$ and $S_3$ is non abelian then this group is isomorphic to $\mathbb{Z}_6$.
Can someone please help with this problem ?