There are two generators: $a$ and $b$. It is known that $a^4 = b^4 = (ab)^{15} = (ab^2)^6 = (ab^3)^9 = 1$.
Is it possible and how to express relations $(ab^2)^6 = (ab^3)^9 = 1$ through relations $a^4 = b^4 = (ab)^{15} = 1$?
PS In this case, the generators are rotations of adjacent faces of the 2x2 Rubik’s Cube. However, I would like to know the answer without considering this information.