To make this polyhedron using Stella 4d (available here), one starts with the icosahedron, drops the symmetry of the model down from icosahedral to tetrahedral, and then stellates it once. The result is a chiral solid featuring four triangular faces and twelve kites:
The dual of this polyhedron, which is also chiral, has four triangular faces, and twelve faces which are isosceles trapezoids. It is a type of faceted dodecahedron — a partial faceting, meaning it is made without using all of the dodecahedron’s vertices.
If the isosceles triangles in this polyhedron were close enough to being equilateral that close inspection would be required to tell the difference, this would be a near-miss to the Johnson Solids. However, in my opinion, this doesn’t meet that test — so I’m calling this a “near-near-miss,” instead.
Software credit: visit this website if you would like to try a free trial download of Stella 4d, the program I used to create this image.
The two types of trapezoid are shown in blue and green. There are twenty-four blue ones (in eights set of three, surrounding each triangle) and twenty-four green ones (in twelve sets of two, with each set in “bowtie” formation).
This symmetrohedron follows logically from one that was already known, and pictured at http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/, with the name “bowtie cube.” Here’s a rotating version of it.
(Images created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
The equiangular hexagons are very nearly regular, with only tiny deviations — probably not visible here — “from equilateralness.”