The structure described by KMg15 crystallizes in the hexagonal P-6m2 space group. K(1) is bonded to six equivalent Mg(2) and six equivalent Mg(4) atoms to form KMg12 cuboctahedra that share corners with six equivalent K(1)Mg12 cuboctahedra, corners with twelve equivalent Mg(3)Mg12 cuboctahedra, edges with six equivalent Mg(2)K2Mg10 cuboctahedra, edges with twelve equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent Mg(1)Mg12 cuboctahedra, faces with six equivalent Mg(2)K2Mg10 cuboctahedra, faces with six equivalent Mg(4)KMg11 cuboctahedra, and faces with six equivalent Mg(5)Mg12 cuboctahedra. There are five inequivalent Mg sites. In the first Mg site, Mg(1) is bonded to six equivalent Mg(3) and six equivalent Mg(4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(1)Mg12 cuboctahedra, corners with twelve equivalent Mg(2)K2Mg10 cuboctahedra, edges with six equivalent Mg(3)Mg12 cuboctahedra, edges with twelve equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent K(1)Mg12 cuboctahedra, faces with six equivalent Mg(4)KMg11 cuboctahedra, faces with six equivalent Mg(3)Mg12 cuboctahedra, and faces with six equivalent Mg(5)Mg12 cuboctahedra. In the second Mg site, Mg(2) is bonded to two equivalent K(1), two equivalent Mg(5), four equivalent Mg(2), and four equivalent Mg(4) atoms to form distorted MgK2Mg10 cuboctahedra that share corners with four equivalent Mg(1)Mg12 cuboctahedra, corners with six equivalent Mg(2)K2Mg10 cuboctahedra, corners with eight equivalent Mg(3)Mg12 cuboctahedra, edges with two equivalent K(1)Mg12 cuboctahedra, edges with four equivalent Mg(2)K2Mg10 cuboctahedra, edges with four equivalent Mg(5)Mg12 cuboctahedra, edges with eight equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent K(1)Mg12 cuboctahedra, faces with two equivalent Mg(3)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with four equivalent Mg(2)K2Mg10 cuboctahedra, and faces with ten equivalent Mg(4)KMg11 cuboctahedra. In the third Mg site, Mg(3) is bonded to two equivalent Mg(1), two equivalent Mg(5), four equivalent Mg(3), and four equivalent Mg(4) atoms to form MgMg12 cuboctahedra that share corners with four equivalent K(1)Mg12 cuboctahedra, corners with six equivalent Mg(3)Mg12 cuboctahedra, corners with eight equivalent Mg(2)K2Mg10 cuboctahedra, edges with two equivalent Mg(1)Mg12 cuboctahedra, edges with four equivalent Mg(3)Mg12 cuboctahedra, edges with four equivalent Mg(5)Mg12 cuboctahedra, edges with eight equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent Mg(2)K2Mg10 cuboctahedra, faces with two equivalent Mg(1)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with four equivalent Mg(3)Mg12 cuboctahedra, and faces with ten equivalent Mg(4)KMg11 cuboctahedra. In the fourth Mg site, Mg(4) is bonded to one K(1), one Mg(1), two equivalent Mg(2), two equivalent Mg(3), two equivalent Mg(5), and four equivalent Mg(4) atoms to form distorted MgKMg11 cuboctahedra that share corners with four equivalent Mg(5)Mg12 cuboctahedra, corners with fourteen equivalent Mg(4)KMg11 cuboctahedra, edges with two equivalent K(1)Mg12 cuboctahedra, edges with two equivalent Mg(1)Mg12 cuboctahedra, edges with two equivalent Mg(5)Mg12 cuboctahedra, edges with four equivalent Mg(2)K2Mg10 cuboctahedra, edges with four equivalent Mg(4)KMg11 cuboctahedra, edges with four equivalent Mg(3)Mg12 cuboctahedra,  a faceface with one K(1)Mg12 cuboctahedra,  a faceface with one Mg(1)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with five equivalent Mg(2)K2Mg10 cuboctahedra, faces with five equivalent Mg(3)Mg12 cuboctahedra, and faces with six equivalent Mg(4)KMg11 cuboctahedra. In the fifth Mg site, Mg(5) is bonded to three equivalent Mg(2), three equivalent