Let $G$ be a group with generators and relations.
I know that in general it is difficult to determine what a group is from its generators and relations. I am interested in learning about techniques for figuring out the order of a group from the given information.
For example, I know that if the number of generators exceeds the number of relations then the group has infinite order. If the number of generators equals the number of relations then the group is cyclic or has infinite order.
Let $G= <x, y|x^2 = y^3 = (xy)^4 = 1>$. My hunch is that G has finite order because $(xy)^4$ is somehow independent of $x^2$ and $y^3$. But if the exponent on $xy$ were bigger, say $(xy)^6=1$ that relation becomes redundant.
My question is: is this sort of thinking correct? Furthermore: my method will only tell me if $G$, or its modification, is finite (or infinite). If $G$ is finite how can I figure out the order of the group? I know that the orders divide the order of the group, but I am looking for a specific number.