Is there a standard name for this index 2 subgroup in an affine group over a finite field of odd char?

2011-09-17T04:33:23Z

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.)

Is there a standard name for this index 2 subgroup in an affine group over a finite field of odd char? - MathOverflow

Is there a standard name for this index 2 subgroup in an affine group over a finite field of odd char?