Same Hall-Senior genus not implies character table-equivalent

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., groups having the same Hall-Senior genus) need not satisfy the second group property (i.e., character table-equivalent groups)
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Statement

It is possible to have two finite groups $G_{1}$ and $G_{2}$ such that:

$G_{1}$ and $G_{2}$ are groups having the same Hall-Senior genus.
$G_{1}$ and $G_{2}$ are not character table-equivalent groups, in fact, we can arrange our example so that the field generated by character values is different for $G_{1}$ and $G_{2}$ .

Proof

The smallest example occurs with the three maximal class groups of order 16: dihedral group:D16, semidihedral group:SD16, and generalized quaternion group:Q16. It turns out that $D_{16}$ and $Q_{16}$ are character table-equivalent, but $S D_{16}$ is not character table-equivalent to either. The field generated by character values for the three groups are described below:

Group	Field generated by character values	Information on linear representation theory
dihedral group:D16	$Q (\sqrt{2})$	linear representation theory of dihedral group:D16
semidihedral group:SD16	$Q (\sqrt{- 2})$	linear representation theory of semidihedral group:SD16
generalized quaternion group:Q16	$Q (\sqrt{2})$	linear representation theory of generalized quaternion group:Q16