12 | S4 x I / D1 x I | μ5
The polyhedron is a compound of 12 cubes and belongs to the symmetry group 12|S4xI|D1xI|μ, using the notation from H.F.Verheyen's book Symmetry Orbits. The angle 'μ' uniquely identifies one compound in the group and may vary between 0 and 45 degrees (with 0 and 45 not included). For some angles μ the compound has some special properties and the book summarises the special angles.
However the special position for this model isn't mentioned in the book, though it is clear that something special happens. For this compound the angle μ=(1/2)atan((4/7)√2) and for that angle the compound can be divided into three Bakos compounds (4|S4xI|D3xI). For each Bakos compound holds that one of the fourfold axes is shared with a fourfold axis of the compound; the other two axes share a twofold axis with the compound.
I built this model in 2005 and I used three colours: each Bakos' compound has one colour.
This compound can be seen as a multiplication between Bakos' compound and the classic compounds of 3 cubes. This multiplication isn't commutative, which means that you won't get the same result if you switch the operands of the multiplication. If you do and you take the multiplication of the classic compound and Bakos' compound you'll get this compound, which can be interpreted as the twin of this one.
Here are some more pictures of the model:
A complete table about cube compounds can be found here here.
All calculations for this model were done algebraically and the summary of the calculation you can see here It also contains the templates and the adjusted templates that I am using, since some faces became so tiny that is was not very practical to build an exact model or I should have built a bigger model. Here you can see several stages of the construction of the model.