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We study the classical Heisenberg model on a recently identified threedimensional corner-shared equilateral triangular lattice, a magnetic sublatticeto a large class of systems with the symmetry group P2$_1$3. Since the degreeof geometric frustration of the nearest neighbor antiferromagnetic model onthis lattice lies on the border between the pyrochlore (not ordered) andhexagonal (ordered) lattices, it is non-trivial to predict its ground state.Using a classical rotor model, we find an ordered ground state with wavevector$(\frac{2\pi}{3a_0},0,0)$ featuring 120$^o$ rotated spins on each triangle.However, a mean field approximation on this lattice fails to find an orderedground state, finding instead a non-trivially degenerate ground state. As themean field approach is known to agree with Monte Carlo on the pyrochlorelattice, the reasons for this discrepancy are discussed. We also discuss thepossible relevance of our results to MnSi.Comment: 14 pages, 9 figures. Accepted for publication in Phys. Rev. B. Minor  change

Geometric frustration inherent to the trillium lattice, a sublattice of the B20 structure