The small triambic icosahedron is the dual polyhedron of the small ditrigonal icosidodecahedron
with Maeder index 30 (Maeder 1997), Weinninger index 70 (Wenninger 1971, p. 106-107),
Coxeter index 39 (Coxeter et al. 1954), and Har'El index 35 (Har'El 1993).
Note that while Wenninger (1989, p. 49) calls this solid the triakis octahedron,
this term more commonly used for the dual of one of the Archimedean solids.

The hull of the small triambic icosahedron can be constructed by augmentation of a unit edge-length icosahedron by a pyramid with
height ,
giving a solid with edge lengths and 1.

The small triambic icosahedron hull has 32 vertices, 90 edges, and 70 faces. Its vertices are arranged in two concentric groups of 12 (indicated in red in the above illustration) and 20 (blue).

For a small triambic icosahedron hull with edge lengths

Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom.22,
321-332, 1999.Coxeter, H. S. M.; Longuet-Higgins, M. S.;
and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy.
Soc. London Ser. A246, 401-450, 1954.Har'El, Z. "Uniform
Solution for Uniform Polyhedra." Geom. Dedicata47, 57-110, 1993.
http://www.math.technion.ac.il/~rl/docs/uniform.pdf. Maeder, R. E. "The Stellated Icosahedra." Mathematica
in Education3, 5-11, 1994. http://library.wolfram.com/infocenter/Articles/2519/.Maeder,
R. E. "30: Small Ditrigonal Icosidodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/30.html.Wenninger,
M. J. Dual
Models. Cambridge, England: Cambridge University Press, pp. 42 and 46-47
1983.Wenninger, M. J. Polyhedron
Models. New York: Cambridge University Press, p. 46, 1989.