Let $1, \omega$ and $\omega^2$ be the cube roots of unity. If $\mathrm{S}$ is the set of all non-singular matrices of the form $\left[\begin{array}{rrr}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]$ where $a, b, c \in\left\{\omega, \omega^2\right\}$, then the number of elements in $\mathrm{S}$ is