A novel 3-,4-connected network is reported in this communication that is a rare example of a three-dimensional (3D) network with a quaternary stoichiometry, in which 4 distinct sites of bonding occur in the unit of pattern of the material. The net lies in the hexagonal space group P-6m2 (#187) and there are 10 vertices (or atoms) in the unit cell, including 4 sites of approximately trigonal planar coordination and 6 sites of approximately tetrahedral coordination. All vertices (atoms) in the network sit on special positions in the P-6m2 space group, giving the network high sixfold symmetry axes parallel to the crystallographic c-axis. The structure has thus been named trigohexagonite in a loose analogy with its hexagonite crystalline homolog, sitting in the hexagonal space group P6/mmm (#191). The Wells point symbol for the trigohexagonite network is given by (73)(6383)3(6.72)3(3.74.8)3 where this symbol indicates the quaternary stoichiometry of the network. The Wells point symbol also reveals that the only structural strain present in the network comes from the presence of the 3-gon, cyclopropane-like moieties in it, built on tetrahedral vertices. This structural motif of cyclopropane-like rings has precedent in organic chemistry and it adds character to the overall sixfold symmetry of the trigohexagonite pattern. Also, the overall network contains rare trimethylenemethane-like clusters of 4 trigonal planar vertices (atoms), bonded together, that constitute the 3-connected component of the network. Both c10 and B10 realizations are briefly described.