  Noncollinear antiferromagnets have recently been attracting considerable
interest partly due to recent surprising discoveries of the anomalous Hall
effect (AHE) in them and partly because they have promising applications in
antiferromagnetic spintronics. Here we study the anomalous Nernst effect (ANE),
a phenomenon having the same origin as the AHE, and also the spin Nernst effect
(SNE) as well as AHE and the spin Hall effect (SHE) in noncollinear
antiferromagnetic Mn$_3X$ ($X$ = Sn, Ge, Ga) within the Berry phase formalism
based on {\it ab initio} relativistic band structure calculations. For
comparison, we also calculate the anomalous Nernst conductivity (ANC) and
anomalous Hall conductivity (AHC) of ferromagnetic iron as well as the spin
Nernst conductivity (SNC) of platinum metal. Remarkably, the calculated ANC at
room temperature (300 K) for all three alloys is huge, being up to 5 times
larger than that of iron. Moreover, the calculated SNC for Mn$_3$Sn and
Mn$_3$Ga is also large, being as large as that of platinum. This suggests that
these anitferromagnets would be useful materials for thermoelectronic devices
and spin caloritronic devices. The calculated ANC of Mn$_3$Sn and iron are in
reasonably good agreement with the very recent experiments. The calculated SNC
of platinum also agrees with the very recent experiments in both sign and
magnitude. The calculated thermoelectric and thermomagnetic properties are
analyzed in terms of the band structures as well as the energy-dependent AHC,
ANC, SNC and spin Hall conductivity via the Mott relations.
