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Fix an odd prime power $q$, fix a generator of the multiplicative group ${\mathbb F}_q^\times$, let $H$ be the subgroup generated by the square of this element, and form the semi-direct product ${\mathbb F}_q \rtimes H$. This is a subgroup of the full affine group of ${\mathbb F}_q$, and in the ongoing work I'm doing with colleagues, it provides a useful example at one point.

More out of curiosity than anything else, I wondered if this group has a standard name in the literature, or is denoted by a standard symbol? In the current draft of our paper it's denoted, unimaginatively, by $G_q$, but I wouldn't be surprised if that clashes with other notation that's standard in finite group theory.

(I've seen the $q=7$ case in several books, usually as an exercise in determining the character table, but it is only described as the non-abelian group of order 21.)

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    $\begingroup$ Either "the Borel subgroup of ${\rm PSL}_2({\bf F}_q)$", or "the $a^2 x + b$ group" — the latter because it acts on the affine line by permutations of the form $x \mapsto a^2 x + b$ for $a \in {\bf F}_q^*$ and $b \in {\bf F}_q$. $\endgroup$
    – Noam D. Elkies
    Commented Sep 17, 2011 at 4:47
  • $\begingroup$ ...so if you need a symbol for it, probably $B$ (for Borel) or $B_q$. $\endgroup$
    – Noam D. Elkies
    Commented Sep 17, 2011 at 5:18
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    $\begingroup$ This group arises as a large subgroup of the automorphism group of a Paley graph. (It's not the full automorphism group, you have to add the field automorphisms to get that.) In any case it appears in many papers and I have never seen it given a name. And $G_q$ is used for so many things in group theory, your contribution to the pile is unlikely to be remarked on. $\endgroup$
    – Chris Godsil
    Commented Sep 17, 2011 at 21:41
  • $\begingroup$ Paley graphs must have $q \equiv 1 \bmod 4$ (though there's a directed version for $q \equiv -1 \bmod 4$).$$ $$ For $q$ prime, this group is also the Galois group of the cover of modular curves ${\rm X}(q) / {\rm X}_0(q)$ over ${\bf C}$. [This fits in with the ${\rm PSL}_2({\bf F}_q)$ picture.] For higher prime powers there's a similar description using Shimura curves.$$ $$ Given Y.Choi's research interests I wonder where this group arises; neither Paley graphs nor modular curves seems a likely context. $\endgroup$
    – Noam D. Elkies
    Commented Sep 18, 2011 at 0:23
  • $\begingroup$ Thanks for the comments - I guess "a^2x+b$ group would work best for us, since the likely readers of the eventual paper will be abstract harmonic analysis people, who are comfortable with talk of the "ax+b" group over various fields. $\endgroup$
    – Yemon Choi
    Commented Sep 19, 2011 at 22:13

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