# Column sum of character table

Consider character table of a finite group, whose rows are indexed by irreducible $\mathbb{C}$-characters and columns by conjugacy classes. Then Huppert says in his book on character theory:

(1) For any row, the sum of entries is always non-negative integer.

(2) The sum of column entries is an integer, and may be negative.

(3) The Mathieu group $M_{11}$ has a conjugacy class such that its column sum is negative.

Q. My question is related to (3); isn't there any smaller order group for which column sum is negative?

## 1 Answer

A routine computer search shows that the smallest example has order $96$. It is the group $\mathtt{SmallGroup}(96,3)$ in the small groups database, and there is an conjugacy class of elements of order $2$ for which the sum of character values is $-2$.

This group has centre of order $2$, with a normal Sylow $2$-subgroup $P$ with structure $2.(4 \times 4)$ with an element of order $3$ acting on $P$ and centralizing only the group centre. So the group has structure $2.(4 \times 4).3$.