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Two compounds, ${\mathrm{Mn}}_{3}\mathrm{CoSi}$ and ${\mathrm{Mn}}_{3}\mathrm{CoGe}$ have been synthesized and found to crystallize in the ${\mathrm{AlAu}}_{4}$ type structure, an ordered form of the $\ensuremath{\beta}\ensuremath{-}\mathrm{Mn}$ structure. The magnetic structure and properties have been studied by magnetometry and neutron powder diffraction and the theoretical work is based on first principles total energy calculations. Comparison is made with the magnetic properties of the isostructural compounds ${\mathrm{Mn}}_{3}\mathrm{IrGe}$ and ${\mathrm{Mn}}_{3}\mathrm{IrSi}$. The solid solutions ${\mathrm{Mn}}_{3}{\mathrm{Ir}}_{1\ensuremath{-}y}{\mathrm{Co}}_{y}\mathrm{Si}$ $(0\ensuremath{\leqslant}y\ensuremath{\leqslant}1)$ and ${\mathrm{Mn}}_{3}{\mathrm{CoSi}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$ $(0\ensuremath{\leqslant}x\ensuremath{\leqslant}1)$ are also studied. A noncollinear antiferromagnetic structure is experimentally observed for $y=0.20$ as well as for $x=0.50$ and 1.0, similar to that of ${\mathrm{Mn}}_{3}\mathrm{IrSi}$ and ${\mathrm{Mn}}_{3}\mathrm{IrGe}$, with 120\ifmmode^\circ\else\textdegree\fi{} angles between magnetic moments on a triangular network of Mn atoms, and this finding is corroborated by theoretical calculations. For $y=0.80\text{--}1.0$ a transformation to a new type of magnetic structure takes place. The magnetic transition temperature decreases on decreasing unit cell dimension, with good qualitative agreement with the decay of the calculated interatomic exchange energy. Both theory and experiments find the magnitude of the Mn magnetic moment to decrease on decreasing unit cell volume, the same trend is found in calculations for $\ensuremath{\beta}\ensuremath{-}\mathrm{Mn}$.

A new ternary Ir-Mn-Si phase with stoichiometry ${\mathrm{Mn}}_{3}\mathrm{IrSi}$ has been synthesized and found to crystallize in the cubic ${\mathrm{AlAu}}_{4}$-type structure, space group ${P2}_{1}3$ with $Z=4,$ which is an ordered form of the \ensuremath{\beta}-Mn structure. The unit cell dimension was determined by x-ray powder diffraction to $a=6.4973(3)\AA{}.$ In addition to the crystal structure, we have determined the magnetic structure and properties using superconducting quantum interference device magnetometry and Rietveld refinements of neutron powder diffraction data. A complex noncollinear magnetic structure is found, with magnetic moments of $2.97(4){\ensuremath{\mu}}_{B}$ at 10 K only on the Mn atoms. The crystal structure consists of a triangular network built up by Mn atoms, on which the moments are rotated $120\ifmmode^\circ\else\textdegree\fi{}$ around the triangle axes. The magnetic unit cell is the same as the crystallographic and carries no net magnetic moment. The N\'eel temperature was determined to be 210 K. A first-principles study, based on density functional theory in a general noncollinear formulation, reproduces the experimental results with good agreement. The observed magnetic structure is argued to be the result of frustration of antiferromagnetic couplings by the triangular geometry.