# Example where character table and Frobenius-Schur index doesn't determine the group.

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.

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.