Looks like a compound of 2 small prismatohecatonicosachora (sophi), and indeed all elements coincide by pairs.

As abstract polytope 2sophi is isomorphic to 2quiphi, thereby replacing pentagons by pentagrams, resp. by replacing pip by stip and gad by sissid.

Incidence matrix according to Dynkin symbol

x5o5/2o5x

. .   . . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
----------+------+-----------+----------------+----------------
x .   . . |    2 | 3600    * |    2    2    0 |   1   2   1   0
. .   . x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
----------+------+-----------+----------------+----------------
x5o   . . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x .   . x |    4 |    2    2 |    * 3600    * |   0   1   1   0
. .   o5x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
----------+------+-----------+----------------+----------------
x5o5/2o . ♦   12 |   30    0 |   12    0    0 | 120   *   *   *
x5o   . x ♦   10 |   10    5 |    2    5    0 |   * 720   *   *
x .   o5x ♦   10 |    5   10 |    0    5    2 |   *   * 720   *
. o5/2o5x ♦   12 |    0   30 |    0    0   12 |   *   *   * 120

snubbed forms: β5o5/2o5x

or
. .   . .    | 1440 |   10 |   10   10 |   2   10
-------------+------+------+-----------+---------
x .   . .  & |    2 | 7200 |    2    2 |   1    3
-------------+------+------+-----------+---------
x5o   . .  & |    5 |    5 | 2880    * |   1    1
x .   . x    |    4 |    4 |    * 3600 |   0    2
-------------+------+------+-----------+---------
x5o5/2o .  & ♦   12 |   30 |   12    0 | 240    *
x5o   . x  & ♦   10 |   15 |    2    5 |   * 1440

x5o5/3o5/4x

. .   .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
------------+------+-----------+----------------+----------------
x .   .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
. .   .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
------------+------+-----------+----------------+----------------
x5o   .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x .   .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
. .   o5/4x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
------------+------+-----------+----------------+----------------
x5o5/3o   . ♦   12 |   30    0 |   12    0    0 | 120   *   *   *
x5o   .   x ♦   10 |   10    5 |    2    5    0 |   * 720   *   *
x .   o5/4x ♦   10 |    5   10 |    0    5    2 |   *   * 720   *
. o5/3o5/4x ♦   12 |    0   30 |    0    0   12 |   *   *   * 120

x5/4o5/2o5/4x

.   .   .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
--------------+------+-----------+----------------+----------------
x   .   .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
.   .   .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
--------------+------+-----------+----------------+----------------
x5/4o   .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x   .   .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
.   .   o5/4x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
--------------+------+-----------+----------------+----------------
x5/4o5/2o   . ♦   12 |   30    0 |   12    0    0 | 120   *   *   *
x5/4o   .   x ♦   10 |   10    5 |    2    5    0 |   * 720   *   *
x   .   o5/4x ♦   10 |    5   10 |    0    5    2 |   *   * 720   *
.   o5/2o5/4x ♦   12 |    0   30 |    0    0   12 |   *   *   * 120

or
.   .   .   .    | 1440 |   10 |   10   10 |   2   10
-----------------+------+------+-----------+---------
x   .   .   .  & |    2 | 7200 |    2    2 |   1    3
-----------------+------+------+-----------+---------
x5/4o   .   .  & |    5 |    5 | 2880    * |   1    1
x   .   .   x    |    4 |    4 |    * 3600 |   0    2
-----------------+------+------+-----------+---------
x5/4o5/2o   .  & ♦   12 |   30 |   12    0 | 240    *
x5/4o   .   x  & ♦   10 |   15 |    2    5 |   * 1440