Say I have a group with the following presentation: $$ G = \langle a,b \mid a^2 = b^3 = (ab)^3 = e \rangle $$
During a conversation someone had mentioned that the order for $G$ must be less than or equal to $12$. I couldn't follow the conversation very well, but on trying to figure out where this bound came from I got confused. They seemed to make it sound like there was some certain property that allowed them to calculate this fairly rapidly. Is there some theorem that gives an upper bound to finite groups that are relatively nicely behaved? (Like those with two or maybe three generators).