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As the smallest exceptional Lie group and the automorphism group of thenon-associative algebra of octonions, G$_2$ is often employed for describingexotic symmetry structures. We prove a G$_2$ symmetry in a Hubbard-like modelwith spin-$\frac{3}{2}$ fermions in a bipartite lattice, which lies in theintersection of two SO(7) algebras connected by the structure constants ofoctonions. Depending on the representations of the order parameters, the G$_2$symmetry can be spontaneously broken into either an SU(3) one associated withan $S^6$ Goldstone manifold, or, into an SU(2)$\times$U(1) with a GrassmannianGoldstone manifold $\mbox{Gr}_5^+(\mathbb{R}^7)$. In the quantum disorderedstates, quantum fluctuations generate the effective SU(3) and SU(2)$\times$U(1)gauge theories for low energy fermions.

Exceptional Symmetry of G$_2$ in Spin-$\frac{3}{2}$ Fermion Systems