⭳

Looks like a compound of 2 icosidodecahedra (id) plus 20 pairs of coincident {3}. And indeed vertices and {5} coincide by pairs, edges coincide by 3, and each {3/2} coincides with 3 {3}.

Incidence matrix according to Dynkin symbol

β5β3o

both( . . .    ) | 60 |  2  2  2 |  1  1  1  3
-----------------+----+----------+------------
sefa( s5s . (r)) |  2 | 60  *  * |  1  0  0  1
sefa( s5s . (l)) |  2 |  * 60  * |  0  1  0  1
sefa( . β3o    ) |  2 |  *  * 60 |  0  0  1  1
-----------------+----+----------+------------
      s5s . (r)  ♦  5 |  5  0  0 | 12  *  *  *
      s5s . (l)  ♦  5 |  0  5  0 |  * 12  *  *
      . β3o      ♦  3 |  0  0  3 |  *  * 20  *
sefa( β5β3o    ) |  3 |  1  1  1 |  *  *  * 60

or
both( . . . ) | 60 |   4  2 |  2  1  3
--------------+----+--------+---------
sefa( s5s . ) |  2 | 120  * |  1  0  1
sefa( . β3o ) |  2 |   * 60 |  0  1  1
--------------+----+--------+---------
both( s5s . ) ♦  5 |   5  0 | 24  *  *
      . β3o   ♦  3 |   0  3 |  * 20  *
sefa( β5β3o ) |  3 |   2  1 |  *  * 60

starting figure: x5x3o

s3/2s5s5*a

demi( .   . .    ) | 60 |  2  2  2 |  1  1  1  3
-------------------+----+----------+------------
sefa( s3/2s .    ) |  2 | 60  *  * |  1  0  0  1
sefa( s   . s5*a ) |  2 |  * 60  * |  0  1  0  1
sefa( .   s5s    ) |  2 |  *  * 60 |  0  0  1  1
-------------------+----+----------+------------
      s3/2s .      ♦  3 |  3  0  0 | 20  *  *  *
      s   . s5*a   |  5 |  0  5  0 |  * 12  *  *
      .   s5s      |  5 |  0  0  5 |  *  * 12  *
sefa( s3/2s5s5*a ) |  3 |  1  1  1 |  *  *  * 60

starting figure: x3/2x5x5*a