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!

Hint: A regular pentagon can be rotated and reflected
