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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?

Maths Rahul
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    Not particularly insightful (or an answer), but using "IsFrobenius(G);" on Magma I get that SmallGroup(1029,i) is Frobenius for i= 6, 9, 10, 17. – AnalysisStudent0414 Sep 26 '23 at 09:16
  • $G[11]$ is a Frobenius group with kernel of order $343$. $G[13]$ has elements of order $21$ and is not a Frobenius group. Why don't you just use GAP to get presentations? – Derek Holt Sep 26 '23 at 10:50

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