  We report the serendipitous discovery of HSC J0904$-$0102, a quadruply-lensed
Lyman break galaxy (LBG) in the Survey of Gravitationally-lensed Objects in
Hyper Suprime-Cam Imaging (SuGOHI). Owing to its point-like appearance, the
source was thought to be a lensed active galactic nucleus. We obtained
follow-up spectroscopic data with the Gemini Multi-Object Spectrographs on the
Gemini South Telescope, which confirmed this to be a lens system. The
deflecting foreground galaxy is a typical early-type galaxy at a high redshift
of $z_{\ell} = 0.957$ with stellar velocity dispersion $\sigma_v=259\pm56$
km~s$^{-1}$. The lensed source is identified as an LBG at $z_{\rm s} = 3.403$,
based on the sharp drop bluewards of Ly$\alpha$ and other absorption features.
A simple lens mass model for the system, assuming a singular isothermal
ellipsoid, yields an Einstein radius of $\theta_{\rm Ein} = 1.
23^{\prime\prime}$ and a total mass within the Einstein radius of $M_{\rm Ein}
= (5.55\pm 0.24) \times 10^{11}M_{\odot}$ corresponding to a velocity
dispersion of $\sigma_{\rm SIE}= 283\pm 3$ km~s$^{-1}$, which is in good
agreement with the value derived spectroscopically. The most isolated lensed
LBG image has a magnification of $\sim 6.5$. In comparison with other lensed
LBGs and typical $z\sim4$ LBG populations, HSC J0904$-$0102 is unusually
compact, an outlier at $>2\sigma$ confidence. Together with a previously
discovered SuGOHI lens, HSC J1152$+$0047, that is similarly compact, we believe
that the HSC Survey is extending LBG studies down to smaller galaxy sizes.
