In the list of groups in GAP of order $1029=7^3\cdot 3$, there are two, with structure description $U_3(\mathbb{F}_7)\rtimes C_3$.
(Among $19$ groups $G[1], G[2], \ldots, G[19]$ of order $1029$, the groups $G[11]$ and $G[13]$ have structure description of above type.)
Question: Which groups among $G[11]$ and $G[13]$ are Frobenius groups? What is presentation of them?
The Frobenius group, whose presentation I know is generated by $x,y,z,t$ where $(\langle z\rangle\times \langle y\rangle)\rtimes \langle x\rangle$ is $U_3(\mathbb{F}_7)$ with $z$ in its center and $t$ normalizes this subgroup by $t^{-1}xt=x^2$, $t^{-1}yt=y^2$. This is the smallest possible order of a Frobenius group with non-abelian kernel.
I was unable to identify this group with $G[11]$ and $G[13]$.
Further, I wanted to know whether both $G[11]$ and $G[13]$ are Frobenius groups?