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

Looks like a compound of 2 small stellated dodecahedra (sissid) plus a great icosahedron (gike), and indeed vertices coincide by five, edges coincide by three.


Incidence matrix according to Dynkin symbol

x5/4x3o

.   . . | 60 |  1  2 |  2  1
--------+----+-------+------
x   . . |  2 | 30  * |  2  0
.   x . |  2 |  * 60 |  1  1
--------+----+-------+------
x5/4x . | 10 |  5  5 | 12  *
.   x3o |  3 |  0  3 |  * 20

x5/4x3/2o

.   .   . | 60 |  1  2 |  2  1
----------+----+-------+------
x   .   . |  2 | 30  * |  2  0
.   x   . |  2 |  * 60 |  1  1
----------+----+-------+------
x5/4x   . | 10 |  5  5 | 12  *
.   x3/2o |  3 |  0  3 |  * 20

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