Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$ Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$

My thoughts:
$|a|=|b|=2\implies a^2=e$ and $b^2=e$
I see that the group cannot be abelian as the order wont be greater than $2$ :  $(ab)^2 = a^2b^2=e$
Not really sure how to proceed further. Greatly appreciate any help. Thanks!

Hints: Suppose the group is finite, what do you know about its order? 
Try writing the relations in terms of $a$ and $c=ab$. I use $1$ for the identity below.

 You should get $a^2=c^5=1$. The third relation $b^2=1$ can be rewritten by noting that $b=a^2b=ac$ so that $acac=1$ and $ca=a^2cac^5=a(acac)c^4=ac^4$ so that every element can be written in the form $a^pc^q$ with $0\le p\lt 2$ and $0\le q \lt 5$

