  We report the first detections of the repeating fast radio burst source FRB
121102 above 5.2 GHz. Observations were performed using the 4$-$8 GHz receiver
of the Robert C. Byrd Green Bank Telescope with the Breakthrough Listen digital
backend. We present the spectral, temporal and polarization properties of 21
bursts detected within the first 60 minutes of a total 6-hour observations.
These observations comprise the highest burst density yet reported in the
literature, with 18 bursts being detected in the first 30 minutes. A few bursts
clearly show temporal sub-structures with distinct spectral properties. These
sub-structures superimpose to provide enhanced peak signal-to-noise ratio at
higher trial dispersion measures. Broad features occur in $\sim 1$ GHz wide
subbands that typically differ in peak frequency between bursts within the
band. Finer-scale structures ($\sim 10-50$ MHz) within these bursts are
consistent with that expected from Galactic diffractive interstellar
scintillation. The bursts exhibit nearly 100% linear polarization, and a large
average rotation measure of 9.359$\pm$0.012 $\times$ 10$^{\rm 4}$ rad m$^{\rm
-2}$ (in the observer's frame). No circular polarization was found for any
burst. We measure an approximately constant polarization position angle in the
13 brightest bursts. The peak flux densities of the reported bursts have
average values (0.2$\pm$0.1 Jy), similar to those seen at lower frequencies
($<3$ GHz), while the average burst widths (0.64$\pm$0.46 ms) are relatively
narrower.
