I'll tentatively call it the J92 hexachoron, since it's similar to the J92 rhombochoron. 
I had trouble figuring out how to typeset the lace city in ASCII, so I decided to let graphviz to do it for me instead:
(In retrospect this is probably quite easy to do in ASCII, but it wasn't obvious just from looking at the coordinates!
) I wasn't sure what node label to use for phi+2, so I assigned W=phi+2. EDIT: Corrected a mistake in the lace city. 
EDIT 2: Corrected more mistakes as pointed out by Klitzing. 
As is obviously, this CRF has trigonal symmetry... What may not be immediately obvious, though, is that it also has trigonal symmetry in perp space, so it actually has 3,3-duoprism symmetry.
There are 18 icosahedra in 3 groups of 6, each group of which has icosahedra sharing faces with the connectivity of a trigonal prism.
In fact, it is possible to augment this CRF with 6 J92 pseudo-pyramids, to produce a polytope with 24 icosahedra -- but it is not the snub 24-cell!! It's a kind of strange, modified 24-cell with 3,3-duoprism symmetry, but not demitesseractic symmetry. It has triangular prisms and square pyramids, and is therefore not uniform.Here's a parallel projection centered on a hexagon shared by two J92's:
The green and yellow cells are J92's, in case it's not obvious (it's not easy to get good renders on this beast). The blue and purple cells are regular icosahedra -- there are 3 pairs visible here. Between the pairs, are clusters of 5 square pyramids, that look like a fragment of the rectified 600-cell o5o3x3o; within each pair, however, there are 5-fold clusters of tetrahedra that look like a fragment of the 600-cell. You can see some of the tetrahedron-trigonal prism-tetrahedron combos that seem to be a recurring theme in these CRFs.
This structure looks deceptively simple... until you look at what goes on in the perpendicular ring to the six J92's:
The J92's are outlined in red here. As you can see, there are 6 icosahedra here, with the 3 in top sharing only vertices with the 3 at the bottom. These are actually two different trigonal-prism clusters of icosahedra; the top and bottom clusters of pentagon-like faces of the projection are images of 6 other icosahedra that share faces with the top 3 icosahedra and the bottom 3 icosahedra, respectively.
Before we get to that, though, let's look at something else: the trigonal prisms in this 4D viewpoint:
As you can see, there are 4 triangular prisms visible here, each of which connects to two tetrahedra (the ones touching the center shown in blue). Now, notice that the pentagonal cross-sections of the icosahedra between the outer (red) triangular prisms form a 5-3-5-3 configuration around the vertices of the central triangular prism. Sounds familiar?
It should, 'cos it suggests that there might be a way to fit J92's here too! I haven't tried it yet, but it might lead to a CRF with two rings of J92's. 
Well, let's look at a slightly shifted viewpoint that shows a cluster of 6 icosahedra in trigonal prism arrangement:
Notice how similar this is to the snub 24-cell? Yet it's not quite the same, because it has trigonal symmetry instead of tetragonal!
Alright, let's finish off with another view of the J92's, this time around a shared triangular ridge:
This time, we show the J62 (metabidiminished icosahedra) wedges between the J92 cells instead. The left side of the projection seems to have the outline of a cube, but that's an illusion. (There is actually no edge between the two "square faces" of the "cube".)
