Acronym ...
Name dodeca-augmented rhombicuboctahedron
  
 ©
Vertex figure [3,4,3,H], [3,h,3,h]
Coordinates
  1. (V/2, 1/2, 1/2)         & all permutations, all changes of sign
    (vertex inscribed a3o4x, a = sqrt(5/2))
  2. (F/2, F/2, 0)           & all permutations, all changes of sign
    (vertex inscribed o3b4o, b = (3+sqrt(5))/sqrt(8))
where F = f+x = ff, V = 2f
Face vector 36, 72, 38
Confer 30-augmented srid   expanded 12-augmented girco  

The non-regular hexagons {(h,H)3} are shield hexagons. Their vertex angles are alternatingly h = arccos[-1/4] = 104.477512° resp. H = arccos[-(3 sqrt(5)-1)/8] = 135.522488°.


Incidence matrix according to Dynkin symbol

((ao3ob4xo))&#zx   → height = 0
                     a = HH = sqrt(5/2) = 1.581139 (pseudo edge)
                     b = hh = (3+sqrt(5))/sqrt(8) = 1.851230 (pseudo edge)
                     x = hH = 1

  o.3o.4o.       | 24  * |  2  2 | 1 1  2  [3,4,3,H]
  .o3.o4.o       |  * 12 |  0  4 | 0 2  2  [3,h,3,h]
-----------------+-------+-------+-------
  .. .. x.       |  2  0 | 24  * | 1 0  1
  oo3oo4oo  &#x  |  1  1 |  * 48 | 0 1  1
-----------------+-------+-------+-------
  .. o.4x.       |  4  0 |  4  0 | 6 *  *
((ao3ob ..))&#xz |  3  3 |  0  6 | * 8  *  {(h,H)3}
  .. .. xo  &#x  |  2  1 |  1  2 | * * 24

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