Consider the group $G=C_3\rtimes C_9$, where $\rtimes$ denotes the semidirect product and $C_n$ denotes the cyclic group with $n$ elements.
Can this group elements be describe using matrices with respect to regular matrix multiplication? In not, is there an easy way to understand its elements? Thanks.
There is only isomorphism class of nontrivial semidirect products $C_9 \rtimes C_3$ (where nontrivial means not isomorphic to the direct product), and such a group is defined by the presentation $\langle x,y | x^9=y^3=1, y^{-1}xy=x^4 \rangle$.
This group can be represented by $3 \times 3$ complex matrices, with $$x \to \left(\begin{array}{ccc}0&0&1\\w&0&0\\0&1&0\end{array}\right), \ \ \ \ y \to \left(\begin{array}{ccc}w&0&0\\0&w^2&0\\0&0&1\end{array}\right),$$ where $w$ is a primitive cube root of $1$.