LuCoGaO4 is Aluminum carbonitride-derived structured and crystallizes in the trigonal R3m space group. Lu(1) is bonded to three equivalent O(1) and three equivalent O(2) atoms to form distorted LuO6 octahedra that share corners with three equivalent Co(1)O5 trigonal bipyramids, corners with three equivalent Ga(1)O5 trigonal bipyramids, and edges with six equivalent Lu(1)O6 octahedra. Co(1) is bonded to one O(1), one O(3), and three equivalent O(4) atoms to form distorted CoO5 trigonal bipyramids that share corners with three equivalent Lu(1)O6 octahedra, corners with six equivalent Co(1)O5 trigonal bipyramids, and edges with three equivalent Ga(1)O5 trigonal bipyramids. The corner-sharing octahedral tilt angles are 63°. Ga(1) is bonded to one O(2), one O(4), and three equivalent O(3) atoms to form GaO5 trigonal bipyramids that share corners with three equivalent Lu(1)O6 octahedra, corners with six equivalent Ga(1)O5 trigonal bipyramids, and edges with three equivalent Co(1)O5 trigonal bipyramids. The corner-sharing octahedral tilt angles are 62°. There are four inequivalent O sites. In the first O site, O(1) is bonded to three equivalent Lu(1) and one Co(1) atom to form OLu3Co tetrahedra that share corners with three equivalent O(4)GaCo3 tetrahedra, corners with three equivalent O(2)Lu3Ga tetrahedra, corners with six equivalent O(1)Lu3Co tetrahedra,  a cornercorner with one O(3)Ga3Co trigonal pyramid, and edges with three equivalent O(2)Lu3Ga tetrahedra. In the second O site, O(2) is bonded to three equivalent Lu(1) and one Ga(1) atom to form OLu3Ga tetrahedra that share  a cornercorner with one O(4)GaCo3 tetrahedra, corners with three equivalent O(1)Lu3Co tetrahedra, corners with six equivalent O(2)Lu3Ga tetrahedra, corners with three equivalent O(3)Ga3Co trigonal pyramids, and edges with three equivalent O(1)Lu3Co tetrahedra. In the third O site, O(3) is bonded to one Co(1) and three equivalent Ga(1) atoms to form distorted OGa3Co trigonal pyramids that share  a cornercorner with one O(1)Lu3Co tetrahedra, corners with three equivalent O(2)Lu3Ga tetrahedra, corners with six equivalent O(3)Ga3Co trigonal pyramids, and edges with three equivalent O(4)GaCo3 tetrahedra. In the fourth O site, O(4) is bonded to three equivalent Co(1) and one Ga(1) atom to form OGaCo3 tetrahedra that share  a cornercorner with one O(2)Lu3Ga tetrahedra, corners with three equivalent O(1)Lu3Co tetrahedra, corners with six equivalent O(4)GaCo3 tetrahedra, and edges with three equivalent O(3)Ga3Co trigonal pyramids.
LuCoGaO4 is Aluminum carbonitride-derived structured and crystallizes in the trigonal R3m space group. Lu(1) is bonded to three equivalent O(1) and three equivalent O(2) atoms to form distorted LuO6 octahedra that share corners with three equivalent Co(1)O5 trigonal bipyramids, corners with three equivalent Ga(1)O5 trigonal bipyramids, and edges with six equivalent Lu(1)O6 octahedra. All Lu(1)-O(1) bond lengths are 2.22 Å. All Lu(1)-O(2) bond lengths are 2.24 Å. Co(1) is bonded to one O(1), one O(3), and three equivalent O(4) atoms to form distorted CoO5 trigonal bipyramids that share corners with three equivalent Lu(1)O6 octahedra, corners with six equivalent Co(1)O5 trigonal bipyramids, and edges with three equivalent Ga(1)O5 trigonal bipyramids. The corner-sharing octahedral tilt angles are 63°. The Co(1)-O(1) bond length is 2.00 Å. The Co(1)-O(3) bond length is 2.38 Å. All Co(1)-O(4) bond lengths are 2.03 Å. Ga(1) is bonded to one O(2), one O(4), and three equivalent O(3) atoms to form GaO5 trigonal bipyramids that share corners with three equivalent Lu(1)O6 octahedra, corners with six equivalent Ga(1)O5 trigonal bipyramids, and edges with three equivalent Co(1)O5 trigonal bipyramids. The corner-sharing octahedral tilt angles are 62°. The Ga(1)-O(2) bond length is 1.93 Å. The Ga(1)-O(4) bond length is 1.96 Å. All Ga(1)-O(3) bond lengths are 1.98 Å. There are four inequivalent O sites. In the first O site, O(1) is bonded to three equivalent Lu(1) and one Co(1) atom to form OLu3Co tetrahedra that share corners with three equivalent O(4)GaCo3 tetrahedra, corners with three equivalent O(2)Lu3Ga tetrahedra, corners with six equivalent O(1)Lu3Co tetrahedra,  a cornercorner with one O(3)Ga3Co trigonal pyramid, and edges with three equivalent O(2)Lu3Ga tetrahedra. In the second O site, O(2) is bonded to three equivalent Lu(1) and one Ga(1) atom to form OLu3Ga tetrahedra that share  a cornercorner with one O(4)GaCo3 tetrahedra, corners with three equivalent O(1)Lu3Co tetrahedra, corners with six equivalent O(2)Lu3Ga tetrahedra, corners with three equivalent O(3)Ga3Co trigonal pyramids, and edges with three equivalent O(1)Lu3Co tetrahedra. In the third O site, O(3) is bonded to one Co(1) and three equivalent Ga(1) atoms to form distorted OGa3Co trigonal pyramids that share  a cornercorner with one O(1)Lu3Co tetrahedra, corners with three equivalent O(2)Lu3Ga tetrahedra, corners with six equivalent O(3)Ga3Co trigonal pyramids, and edges with three equivalent O(4)GaCo3 tetrahedra. In the fourth O site, O(4) is bonded to three equivalent Co(1) and one Ga(1) atom to form OGaCo3 tetrahedra that share  a cornercorner with one O(2)Lu3Ga tetrahedra, corners with three equivalent O(1)Lu3Co tetrahedra, corners with six equivalent O(4)GaCo3 tetrahedra, and edges with three equivalent O(3)Ga3Co trigonal pyramids.