Integer Group Determinants for All Groups of Order 27

We determine the integer group-determinant sets of the elementary abelian group C₃³, the group C₉ × C₃, and the nonabelian group of order 27 and exponent 9. For C₃³, the two exceptional divisible strata have 3-adic valuations 18 and 19, and every multiple of 3²⁰ occurs. For C₉ × C₃, we give prime-divisor criteria at valuations 9 and 10, an unrestricted tail 3¹¹ℤ, and an exact coprime criterion on 81 states with a uniform bound of eight field-prime factors. For the nonabelian group, the divisible part is exactly 3⁹ℤ, and the coprime part is characterized by an explicit condition on rational prime exponents. The proofs combine integral Fourier inversion, complete unit images modulo conductors, valuation arguments, and global lifting. All required finite tables and constructions are supplied with exact reproducible certificates. We also give a corrected ordering for one explicit Heisenberg-group construction in the existing preprint literature, without changing its determinant set. Together with Mossinghoff–Pinner's classifications of C₂₇ and the Heisenberg group, these results determine the integer group determinants for every group of order 27.