On sharp characters of type {−1, 0, 2}

For a complex character χ of a finite group G , it is known that the product sh ( χ ) = ∏ l ∈ L ( χ ) ( χ ( 1 ) − l ) is a multiple of ∣ G ∣, where L (χ) is the image of χ on G − {1} The character χ is said to be a sharp character of type L if L = L (χ) and sh(χ) = ∣ G ∣. If the principal character of G is not an irreducible constituent of χ, then the character χ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups G with normalized sharp characters of type {−1, 0, 2}. Here we prove that such a group with nontrivial center is isomorphic to the dihedral group of order 12.