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.



Character table for the affine group of Z/p^nZ
The groups you are interested in are sometimes called false Tate extensions in number theorists' jargon. They are Galois groups of the Galois closures of extensions of $\mathbb{Q}$ obtained by adjoining the $p^n$-th roots of a $p$-th power free element. The irreducible representations are very explicitly described in [Vladimir Dokchitser's PhD thesis](http://www.dpmms.cam.ac.uk/~vd209/phdweb.ps), beginning of chapter 3.


Alternatively, the characters, together with their values, are easy to compute using a procedure described in Serre's Linear representation book, part II, Section 8.2. He explains how to obtain all the irreducible characters of any group that is a semi-direct product when the normal subgroup is abelian. Note that the character of an induced representation is easy to compute in terms of the original character and the coset-action.


And of course one more remark: for any given $p$ and $n$, MAGMA will just give you the character table.


Edit: Since you seem unsure, how to ask a computer for the character table, here is MAGMA code as an example. You will easily adopt it to any other package that handles character tables:



```
p:=3; n:=2;
Z:=Integers();
gl:=GL(2,quo<Z|p^n*Z>);
pr:=PrimitiveRoot(p^n);

G:=sub<gl|gl![[1,1],[0,1]],gl![[pr,0],[0,1]]>;
CharacterTable(G);

```

