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. 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 ?
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.