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Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is prime? I can't find anything about it.

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    $\begingroup$ What do you want to know about these group actions? Please try to find a more focused question $\endgroup$
    – Yemon Choi
    Apr 22, 2015 at 11:50
  • $\begingroup$ $C_p^*$ is particularly awkward because it makes use of a choice of field structure on $C_p$ (which is determined only after the choice of some generator devoted to be the multiplicative unit) $\endgroup$
    – YCor
    Apr 22, 2015 at 15:09
  • $\begingroup$ I think the most natural action of the group you say is permutations of a set with $p$ elements (it's the "affine general linear group" AGL(1,$p$)). This action is easy to understand. So the group could permute $p$ of the variables in a nice way... $\endgroup$ Apr 22, 2015 at 15:34

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