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Optical properties of carbon nanotubes (CNTs) have been extensively studied, and much basic knowledge has been accumulated on how light emission, scattering, and absorption occur in CNTs. However, their nonlinear optical properties remain largely unexplored. Here, we have observed third harmonic generation from highly-aligned CNTs on sapphire with linearly-polarized intense mid-infrared radiation. Through polarization-dependent experiments, we extracted the nonzero tensor elements of χ(3). The contribution of the weaker tensor elements to the overall χ(3) signal has been calculated to be approximately 1/3 of that of the dominant χ(3)zzzz component, which is consistent with theory and other measured values.