Groups with same character table need not be isomorphic

This fact is related to: linear representation theory
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Statement

Two groups with the same values in their character tables need not be isomorphic.

In other words, it is possible to have two character table-equivalent groups that are not isomorphic.

Example

The dihedral group:D8 and quaternion group:Q8 have the same values in their character tables but are not isomorphic groups. See linear representation theory of quaternion group and linear representation theory of dihedral group:D8 for further details.