Acronym ...
Name Leonardo polyhedron of type {4,5;19}
 
 ©
Vertex figure [ro5], [ri5], [ro,Ro2,t2], [ri,Ri2,T2]
Confer rhombi-propello-icosahedron

Incidence matrix

12  *  *  * |  5  0  0  0  0  0  0 |  5  0  0  [ro5]
 * 12  *  * |  0  5  0  0  0  0  0 |  0  5  0  [ri5]
 *  * 60  * |  1  0  2  1  1  0  0 |  3  0  2  [ro,Ro2,t2]
 *  *  * 60 |  0  1  0  0  1  2  1 |  0  3  2  [ri,Ri2,T2]
------------+----------------------+---------
 1  0  1  0 | 60  *  *  *  *  *  * |  2  0  0
 0  1  0  1 |  * 60  *  *  *  *  * |  0  2  0
 0  0  2  0 |  *  * 60  *  *  *  * |  1  0  1  (around outer ends of tubes)
 0  0  2  0 |  *  *  * 30  *  *  * |  2  0  0  (connecting outer ends of tubes)
 0  0  1  1 |  *  *  *  * 60  *  * |  0  0  2
 0  0  0  2 |  *  *  *  *  * 60  * |  0  1  1  (around inner ends of tubes)
 0  0  0  2 |  *  *  *  *  *  * 30 |  0  2  0  (connecting inner ends of tubes)
------------+----------------------+---------
 1  0  3  0 |  2  0  1  1  0  0  0 | 60  *  *  {(roRo)2}
 0  1  0  3 |  0  2  0  0  0  1  1 |  * 60  *  {(riRi)2}
 0  0  2  2 |  0  0  1  0  2  1  0 |  *  * 60  {(t2T2)}

(combinatorically)

144 |   5 |   5
----+-----+----
  2 | 360 |   2
----+-----+----
  4 |   4 | 180

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