Finite [p-groups](http://en.wikipedia.org/wiki/P-group) - have p^n elements by definition. According to WP there is rich structure theory.


**Question:** How far is representation theory of p-groups is understood? 


In case this question is too broad let me restrict it to upper triangular matrices over F\_q.
What is known about its irreps ? Dimensions ? Constructions ? Action of Out(G) ? Characters ? 


E.g. 4x4 matrices over F\_2 - this group has order 64. So dimensions of irreps over C are 1,2,4 by trivial reasons that 1) dim | order of group and 2) \sum dim^2 = order of group. 
Is it possible to illustrate the general theory (if it exists) on this example to compute number
of different irreps of for this group ? 




---


Remarks:


Notation: let me denote U\_n(F\_q) = upper triangular matrices over F\_q with units on the diagonal.


Some trivial examples: U\_2(F\_2) = Z/2; U\_3(F\_2) = D\_8 - dihedral group with 8 elements.


Remark: U\_n(F\_q) is p-Sylow subgroup of GL\_n(F\_q). Over F\_2 it will also be Borel subgroup. 



Representation theory of p-groups in particular upper tringular matrices over F_p 
(This answer is only about *representation theory over $\mathbb{C}$*, that is, character theory.) You should take a look at §26: *Characters of $p$-groups* of Bertram Huppert's book *Character Theory of Finite Groups*. In particular, it contains a proof of Isaacs' results that the set of character degrees of a $p$-group is rather arbitrary (Proc. Amer. Math. Soc. 96 (1986), p.551-552, [doi: 10.2307/2046302](http://dx.doi.org/10.2307/2046302)) and that character degrees of groups of the form $1+ \mathbf{J}(A)$, where $A$ is a finite algebra over the field $F$ with $q$ elements, are powers of $q$ (J. Algebra 177 (1995), p. 708-730, [doi: 10.1006/jabr.1995.1325](http://dx.doi.org/10.1006/jabr.1995.1325)), and a proof that the character degrees of $U\_n(F\_q)$ are the powers $q^i$ for $0\leq i\leq \lfloor \frac{(n-1)^2}{4}\rfloor$. (A result of Huppert, Arch. Math. 59 (1992), p. 313-318, [doi: 10.1007/BF01197044](http://dx.doi.org/10.1007/BF01197044).)  

There is also an elaborate theory of "supercharacters" and "superclasses" for the upper triangular group (and other groups). A good place to start reading is, I think (but I'm not an expert), the paper of Diaconis and Isaacs: *Supercharacters and superclasses for algebra groups*, Trans. Amer. Math. Soc. 360 (2008), p.2359-2392, [MR2373317](http://www.ams.org/mathscinet-getitem?mr=2373317) ([doi](http://dx.doi.org/10.1090/S0002-9947-07-04365-6)). There are, by now, many papers about the character theory of the upper triangular group and related topics, which is in part motivated by Higman's conjecture that for every $n$, the number of conjugacy classes of $U\_n(F\_q)$ is a polynomial in $q$ with integer coefficients.


