Pairs of non-isomorphic monomial groups with identical character tables (Q1898863)

\textit{S. Mattarei} [J. Lond. Math. Soc., II. Ser. 46, 92-100 (1992; Zbl 0792.20007)] constructed a family of pairs of non isomorphic solvable groups that have identical character tables. He also mentions that his ``character table isomorphism'' does also commute with power maps, which can be seen easily when using his Ph.D. thesis [``Retrieving information about a group from its character table'' (Univ. Warwick, 1992)]. This paper applies \textit{S. B. Conlon}'s explicit algorithm [J. Symb. Comput. 9, 551-570 (1990; Zbl 0733.20006)] on his abstractly given groups and computes the number and degrees of all regular irreducible characters of each pair of them. This information is then used to show that Mattarei did actually construct pairs of monomial groups and linear characters of which groups can be used to construct the irreducible characters of his groups.