We use a Fourier transform spectrometer based on a difference frequency generation optical frequency comb to measure high-resolution, low-pressure, room-temperature spectra of methane in the 1250 - 1380 cm−1 range. From these spectra, we retrieve line positions and intensities of 678 lines of two isotopologues: 157 lines from the 12CH4 ν4 fundamental band, 131 lines from the 13CH4 ν4 fundamental band, as well as 390 lines from two 12CH4 hot bands, ν2 + ν4 - ν2 and 2ν4 - ν4. For another 165 lines from the 12CH4 ν4 fundamental band we retrieve line positions only. The uncertainties of the line positions range from 0.19 to 2.3 MHz, and their median value is reduced by a factor of 18 and 59 compared to the previously available data for the 12CH4 fundamental and hot bands, respectively, obtained from conventional FTIR absorption measurements. The new line positions are included in the global models of the spectrum of both methane isotopologues, and the fit residuals are reduced by a factor of 8 compared to previous absorption data, and 20 compared to emission data. The experimental line intensities have relative uncertainties in the range of 1.5 - 7.7%, similar to those in the previously available data; 235 new 12CH4 line intensities are included in the global model.