> 
> $G=\langle a,b\rangle$. I have to prove that $|G|=8$ if 
> 
> 
> 1. $|a|=|b|=4$
> 2. $a^2=b^2$
> 3. $ba=a^3 b=a^{-1}b$
> 
> 
> 

$G$ is not abelian. I tried generating all terms of $G$ and then equating some of them, but the number of terms I would have to generate grew extremely rapidly. 

I then thought this is equivalent to proving $$\left|\frac{\langle a,b\rangle}{\langle a^4,b^4,a^2b^{-2},bab^{-1}a^{-3},baba,a^3 b^2 a\rangle}\right|=8$$

Is this assertion correct. Also, how can I go about proving this?