By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fai...

Let G be a finite group of order n and V a simple C[G]-module of dimension d. For some nonnegative number e, we have n = d(d + e). If e is small, then the character of V has unusually large degree. We fix e and attempt to classify such groups. For e ≤ 3 we give a complete classification. For any other fixed e we show that there are only finitely many examples.