Looks like a compound of 4 coincident pentachora (pen) plus 160 additional triangles coincident by 16 in every position of the pentachoral ones. Indeed those additional ones are 2 triangular retrograde antiprisms (3/2-ap) each.

Incidence matrix according to Dynkin symbol

β3o3o3β

both( . . . .    ) | 20 |  6  6  6 |  3  3  3  9  9  3 | 1  3  3 1  6
-------------------+----+----------+-------------------+-------------
sefa( β3o . .    ) |  2 | 60  *  * |  1  0  1  1  0  0 | 1  1  0 0  1
sefa( β . . β2*a ) |  2 |  * 60  * |  0  0  0  2  2  0 | 0  1  1 0  2
sefa( . . o3β    ) |  2 |  *  * 60 |  0  1  0  0  1  1 | 0  0  1 1  1
-------------------+----+----------+-------------------+-------------
      β3o . .      ♦  3 |  3  0  0 | 20  *  *  *  *  * | 1  1  0 0  0
      . . o3β      ♦  3 |  0  0  3 |  * 20  *  *  *  * | 0  0  1 1  0
sefa( β3o3o .    ) |  3 |  3  0  0 |  *  * 20  *  *  * | 1  0  0 0  1
sefa( β3o . β2*a ) |  3 |  1  2  0 |  *  *  * 60  *  * | 0  1  0 0  1
sefa( β . o3β2*a ) |  3 |  0  2  1 |  *  *  *  * 60  * | 0  0  1 0  1
sefa( . o3o3β    ) |  3 |  0  0  3 |  *  *  *  *  * 20 | 0  0  0 1  1
-------------------+----+----------+-------------------+-------------
      β3o3o .      ♦  4 | 12  0  0 |  4  0  4  0  0  0 | 5  *  * *  *
      β3o . β2*a   ♦  6 |  6  6  0 |  2  0  0  6  0  0 | * 10  * *  *
      β . o3β2*a   ♦  6 |  0  6  6 |  0  2  0  0  6  0 | *  * 10 *  *
      . o3o3β      ♦  4 |  0  0 12 |  0  4  0  0  0  4 | *  *  * 5  *
sefa( β3o3o3β    ) ♦  6 |  3  6  3 |  0  0  1  3  3  1 | *  *  * * 20

or
both( . . . .    )    | 20 |  12  6 |  6  6  18 |  2  6  6
----------------------+----+--------+-----------+---------
sefa( β3o . .    )  & |  2 | 120  * |  1  1   1 |  1  1  1
sefa( β . . β2*a )  & |  2 |   * 60 |  0  0   4 |  0  2  2
----------------------+----+--------+-----------+---------
      β3o . .       & ♦  3 |   3  0 | 40  *   * |  1  1  0
sefa( β3o3o .    )  & |  3 |   3  0 |  * 40   * |  1  0  1
sefa( β3o . β2*a )  & |  3 |   1  2 |  *  * 120 |  0  1  1
----------------------+----+--------+-----------+---------
      β3o3o .       & ♦  4 |  12  0 |  4  4   0 | 10  *  *
      β3o . β2*a    & ♦  6 |   6  6 |  2  0   6 |  * 20  *
sefa( β3o3o3β    )    ♦  6 |   6  6 |  0  2   6 |  *  * 20

starting figure: x3o3o3x