Mg(3), and six equivalent Mg(4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(5)Mg12 cuboctahedra, corners with twelve equivalent Mg(4)KMg11 cuboctahedra, edges with six equivalent Mg(2)K2Mg10 cuboctahedra, edges with six equivalent Mg(4)KMg11 cuboctahedra, edges with six equivalent Mg(3)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with three equivalent K(1)Mg12 cuboctahedra, faces with three equivalent Mg(2)K2Mg10 cuboctahedra, faces with three equivalent Mg(1)Mg12 cuboctahedra, faces with three equivalent Mg(3)Mg12 cuboctahedra, and faces with six equivalent Mg(4)KMg11 cuboctahedra. is represented by the Crystallographic Information File (CIF) [CIF]
data_KMg15
_symmetry_space_group_name_H-M   'P 1'
_cell_length_a   6.589
_cell_length_b   6.589
_cell_length_c   10.487
_cell_angle_alpha   90.000
_cell_angle_beta   90.000
_cell_angle_gamma   120.000
_symmetry_Int_Tables_number   1
_cell_volume   394.348
_cell_formula_units_Z   1
loop_
 _symmetry_equiv_pos_site_id
 _symmetry_equiv_pos_as_xyz
  1  'x, y, z'
loop_
 _atom_site_type_symbol
 _atom_site_label
 _atom_site_symmetry_multiplicity
 _atom_site_fract_x
 _atom_site_fract_y
 _atom_site_fract_z
 _atom_site_occupancy
  K  K0  1  0.667  0.333  0.000  1.0
  Mg  Mg1  1  0.667  0.333  0.500  1.0
  Mg  Mg2  1  0.170  0.340  0.000  1.0
  Mg  Mg3  1  0.165  0.330  0.500  1.0
  Mg  Mg4  1  0.170  0.830  0.000  1.0
  Mg  Mg5  1  0.165  0.835  0.500  1.0
  Mg  Mg6  1  0.660  0.830  0.000  1.0
  Mg  Mg7  1  0.670  0.835  0.500  1.0
  Mg  Mg8  1  0.010  0.505  0.263  1.0
  Mg  Mg9  1  0.010  0.505  0.737  1.0
  Mg  Mg10  1  0.495  0.505  0.263  1.0
  Mg  Mg11  1  0.495  0.505  0.737  1.0
  Mg  Mg12  1  0.495  0.990  0.263  1.0
  Mg  Mg13  1  0.495  0.990  0.737  1.0
  Mg  Mg14  1  0.000  0.000  0.247  1.0
  Mg  Mg15  1  0.000  0.000  0.753  1.0
[/CIF]
.
The structure described by KMg15 crystallizes in the hexagonal P-6m2 space group. K(1) is bonded to six equivalent Mg(2) and six equivalent Mg(4) atoms to form KMg12 cuboctahedra that share corners with six equivalent K(1)Mg12 cuboctahedra, corners with twelve equivalent Mg(3)Mg12 cuboctahedra, edges with six equivalent Mg(2)K2Mg10 cuboctahedra, edges with twelve equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent Mg(1)Mg12 cuboctahedra, faces with six equivalent Mg(2)K2Mg10 cuboctahedra, faces with six equivalent Mg(4)KMg11 cuboctahedra, and faces with six equivalent Mg(5)Mg12 cuboctahedra. There are five inequivalent Mg sites. In the first Mg site, Mg(1) is bonded to six equivalent Mg(3) and six equivalent Mg(4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(1)Mg12 cuboctahedra, corners with twelve equivalent Mg(2)K2Mg10 cuboctahedra, edges with six equivalent Mg(3)Mg12 cuboctahedra, edges with twelve equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent K(1)Mg12 cuboctahedra, faces with six equivalent Mg(4)KMg11 cuboctahedra, faces with six equivalent Mg(3)Mg12 cuboctahedra, and faces with six equivalent Mg(5)Mg12 cuboctahedra. In the second Mg site, Mg(2) is bonded to two equivalent K(1), two equivalent Mg(5), four equivalent Mg(2), and four equivalent Mg(4) atoms to form distorted MgK2Mg10 cuboctahedra that share corners with four equivalent Mg(1)Mg12 cuboctahedra, corners with six equivalent Mg(2)K2Mg10 cuboctahedra, corners with eight equivalent Mg(3)Mg12 cuboctahedra, edges with two equivalent K(1)Mg12 cuboctahedra, edges with four equivalent Mg(2)K2Mg10 cuboctahedra, edges with four equivalent Mg(5)Mg12 cuboctahedra, edges with eight equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent K(1)Mg12 cuboctahedra, faces with two equivalent Mg(3)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with four equivalent