Acronym ...
Name 3sissid+gike (?)
Circumradius sqrt[(5-sqrt(5))/8] = 0.587785
Vertex figure 5[10/4,5/2,10/4,3]
General of army ike
Colonel of regiment sissid
Confer
non-Grünbaumian masters:
gike   sissid  
Grünbaumian relatives:
gacid   sissid+3gike   2sissid+gike   2gike+sissid   2sissid+4gike   4sissid+2gike  

Looks like a compound of 3 small stellated dodecahedra (sissid) plus a great icosahedron (gike), and indeed vertices coincide by five, edges coincide by four, {5/2} and {10/2} coincide.


Incidence matrix according to Dynkin symbol

x5/4x5/2o3*a

.   .   .    | 60 |  2  2 |  2  1  1
-------------+----+-------+---------
x   .   .    |  2 | 60  * |  1  1  0
.   x   .    |  2 |  * 60 |  1  0  1
-------------+----+-------+---------
x5/4x   .    | 10 |  5  5 | 12  *  *
x   .   o3*a |  3 |  3  0 |  * 20  *
.   x5/2o    |  5 |  0  5 |  *  * 12

x5/4x3/2o5/3*a

.   .   .      | 60 |  2  2 |  2  1  1
---------------+----+-------+---------
x   .   .      |  2 | 60  * |  1  1  0
.   x   .      |  2 |  * 60 |  1  0  1
---------------+----+-------+---------
x5/4x   .      | 10 |  5  5 | 12  *  *
x   .   o5/3*a |  5 |  5  0 |  * 12  *
.   x3/2o      |  3 |  0  3 |  *  * 20

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