  We generalize Regev's result on a virtual character of $S_n$. Suppose that
$\lambda$ and $\mu$ are integer partitions of $n$. For the associated
irreducible character $\chi^\lambda$ of $S_n$, when $\chi^\lambda(\mu)\neq0$ we
find another partition $\tau$ related to $\lambda$ such that
$\chi^\tau(\mu)\neq0$ by the virtual character. Applying this result, we obtain
a class of nonzero Kronecker coefficients by Pak et al.'s character criterion.
Moreover, we discuss the effectiveness of Pak et al.'s character criterion by a
concrete example.
