A Semi-Abelian Group of Order 768 That Is Not an M-Group

Abstract

M. Kida conjectured that every finite semi-abelian group is monomial (J. Group Theory, 2025; Kourovka Notebook, Problem 21.68). We show that the conjecture is false. If a finite group W acts on a finite abelian group N, a linear character of N has stabiliser T in W, and T has an irreducible character that is not monomial, then N ⋊ W is not an M-group. We apply this to the augmentation submodule of the permutation module F_2[W/T], where W = E ⋊ A_4 is the index-two subgroup of C_2 ≀ A_4 and T ≅ SL(2,3) is the binary tetrahedral group acting on the eight quaternion units. The result is a semi-abelian group of order 768 = 2^8·3 with a non-monomial irreducible character of degree 8. The proof uses only Clifford theory. The example was also checked by exact computation, including every subgroup of index 8. This is an unrefereed note; minimality of the order is not claimed. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-21.68. Paper page: https://eulersolve.org/papers/kou-21-68/