Q:

Showing that a quotient group $G/N$ is isomorphic to $\mathbb{Z}_3$

I have permutations $\sigma=(135)(27)$, and $\tau = (27)(468)$. $G =\langle \sigma,\tau \rangle$ and $N$ is the smallest subgroup of $G$ that contains $\tau$, so $N = \langle \tau \rangle$. $|\sigma| = |\tau| = 6$. I have to show that $G/N$ is isomorphic to $\mathbb{Z}_3$, but I don't know how to do this in an 'easy' practical way (not theoretical approach). 
Normally I start by looking at the quotient groups order, and go by that but I don't see any obvious approach to determine $G$'s order. To start off with, how do I find $G$'s order in a quick way? 

A:

Hint: $G = \langle (135),(27),(468)\rangle$ and $N = \langle (27),(468)\rangle$.

