Magnetic properties of the honeycomb oxide <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="bold">Na</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="bold">Co</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="bold">TeO</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>

We have studied the magnetic properties of ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$, which features a honeycomb lattice of magnetic ${\mathrm{Co}}^{2+}$ ions, through macroscopic characterization and neutron diffraction on a powder sample. We have shown that this material orders in a zigzag antiferromagnetic structure. In addition to allowing a linear magnetoelectric coupling, this magnetic arrangement displays very peculiar spatial magnetic correlations, larger in the honeycomb planes than between the planes, which do not evolve with the temperature. We have investigated this behavior by classical Monte Carlo calculations using the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ model on a honeycomb lattice with a small interplane interaction. Our model reproduces the experimental neutron structure factor, although its absence of temperature evolution must be due to additional ingredients, such as chemical disorder or quantum fluctuations enhanced by the proximity to a phase boundary.