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We realize the simple Lie superalgebra G(3) as supersymmetry of variousgeometric structures, most importantly super-versions of the Hilbert-Cartanequation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry.We provide the symmetries explicitly and compute, via the first Spencercohomology groups, the Tanaka-Weisfeiler prolongation of the negatively gradedLie superalgebras associated with two particular choices of parabolics. Wediscuss non-holonomic superdistributions with growth vector (2|4,1|2,2|0)deforming the flat model SHC, and prove that the second Spencer cohomologygroup gives a binary quadratic form, thereby providing a "square-root" ofCartan's classical binary quartic invariant for generic rank 2 distributions ina 5-dimensional space. Finally, we obtain super-extensions of Cartan'sclassical submaximally symmetric models, compute their symmetries and observe asupersymmetry dimension gap phenomenon.

G(3)-supergeometry and a supersymmetric extension of the Hilbert-Cartan equation