What is this group $G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle$

I don't know about "known groups". I can answer questions about its properties.

For example it is virtually abelian: the subgroup $H = \langle ab, cac^{-1}a \rangle$ is free abelian and normal in $G$ with $G/H \cong C_2^2$.

You can see directly from the presentation that the subgroup $\langle ab \rangle$ is normal with quotient group the infinite dihedral group generated by the images of $a$ and $c$.