Looks like a compound of 2 hi with vertices and edges identified, while faces and cells coincide by pairs.

As abstract polytope 2hi is isomorphic to 2gogishi, thereby replacing pentagons by pentagrams resp. replacing doe by gissid.

Incidence matrix according to Dynkin symbol

x5o3o3o3/2*b

. . . .      | 600 ♦    4 |   12 |   4   4
-------------+-----+------+------+--------
x . . .      |   2 | 1200 |    6 |   3   3
-------------+-----+------+------+--------
x5o . .      |   5 |    5 | 1440 |   1   1
-------------+-----+------+------+--------
x5o3o .      ♦  20 |   30 |   12 | 120   *
x5o . o3/2*b ♦  20 |   30 |   12 |   * 120

x5o3o3/2o3*b

. . .   .    | 600 ♦    4 |   12 |   4   4
-------------+-----+------+------+--------
x . .   .    |   2 | 1200 |    6 |   3   3
-------------+-----+------+------+--------
x5o .   .    |   5 |    5 | 1440 |   1   1
-------------+-----+------+------+--------
x5o3o   .    ♦  20 |   30 |   12 | 120   *
x5o .   o3*b ♦  20 |   30 |   12 |   * 120

or
. . .   .       | 600 ♦    4 |   12 |   8
----------------+-----+------+------+----
x . .   .       |   2 | 1200 |    6 |   6
----------------+-----+------+------+----
x5o .   .       |   5 |    5 | 1440 |   2
----------------+-----+------+------+----
x5o3o   .     & ♦  20 |   30 |   12 | 240

x5o3/2o3/2o3/2*b

. .   .   .      | 600 ♦    4 |   12 |   4   4
-----------------+-----+------+------+--------
x .   .   .      |   2 | 1200 |    6 |   3   3
-----------------+-----+------+------+--------
x5o   .   .      |   5 |    5 | 1440 |   1   1
-----------------+-----+------+------+--------
x5o3/2o   .      ♦  20 |   30 |   12 | 120   *
x5o   .   o3/2*b ♦  20 |   30 |   12 |   * 120

or
. .   .   .         | 600 ♦    4 |   12 |   8
--------------------+-----+------+------+----
x .   .   .         |   2 | 1200 |    6 |   6
--------------------+-----+------+------+----
x5o   .   .         |   5 |    5 | 1440 |   2
--------------------+-----+------+------+----
x5o3/2o   .       & ♦  20 |   30 |   12 | 240

x5/4o3o3o3/2*b

.   . . .      | 600 ♦    4 |   12 |   4   4
---------------+-----+------+------+--------
x   . . .      |   2 | 1200 |    6 |   3   3
---------------+-----+------+------+--------
x5/4o . .      |   5 |    5 | 1440 |   1   1
---------------+-----+------+------+--------
x5/4o3o .      ♦  20 |   30 |   12 | 120   *
x5/4o . o3/2*b ♦  20 |   30 |   12 |   * 120

x5/4o3o3/2o3*b

.   . .   .    | 600 ♦    4 |   12 |   4   4
---------------+-----+------+------+--------
x   . .   .    |   2 | 1200 |    6 |   3   3
---------------+-----+------+------+--------
x5/4o .   .    |   5 |    5 | 1440 |   1   1
---------------+-----+------+------+--------
x5/4o3o   .    ♦  20 |   30 |   12 | 120   *
x5/4o .   o3*b ♦  20 |   30 |   12 |   * 120

or
.   . .   .       | 600 ♦    4 |   12 |   8
------------------+-----+------+------+----
x   . .   .       |   2 | 1200 |    6 |   6
------------------+-----+------+------+----
x5/4o .   .       |   5 |    5 | 1440 |   2
------------------+-----+------+------+----
x5/4o3o   .     & ♦  20 |   30 |   12 | 240

x5/4o3/2o3/2o3/2*b

.   .   .   .      | 600 ♦    4 |   12 |   4   4
-------------------+-----+------+------+--------
x   .   .   .      |   2 | 1200 |    6 |   3   3
-------------------+-----+------+------+--------
x5/4o   .   .      |   5 |    5 | 1440 |   1   1
-------------------+-----+------+------+--------
x5/4o3/2o   .      ♦  20 |   30 |   12 | 120   *
x5/4o   .   o3/2*b ♦  20 |   30 |   12 |   * 120

or
.   .   .   .         | 600 ♦    4 |   12 |   8
----------------------+-----+------+------+----
x   .   .   .         |   2 | 1200 |    6 |   6
----------------------+-----+------+------+----
x5/4o   .   .         |   5 |    5 | 1440 |   2
----------------------+-----+------+------+----
x5/4o3/2o   .       & ♦  20 |   30 |   12 | 240