Initial caveat: the following question could probably be answered by Google, MathSciNet or my library, if I could find the right search terms or book... but I've not had any luck today, so I hope someone can point me to a reference.

(The question is related to some of my older questions concerning characters of finite groups. All representations/characters are over the complex field.)

I am trying to estimate a certain invariant associated to finite groups, and recently thought that a useful toy example to play with would be $$ G = \left\{ \left( \begin{matrix} a & b \\ 0 & 1 \end{matrix} \right) \;\mid\; a,b\in {\mathbb Z}/p^n{\mathbb Z}, p \nmid a \right\} $$ where $p$ is a prime and $n\geq 1$. I guess this might be called the "affine group" of the ring $R={\mathbb Z}/p^n{\mathbb Z}$?

Now the invariant can be calculated pretty easily (modulo tedious sums) once we know the character table of $G$, but this means more than knowing the degrees of the irreducible characters; I need to know the values they take on the various conjugacy classes inside $G$.

For $n=1$ this does not take long to do directly and can also be found as an example in various introductory-level textbooks on representation theory. However, for $n=2$ the best I could find was a section in a paper of several authors, where they just work out the character table by hand after first finding the characters via induction ($G$ is a semi-direct product arising from the action of the group of units in $R$ on the additive group of $R$). Now since I want to continue to higher $n$, I seem to be faced with three options:

1) Slog through the computation myself (which is probably good for my mathematical soul, but takes time & brainpower I need to spend on other things)

2) Learn how to ask a computer to do this (see previous parenthetical remark)

3) Find a reference which just gives the table.

So before embarking on 1), I thought I'd ask here. Most sources I could find from a crude skim online and in my library only discussed linear groups over finite fields; but I'm hoping that the construction here is sufficiently natural that it might have been treated already and written up somewhere.

being bold(wait, that's not our motto! but anyway) I changed it back to \nmid to see what that looks like. My answer: it looks much better. – Pete L. Clark Oct 30 '10 at 8:03ifthe post becomes CW, which hasn't happened yet, I would hesitate to call it "injustice". After all, this is a Q&A site, he asked a question, and he has already received an excellent answer. That's justice, right? – Pete L. Clark Oct 30 '10 at 8:24