The K phase, Mn77Fe4Si~9, is monoclinic, a = 13.362 (1), b = 11.645 (1), c = 8.734 (1) × 2 ,~,,/~ = 90-53 (1) °, 110 x 2 atoms per unit cell, space group C2 (C~). Reflections with odd / are quite weak, indicating that the full cell represents a superstructure. The structure model for the substructure was derived from the aspects of the diffraction pattern that resemble the o-FeCr and the 6-MoNi patterns, and with the help of a threedimensional Patterson function. It was refined by full-matrix anisotropic least squares with Mo Kc~ diffractometer data to R(F) = 0.13 (all 3244 reflections) or 0-06 (1932 reflections with I > 2o). The substructure has rumpled layers and is entirely 'tetrahedrally close-packed' (tcp), with Mn(Fe) in the positions of coordination numbers (CN) 16, 15, and 14, and with mixtures of Mn(Fe) and Si in the positions of CN 12. Since the substructure did not show evidence of 'elongated atoms' (marked apparent thermal anisotropies) the superstructure was assumed to be substitutional in regard to occupancy of CN 12 sites by Mn(Fe) or Si. The superstructure was solved with the aid of a Patterson function calculated with superstructure data only, and with least-squares refinement of site occupancies for the CN 12 atoms. Different CN 12 sites were found to be occupied by Mn(Fe) atoms, Si atoms, or mixtures of Mn(Fe) and Si. The alteration between the two subcells is such as to allow Si atoms to be everywhere coordinated only by Mn(Fe). In this the K phase resembles the v phase, Mn82Si~8, and the X phase, MN45Co40Sils, with plane-layered tcp structures, but differs from the D phase, MnsSi 2, with a somewhat 'non-ideal' tcp structure in which there are a few Si-Si contacts.