[GAP Forum] Character table of D_{2n}, n odd

[Thomas Breuer]

Dear GAP Forum,

Dan Lanke wrote

> Let n be an odd integer.
> Let D_{2n} denote the dihedral group of order 2n.
> Let T denote the character table of D_{2n}.
> 
> Is it true that, up to isomorphism, D_{2n} is the only group with character
> table T? If no, any suggestions on how to use GAP for finding an example please?

Let $G$ be a group with the given table as character table.
- The kernel of the nontrivial linear character of $G$
  is a normal subgroup $N$, say, of index two.
- $N$ is cyclic:
  For $n = 3$ this is clear,
  for larger $n$ there is a nonlinear irreducible character of $G$
  that takes values of the form $\zeta + \zeta^{-1}$ on $N$,
  where $\zeta$ is a primitive $n$-th root of unity,
  and such values do not lie in a smaller cyclotomic field.
  So $N$ contains elements of order $n$.
- The group is a semidirect product of $N$ and a group of order two.
- Finally, each complement of $N$ in $G$ acts by inverting.
This determines $G$ up to isomorphism.

All the best,
Thomas