We present a spectroscopic study of the globular clusters (GCs) in the giant elliptical galaxy NGC 4636 in the Virgo cluster. We selected target GC candidates using the Washington photometry derived from the deep CCD images taken at the KPNO 4m. Then we obtained the spectra of 164 target objects in the field of NGC 4636 using the Multi-Object Spectroscopy (MOS) mode of Faint Object Camera and Spectrograph (FOCAS) on the SUBARU 8.2m Telescope. We have measured the velocities for 122 objects: 105 GCs in NGC 4636, the nucleus of NGC 4636, 11 foreground stars, 2 background galaxies, and 3 probable intracluster GCs in the Virgo cluster. The GCs in NGC 4636 are located in the projected galactocentric radius within 10arcmin (corresponding to 43 kpc). The measured velocities for the GCs range from 300km/s to 1600km/s, with a mean value of 932_{-22}^{+25} km/s, which is in good agreement with the velocity for the nucleus of NGC 4636, 928pm 45 km/s. The velocity dispersion of the GCs in NGC 4636 is derived to be 231_{-17}^{+15} km/s and the velocity dispersion of the blue GCs is slightly larger than that of the red GCs. Combining our results with data in the literature, we produce a master catalog of radial velocities for 238 GCs in NGC 4636. The velocity dispersion of the GCs in the master catalog is found to be 225_{-9}^{+12} km/s for the entire sample, 251_{-12}^{+18} km/s for 108 blue GCs, and 205_{-13}^{+11} km/s for 130 red GCs.
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