1
$\begingroup$

Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise 28.2 of Representations of characters of groups by James and Liebeck).

Let $\phi$ be a field automorphism of $\mathrm{SL}_2(q)$ of order $f$, so $\mathrm{P\Gamma L}_2(q) = \mathrm{SL}_2(q).\langle\phi\rangle$.

Question 1. Do we know the explicit character table of $\mathrm{P\Gamma L}_2(q)$?

More generally, if $G$ is an almost simple group of socle $\mathrm{SL}_2(q)$, then we have $\mathrm{SL}_2(q)\lhd G\leqslant\mathrm{P\Gamma L}_2(q)$ and $G = \mathrm{SL}_2(q).\langle\phi^s\rangle$ for some $s$.

Question 2. Do we know the explicit character table of $\mathrm{SL}_2(q).\langle\phi^s\rangle$?

All of the above character tables are over $\mathbb{C}$.

Improve this question
$\endgroup$
1
  • $\begingroup$ There is a standard method to enumerate the irreps of a semidirect product $H \rtimes A$ with $A$ abelian, which you can read about in many places. $\endgroup$
    – Satan's Minion
    Commented Jan 2 at 19:55

0

Reset to default

You must log in to answer this question.