Acronym ...
Name Leonardo polyhedron of type {5,4;13}
 
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Vertex figure [(poa,Poa)2], [(pia,Pia)2], [pob2,Pob2], [pib2,Pib2], [poc2,pic2], [Poc2,Pic2]

Incidence matrix

12  *  *  *  *  * |  4  0  0  0  0  0  0  0 |  2  2  0  0  oa : [(poa,Poa)2]
 * 12  *  *  *  * |  0  4  0  0  0  0  0  0 |  0  0  2  2  ia : [(pia,Pia)2]
 *  * 24  *  *  * |  2  0  1  1  0  0  0  0 |  2  2  0  0  ob: [pob2,Pob2]
 *  *  * 24  *  * |  0  2  0  0  1  1  0  0 |  0  0  2  2  ib: [pib2,Pib2]
 *  *  *  * 24  * |  0  0  1  0  1  0  2  0 |  2  0  2  0  oc=ic at square tubes : [poc2,pic2]
 *  *  *  *  * 24 |  0  0  0  1  0  1  0  2 |  0  2  0  2  oc=ic at triangular tubes : [Poc2,Pic2]
------------------+-------------------------+------------
 1  0  1  0  0  0 | 48  *  *  *  *  *  *  * |  1  1  0  0
 0  1  0  1  0  0 |  * 48  *  *  *  *  *  * |  0  0  1  1
 0  0  1  0  1  0 |  *  * 24  *  *  *  *  * |  2  0  0  0
 0  0  1  0  0  1 |  *  *  * 24  *  *  *  * |  0  2  0  0
 0  0  0  1  1  0 |  *  *  *  * 24  *  *  * |  0  0  2  0
 0  0  0  1  0  1 |  *  *  *  *  * 24  *  * |  0  0  0  2
 0  0  0  0  2  0 |  *  *  *  *  *  * 24  * |  1  0  1  0
 0  0  0  0  0  2 |  *  *  *  *  *  *  * 24 |  0  1  0  1
------------------+-------------------------+------------
 1  0  2  0  2  0 |  2  0  2  0  0  0  1  0 | 24  *  *  *  {po}
 1  0  2  0  0  2 |  2  0  0  2  0  0  0  1 |  * 24  *  *  {Po}
 0  1  0  2  2  0 |  0  2  0  0  2  0  1  0 |  *  * 24  *  {pi}
 0  1  0  2  0  2 |  0  2  0  0  0  2  0  1 |  *  *  * 24  {Pi}

(combinatorically)

120 |   4 |  4
----+-----+---
  2 | 240 |  2
----+-----+---
  5 |   5 | 96

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