Mg(2)K2Mg10 cuboctahedra, and faces with ten equivalent Mg(4)KMg11 cuboctahedra. In the third Mg site, Mg(3) is bonded to two equivalent Mg(1), two equivalent Mg(5), four equivalent Mg(3), and four equivalent Mg(4) atoms to form MgMg12 cuboctahedra that share corners with four equivalent K(1)Mg12 cuboctahedra, corners with six equivalent Mg(3)Mg12 cuboctahedra, corners with eight equivalent Mg(2)K2Mg10 cuboctahedra, edges with two equivalent Mg(1)Mg12 cuboctahedra, edges with four equivalent Mg(3)Mg12 cuboctahedra, edges with four equivalent Mg(5)Mg12 cuboctahedra, edges with eight equivalent Mg(4)KMg11 cuboctahedra, faces with two equivalent Mg(2)K2Mg10 cuboctahedra, faces with two equivalent Mg(1)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with four equivalent Mg(3)Mg12 cuboctahedra, and faces with ten equivalent Mg(4)KMg11 cuboctahedra. In the fourth Mg site, Mg(4) is bonded to one K(1), one Mg(1), two equivalent Mg(2), two equivalent Mg(3), two equivalent Mg(5), and four equivalent Mg(4) atoms to form distorted MgKMg11 cuboctahedra that share corners with four equivalent Mg(5)Mg12 cuboctahedra, corners with fourteen equivalent Mg(4)KMg11 cuboctahedra, edges with two equivalent K(1)Mg12 cuboctahedra, edges with two equivalent Mg(1)Mg12 cuboctahedra, edges with two equivalent Mg(5)Mg12 cuboctahedra, edges with four equivalent Mg(2)K2Mg10 cuboctahedra, edges with four equivalent Mg(4)KMg11 cuboctahedra, edges with four equivalent Mg(3)Mg12 cuboctahedra,  a faceface with one K(1)Mg12 cuboctahedra,  a faceface with one Mg(1)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with five equivalent Mg(2)K2Mg10 cuboctahedra, faces with five equivalent Mg(3)Mg12 cuboctahedra, and faces with six equivalent Mg(4)KMg11 cuboctahedra. In the fifth Mg site, Mg(5) is bonded to three equivalent Mg(2), three equivalent Mg(3), and six equivalent Mg(4) atoms to form MgMg12 cuboctahedra that share corners with six equivalent Mg(5)Mg12 cuboctahedra, corners with twelve equivalent Mg(4)KMg11 cuboctahedra, edges with six equivalent Mg(2)K2Mg10 cuboctahedra, edges with six equivalent Mg(4)KMg11 cuboctahedra, edges with six equivalent Mg(3)Mg12 cuboctahedra, faces with two equivalent Mg(5)Mg12 cuboctahedra, faces with three equivalent K(1)Mg12 cuboctahedra, faces with three equivalent Mg(2)K2Mg10 cuboctahedra, faces with three equivalent Mg(1)Mg12 cuboctahedra, faces with three equivalent Mg(3)Mg12 cuboctahedra, and faces with six equivalent Mg(4)KMg11 cuboctahedra. is represented by the Crystallographic Information File (CIF) [CIF]
data_KMg15
_symmetry_space_group_name_H-M   'P 1'
_cell_length_a   6.589
_cell_length_b   6.589
_cell_length_c   10.487
_cell_angle_alpha   90.000
_cell_angle_beta   90.000
_cell_angle_gamma   120.000
_symmetry_Int_Tables_number   1
_cell_volume   394.348
_cell_formula_units_Z   1
loop_
 _symmetry_equiv_pos_site_id
 _symmetry_equiv_pos_as_xyz
  1  'x, y, z'
loop_
 _atom_site_type_symbol
 _atom_site_label
 _atom_site_symmetry_multiplicity
 _atom_site_fract_x
 _atom_site_fract_y
 _atom_site_fract_z
 _atom_site_occupancy
  K  K0  1  0.667  0.333  0.000  1.0
  Mg  Mg1  1  0.667  0.333  0.500  1.0
  Mg  Mg2  1  0.170  0.340  0.000  1.0
  Mg  Mg3  1  0.165  0.330  0.500  1.0
  Mg  Mg4  1  0.170  0.830  0.000  1.0
  Mg  Mg5  1  0.165  0.835  0.500  1.0
  Mg  Mg6  1  0.660  0.830  0.000  1.0
  Mg  Mg7  1  0.670  0.835  0.500  1.0
  Mg  Mg8  1  0.010  0.505  0.263  1.0
  Mg  Mg9  1  0.010  0.505  0.737  1.0
  Mg  Mg10  1  0.495  0.505  0.263  1.0
  Mg  Mg11  1  0.495  0.505  0.737  1.0
  Mg  Mg12  1  0.495  0.990  0.263  1.0
  Mg  Mg13  1  0.495  0.990  0.737  1.0
  Mg  Mg14  1  0.000  0.000  0.247  1.0
  Mg  Mg15  1  0.000  0.000  0.753  1.0
[/CIF]
.