Non-convex polyhedron with 18 edges, 12 faces and 8 vertices

Question

Which non-convex polyhedron has 8 vertices, 12 faces and 18 edges?

Answer 1

The dual of @lhf's construction will work. Take a (say, nonconvex) hexagon and form the (double) cone over the plane figure, to get a solid.

The two apexes and six points of the hexagon make eight vertices, there are twelve triangular faces (six meeting at each apex), and of course $18$ edges.

There is another way to make the solid nonconvex, as one cone could be "everted" inside the other.

Answer 2

Besides the hexagonal bipyramid mentioned by hardmath, you can also form such a solid by adding a pyramid over each face of a tetrahedron. This "triakis tetrahedron" has 8 vertices (the four original vertices, and the four new apices), $4 \cdot 3 = 12$ faces, and 18 edges (the six original edges, and 12 new ones.) If you make the pyramids sufficiently tall, or if you indent them inside the tetrahedron, then you have a non-convex solid.