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The elongated square gyrobicupola is a nonuniform polyhedron obtained by rotating the bottom third of a small
rhombicuboctahedron (Ball and Coxeter 1987, p. 137). It is also called Miller's
solid, the Miller-aškinuze solid, or the pseudorhombicuboctahedron, and is
Johnson solid 
.
The crossed graph of the elongated square gyrobicupola is the crossed elongated square gyrobicupola graph, which is the 24-Fabrici-Madaras graph.
Although some writers have suggested that the elongated square gyrobicupola should be considered a fourteenth Archimedean solid, its twist allows vertices "near the equator" and those "in the polar regions" to be distinguished. Therefore, it is not a true Archimedean solid like the small rhombicuboctahedron, whose vertices cannot be distinguished (Cromwell 1997, pp. 91-92).
The elongated square gyrobicupola has volume
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(1)
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and Dehn invariant
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(2)
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(3)
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where the first expression uses the basis of Conway et al. (1999). It can be dissected into the small rhombicuboctahedron, from which it differs only by relative rotation of the top and bottom cupolas.
