It is well known that the complex character table does not determine the group. The classical example always given is of a pair of groups of order $8$, the dihedral group $D_8$ and the Quaternion group $Q_8$.

However these have different real representation theory - $\mathbb{R}[Q_8] = \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{H}$ and $\mathbb{R}[D_8] = \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times M_2(\mathbb{R})$.

This can be deduced from the complex character table as the Frobenius-Schur index of the 2 dimensional representation is 1 for $D_8$ and -1 for $Q_8$.

This then gives that the groups are non isomorphic.

I was wondering what the smallest (or just any) example of two non isomorphic groups with identical character tables and Frobenius Schur indicators was.

Many thanks in advance!

  • $\begingroup$ @peterag no problem, thanks $\endgroup$ – James Dec 31 '19 at 15:58

The two nonisomorphic nonabelian groups of order $p^3$ for odd primes $p$ have the same character tables and the same Frobenius-Schur indicators - these are $0$ for all except the trivial character. I would guess that the nonabelian groups of order $27$ are the smallest such examples, but I haven't checked.

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