Looks like a compound of a fix with gahi. Cells are connected by further 120 diametral id. Its vertex figure is a variation of gidtid, in fact f3o5/4x or f3/2o5x (which degenerates to something what looks like a gad plus an inscribed f-gike, but diametral rectangles will there be in addition). Accordingly edges coincide by 5, all faces come in coincident pairs, and the id also come in coincident pairs.

As abstract polytope fix+gahi+120id is isomorphic to gofix+gishi+120gid, thereby replacing pentagons by pentagrams resp. replacing doe by gissid, ike by gike and id by gid.

Incidence matrix according to Dynkin symbol

x3o5o3o5/4*a

. . . .      | 120 |   60 |   60   60 |  12  30  20
-------------+-----+------+-----------+------------
x . . .      |   2 | 3600 |    2    2 |   1   2   1
-------------+-----+------+-----------+------------
x3o . .      |   3 |    3 | 2400    * |   1   1   0
x . . o5/4*a |   5 |    5 |    * 1440 |   0   1   1
-------------+-----+------+-----------+------------
x3o5o .      ♦  12 |   30 |   20    0 | 120   *   *
x3o . o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x . o3o5/4*a ♦  20 |   30 |    0   12 |   *   * 120

x3o5o3/2o5*a

. . .   .    | 120 |   60 |   60   60 |  12  30  20
-------------+-----+------+-----------+------------
x . .   .    |   2 | 3600 |    2    2 |   1   2   1
-------------+-----+------+-----------+------------
x3o .   .    |   3 |    3 | 2400    * |   1   1   0
x . .   o5*a |   5 |    5 |    * 1440 |   0   1   1
-------------+-----+------+-----------+------------
x3o5o   .    ♦  12 |   30 |   20    0 | 120   *   *
x3o .   o5*a ♦  30 |   60 |   20   12 |   * 120   *
x . o3/2o5*a ♦  20 |   30 |    0   12 |   *   * 120

x3o5/4o3o5*a

. .   . .    | 120 |   60 |   60   60 |  12  30  20
-------------+-----+------+-----------+------------
x .   . .    |   2 | 3600 |    2    2 |   1   2   1
-------------+-----+------+-----------+------------
x3o   . .    |   3 |    3 | 2400    * |   1   1   0
x .   . o5*a |   5 |    5 |    * 1440 |   0   1   1
-------------+-----+------+-----------+------------
x3o5/4o .    ♦  12 |   30 |   20    0 | 120   *   *
x3o   . o5*a ♦  30 |   60 |   20   12 |   * 120   *
x .   o3o5*a ♦  20 |   30 |    0   12 |   *   * 120

x3/2o5o3o5*a

.   . . .    | 120 |   60 |   60   60 |  12  30  20
-------------+-----+------+-----------+------------
x   . . .    |   2 | 3600 |    2    2 |   1   2   1
-------------+-----+------+-----------+------------
x3/2o . .    |   3 |    3 | 2400    * |   1   1   0
x   . . o5*a |   5 |    5 |    * 1440 |   0   1   1
-------------+-----+------+-----------+------------
x3/2o5o .    ♦  12 |   30 |   20    0 | 120   *   *
x3/2o . o5*a ♦  30 |   60 |   20   12 |   * 120   *
x   . o3o5*a ♦  20 |   30 |    0   12 |   *   * 120

x3o5/4o3/2o5/4*a

. .   .   .      | 120 |   60 |   60   60 |  12  30  20
-----------------+-----+------+-----------+------------
x .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3o   .   .      |   3 |    3 | 2400    * |   1   1   0
x .   .   o5/4*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3o5/4o   .      ♦  12 |   30 |   20    0 | 120   *   *
x3o   .   o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x .   o3/2o5/4*a ♦  20 |   30 |    0   12 |   *   * 120

x3/2o5o3/2o5/4*a

.   . .   .      | 120 |   60 |   60   60 |  12  30  20
-----------------+-----+------+-----------+------------
x   . .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3/2o .   .      |   3 |    3 | 2400    * |   1   1   0
x   . .   o5/4*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3/2o5o   .      ♦  12 |   30 |   20    0 | 120   *   *
x3/2o .   o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x   . o3/2o5/4*a ♦  20 |   30 |    0   12 |   *   * 120

x3/2o5/4o3o5/4*a

.   .   . .      | 120 |   60 |   60   60 |  12  30  20
-----------------+-----+------+-----------+------------
x   .   . .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3/2o   . .      |   3 |    3 | 2400    * |   1   1   0
x   .   . o5/4*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3/2o5/4o .      ♦  12 |   30 |   20    0 | 120   *   *
x3/2o   . o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x   .   o3o5/4*a ♦  20 |   30 |    0   12 |   *   * 120

x3/2o5/4o3/2o5*a

.   .   .   .    | 120 |   60 |   60   60 |  12  30  20
-----------------+-----+------+-----------+------------
x   .   .   .    |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3/2o   .   .    |   3 |    3 | 2400    * |   1   1   0
x   .   .   o5*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3/2o5/4o   .    ♦  12 |   30 |   20    0 | 120   *   *
x3/2o   .   o5*a ♦  30 |   60 |   20   12 |   * 120   *
x   .   o3/2o5*a ♦  20 |   30 |    0   12 |   *   * 120