This page is linked from here; go there for some background.

This is the "augmented octahedron" (I'll also call it "7V5"),

which is what you get if you snap together an octahedron and a tetrahedron.

See the first page in the list, 7V1, for a philosophical discussion concerning this nomenclature.

There are four ways to add a tetrahedron to the augmented octahedron (7V5) to get a new (8-vertex) deltahedron. Which ones among the following 14 are they? (Source for image: this paper by N. Tsuruta, J. Mitani, Y. Kanamori, Y. Fukui (2015), Random Realization of Polyhedral Graphs as Deltahedra, Journal for Geometry and Graphics, Volume 19 (2015), No. 2, 227–236.)