1. **State the problem:** We are given a complex number $\omega \neq 1$ that satisfies the equation $$z^7 - 1 = 0.$$ We want to factorize the expression $$\omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1.$$\n\n2. **Recall the roots of unity and factorization:** The equation $$z^7 - 1 = 0$$ has 7 roots, the 7th roots of unity, given by $$z = e^{2\pi i k/7}$$ for $k=0,1,2,\ldots,6$. The root $z=1$ corresponds to $k=0$. The other roots satisfy $$\frac{z^7 - 1}{z - 1} = z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0.$$\n\n3. **Use the factorization formula:** We know that $$z^7 - 1 = (z - 1)(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1).$$ Since $\omega \neq 1$ is a root of $z^7 - 1 = 0$, it must satisfy $$\omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0.$$\n\n4. **Factorization of the expression:** The expression $$\omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1$$ is exactly the factor that appears when dividing $z^7 - 1$ by $z - 1$. It can be factorized into linear factors corresponding to the other 6 roots of unity: $$z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = \prod_{k=1}^6 (z - \omega^k).$$\n\n5. **Final answer:** Since $\omega$ is one of these roots (with $\omega \neq 1$), the expression factorizes as $$\boxed{\omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0}.$$