On sharp characters of rank 2 with rational values

For a finite group G and its character χ, let L be the set of distinct values of χ on non-identity elements of G and fL(χ(1)) denote the monic polynomial of least degree having L as its set of roots. Blichfeldt [A theorem concerning the invariants of linear homogeneous groups with some applications to substitution groups, Trans. Amer. Math. Soc. 5 (1904) 461–466. [2]] showed that fL(χ(1))/G is a rational integer. Cameron and Kiyota [Sharp character of finite groups, J. Algebra 115 (1988) 125–143] called the pair (G,χ) L-sharp if G=fL(χ(1)) and posed the problem of determining all the L-sharp pairs (G,χ) for a given set L. For several cases those problems have already been studied by many authors. Our purpose is to determine the sharp pairs of types {−2,1} and {−1,2} having non-trivial center.