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In 4-dimensional geometry, the cuboctahedral pyramid is bounded by one cuboctahedron on the base, 6 square pyramid, and 8 triangular pyramid cells which meet at the apex. It has 38 faces: 32 triangles and 6 squares. It has 32 edges, and 13 vertices.
Since a cuboctahedron has a circumradius divided by edge length equal to one, the triangles must be taller than equilateral to create a positive height.
The dual to the cuboctahedral pyramid is a rhombic dodecahedral pyramid, seen as an dodecahedral base, and 8 rhombic pyramids meeting at an apex.
References
Cuboctahedral pyramid Wikipedia(Text) CC BY-SA