The Brauer–Picard Groupoid of Pointed Fusion Categories of Dimension 27

We give an explicit description of the ordinary Brauer–Picard 1-groupoid of pointed complex fusion categories of dimension 27. Starting from the established 61-object tensor census and 47-component Morita partition, we realize the full groupoid through faithful models for the base groups and explicit connecting bimodules in compatible cocycle conventions. The description rests on three constructions: finite metric isometries, determinant-one quadratic stabilizers, and induction with Fourier completion. The metric models preserve the nonsplit charge–flux extensions, while four cyclic restrictions recognize the Heisenberg classes in characteristic three. Fourteen invertible connectors form a spanning forest, with support order 81 on every edge; exact center spectra give an independent check of its components. Choosing paths then gives every arrow a unique coordinate in a base group, with chronological composition given by hg. We also give a restricted relative-product construction that makes realization and recognition effective for exact algebraic records. Throughout, we work with pointed objects and bimodule equivalence classes at the fixed prime 3.