Let $G$ be a finite group (with only two generators and $m=n$) presented as
$$ G = \langle a, b : a^m = b^n = (W(a,b))^p= \ldots\text{other-such-relations}\ldots= 1 \rangle $$
where $m,n,p>1$ , and taking the smallest $p$ for each $W(a,b)$ which is made out of products of $a$ and $b$, e.g. $(ab)^2$, $(ab^2ab^{-1})^3$ etc.
I know three examples
1) Dihedral groups of order $n$: $ G = \langle a, b : a^2 = b^2 = (ab)^n= 1 \rangle $
2) Another two from the following paper (page 2) and presented as :
J. Howie, V. Metaftsis, and R. M. Thomas. Finite generalized triangle groups. Trans. Amer. Math. Soc., 347(9):3613–3623, 1995
$$ G = \langle a, b : a^3 = b^3 = (abab^2)^2= 1 \rangle $$ of order 180 and
$$ G = \langle a, b : a^3 = b^3 = (aba^2b^2)^2= 1 \rangle $$ of order 288.
Now, after going through the list of finite group presentations, I could not find any other finite group with such a presentation (i.e only two generators and $m=n$).
So, are there any other examples? Or is it possible to give arguments why there might not exist any other example?
References will also be useful.
Thank you.