Character table-equivalent groups

Definition
Suppose $$G_1$$ and $$G_2$$ are finite groups and $$K$$ is a splitting field in characteristic zero for both (we can take $$K = \mathbb{C}$$ or $$\overline{\mathbb{Q}}$$. We say that $$G_1$$ and $$G_2$$ are character table-equivalent groups if there exists a bijection $$\alpha$$ between the set of conjugacy classes in $$G_1$$ and that of $$G_2$$, and a bijection $$\beta$$ between the set of irreducible characters (up to equivalence) of $$G_1$$ and that of $$G_2$$, such that under this pair of bijections, the character table of $$G_1$$ is identified with that of $$G_2$$. In other words, for any character $$\chi$$ of $$G_1$$ and any conjugacy class $$c$$ of $$G_1$$, we have $$\beta(\chi)(\alpha(c)) = \chi(c)$$.

Invariants
Character table-equivalent groups share a number of properties and arithmetic function values.

On the other hand, the following are not determined:


 * Minimal splitting field: We could have two character table-equivalent groups such that they do not have isomorphic minimal splitting fields. For instance, dihedral group:D8 and quaternion group (see linear representation theory of dihedral group:D8 and linear representation theory of quaternion group).