Groups with Sharp Character of Type $$\lbrace -1,1,3\rbrace $$

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