Finite p-groups – 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. 4×4 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.
Best Answer
(This answer is only about representation theory over $\mathbb{C}$, that is, character theory.) You should take a look at ยง26: 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) 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), 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.)
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 (doi). 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.