Conjugacy class of more than average size has character value zero for some irreducible character
From Groupprops
Statement
Suppose is a finite group and
is a conjugacy class in
such that
has size more than the average conjugacy class size, i.e.,
is less than the number of conjugacy classes in
.
Then, there exists an irreducible representation of over the complex numbers with character
such that
for some
.