The Conway-Wales lattice and the Rudvalis and Tits groups

Abstract

We construct the Conway-Wales lattice from a self-dual even code C over ℤ4 whose binary layers interweave the Hamming codes ℋ7 and ℋ8. Explicit signed symmetries lift every automorphism of the binary residue, and the full signed code group is the coordinate-frame stabilizer of the Gaussian lattice. An integral reflection in an embedded scaled self-dual sublattice supplies an additional symmetry, from which the classical frame exchange is recovered. Counting intrinsically defined frames determines the full Gaussian isometry group, of order 583,704,576,000. Its quotient by the four Gaussian scalars is the Rudvalis simple group, and the derived subgroup of a cross stabilizer is the Tits simple group of order 17,971,200. The construction, automorphism-group determination, and both simplicity proofs are carried out within the lattice, without assuming the known group orders or using a recognition or classification theorem. This deposit contains the unchanged 40-page PDF submitted to arXiv on September 27, 2026, as submission 8136489 (math.GR, math.CO). The PDF bears the arXiv submission stamp. The original arXiv submission uses the arXiv.org perpetual, non-exclusive distribution license.