Two subgroups of observations have sizes $n_1 = 10$ and $n_2 = 10$, with means $\bar{x}_1 = 100$ and $\bar{x}_2 = 120$, and standard deviations $s_1 = 8$ and $s_2 = 6$. What is the population variance $\sigma_c^2$ of the combined group of $20$ observations?