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What are the names of the convex polyhedron with five, six, or seven vertices, where all vertices lie at distinct corners of a cube? I'm particularly interested in the five vertex case.

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With five vertices, depending on which ones you choose, you can get an oblique square pyramid, an oblique rectangular pyramid, or an irregular triangular bipyramid with $S_3$ symmetry.

With six vertices, you can get a right isosceles triangular prism, an equilateral triangular right antiprism, or an irregular octahedron with $S_2$ symmetry.

I don't think the seven-vertex figure has any particular name.

Neither of these names are precise enough to express the fact that the vertices are chosen among the vertices of a cube.

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