interesting topic about representation theory I have to develop a discussion about representation theory (about 30 pages). My knowledge is very superficial and limited to general representations theory of groups and characters theory. Do you know recommend some interesting topic to be treated?
 A: As you have said "about 30 pages"... I would recommend to prove Burnside theorem

If order of a group $G$ is divisible by at most two primes then $G$ is solvable

using character theory of finite groups...
A: Thirty pages sounds more like a term paper as opposed to a takehome exam :-)
Burnside's $p^aq^b$ Theorem is a fine suggestion, but may be covered in a more advanced course. I once gave myself as an exercise (self-study mode) the task of constructing the character table(s) of a few Mathieu groups. It was fun to do without help from any texts (apart from the construction of the ternary Golay code). Not the most efficient use of my time for sure, but fun anyway. IIRC you need relatively little beyond orthogonality of characters, the construction of the group (together with identifying conjugacy classes, expect to spend some time here). The method for calculating the character of a permutation representation and its (anti)symmetric powers comes in really handy. Something a bit extra was possibly needed, but I don't remember what. 
$M_{11}$ is of order 7920 and $M_{12}$ of order 95040. It's so-so, whether this is a suitable project. Doing $M_{11}$ might be enough. While doing that you would also learn a few tricks about some nice subgroups in there (IIRC some projective linear groups appear). If that sounds interesting, and your professor
doesn't think it would overwhelm you, you could give it a go.
If that sounds like it's too much, then working out the character tables of symmeric groups, say, up to $S_7$ or $S_8$ could also be fun. That has a higher risk of being included in a more advanced course. But also has been carried out in many texts (though probably at $S_6$ or $S_7$ they turn into Young tableau and such, which may, again be a bit too much), so if you prefer to heavily rely on the literature, then this is an option.
Still another possibility would be to produce the character tables of the symmetry groups of the Platonic polyhedra. The groups themselves won't be too daunting (might be best here), but not too trivial either. The interplay of combinatorial aspects (permutation representations on various parts of the polyhedra) as well as the geometric aspects of the "defining" 3-dimensional representation make this IMVHO a nice topic. Undoubtedly the dihedral groups were
done in class, so moving up a dimension would be logical..
But you should consult your professor. For example, your projected grade may depend on how difficult a project you undertake. That is between you and your professor - can't help you there.
