Groups with Sharp Character of Type { -1,1,3}

For a finite group G and its character χ , let L_χ be the image of χ on G-{ 1} . The pair (G,χ ) is said to be sharp of type L if |G|=Π _a∈ L(χ (1)-a) , where L=L_χ . The pair (G,χ ) is said to be normalized if the principal character of G is not an irreducible constituent of χ . In this paper, we study normalized sharp pairs of type L={ -1,1,3} proposed by Cameron and Kiyota in [J Algebra 115(1):125–143, 1988], under some additional hypotheses.