Groups with representation $\langle a, b; a^n=1, ab=b^3a^3\rangle$

Established that for $n = 4$ and $n\ge 7$ group $G(n) = \langle a, b; a^n=1, ab=b^3a^3\rangle$ are infinite, and for the remaining $n$ evaluated the procedure and investigate the structure of the group $G(n)$.