Acronym  ... 
Name  2gike (?) 
Circumradius  sqrt[(5sqrt(5))/8] = 0.587785 
Vertex figure  [3^{10}]/4 
General of army  ike 
Colonel of regiment  sissid 
Confer 

Looks like a compound of 2 great icosahedra (gike), and indeed edges and faces both coincide by pairs, but vertices are identified.
As abstract polytope 2gike is isomorphic to 2ike.
Incidence matrix according to Dynkin symbol
o3/2x3o5*a . . .  12  10  5 5 +++ . x .  2  60  1 1 +++ o3/2x .  3  3  20 * . x3o  3  3  * 20
o5/4o3x3*a . . .  12  10  5 5 +++ . . x  2  60  1 1 +++ . o3x  3  3  20 * o . x3*a  3  3  * 20
o5/4o3/2x3/2*a . . .  12  10  5 5 +++ . . x  2  60  1 1 +++ . o3/2x  3  3  20 * o . x3/2*a  3  3  * 20
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