You ask the question as if there were only one possible answer. This is not at all the case, but one example of such a shape is a cube. The unit cell of the triclinic crystal system is another, and there are even less symmetric possibilities (for example, there's nothing in your description requiring the opposite faces to be parallel), but I can't think of any simple name to give them.
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cube!
6 faces, 12 edges
A cuboid seems to fit the given description because it has 6 faces, 12 edges and 8 vertices.
That 3D shape is called triangular pyramid.
A 3D 5 sided shape is called a dodecahedron. It is a solid with 20 vertices, 30 edges and 12 equal pentagonal triangular faces.