Lower Bounds for the Finite Quotients of Conder's Group on the Cubic Tree (Kourovka Notebook Problem 16.46)

Problem 16.46 of the Kourovka Notebook, posed by M. Conder and going back to his paper of 1992, concerns the free product G = G₃ ∗G₁ G₄ of two of the Djoković–Miller groups that act arc-transitively on the cubic tree, amalgamated along the modular group G₁, and the normal subgroup K of index 8 of G, which has a presentation with six generators and 22 relators: does K have a non-trivial finite quotient? Conder found no proper subgroup of finite index. We do not decide the question. We prove, without computation, that K is perfect, that in every non-trivial image of K the generators keep their orders and the symmetric group S₄ embeds, and that K is the amalgamated free product of four copies of L = S₄ ∗D₈ S₄ over a common subgroup C₃ ∗ C₃, glued along its four sign automorphisms; a non-trivial finite quotient of K gives one of the subgroup J generated by two of these copies. By exhaustive computer searches we show that J, and hence K, has no non-trivial action on at most 36 points, so K has no subgroup of index between 2 and 36, and, with GAP, that no non-abelian finite simple group of order less than 10⁸ receives a non-trivial homomorphism from J. Hence every non-trivial finite quotient of K has order at least 10⁸. Jambor (2015) proved, by a machine computation, that K has no quotient PSL₂(q) or PGL(2,q); together with a search of the twelve other simple groups of order between 10⁸ and 10⁹, every non-trivial finite quotient of K has order at least 10⁹. In contrast, the amalgam of two copies differing by the twist (A,B) ↦ (A⁻¹,B⁻¹) maps onto A₆ and PSL(2,7). The bounds are computer-assisted; the question itself remains open. This is an unrefereed note. 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-16.46 (Kourovka Notebook, Problem 16.46, M. Conder, 16th issue, 2006).