Looks like a compound of a gaghi with idhi. Its vertex figure is a variation of 2gike+sissid, in fact v3/2f5/2o. Edges coincide by 3 and pentagons come in coincident pairs.

As abstract polytope gaghi+idhi is isomorphic to sishi+idhi, thereby interchanging pentagrams and pentagons resp. replacing gad by sissid.

Incidence matrix according to Dynkin symbol

o5/2o3o5/2x5/4*b

.   . .   .      | 120 |   60 |   60  30 |  12  20
-----------------+-----+------+----------+--------
.   . .   x      |   2 | 3600 |    2   1 |   1   2
-----------------+-----+------+----------+--------
.   o .   x5/4*b |   5 |    5 | 1440   * |   1   1
.   . o5/2x      |   5 |    5 |    * 720 |   0   2
-----------------+-----+------+----------+--------
o5/2o .   x5/4*b ♦  12 |   30 |   12   0 | 120   *
.   o3o5/2x5/4*b ♦  20 |   60 |   12  12 |   * 120

o5/2o3o5/3x5*b

.   . .   .    | 120 |   60 |   60  30 |  12  20
---------------+-----+------+----------+--------
.   . .   x    |   2 | 3600 |    2   1 |   1   2
---------------+-----+------+----------+--------
.   o .   x5*b |   5 |    5 | 1440   * |   1   1
.   . o5/3x    |   5 |    5 |    * 720 |   0   2
---------------+-----+------+----------+--------
o5/2o .   x5*b ♦  12 |   30 |   12   0 | 120   *
.   o3o5/3x5*b ♦  20 |   60 |   12  12 |   * 120

o5/2o3/2o5/2x5*b

.   .   .   .    | 120 |   60 |   60  30 |  12  20
-----------------+-----+------+----------+--------
.   .   .   x    |   2 | 3600 |    2   1 |   1   2
-----------------+-----+------+----------+--------
.   o   .   x5*b |   5 |    5 | 1440   * |   1   1
.   .   o5/2x    |   5 |    5 |    * 720 |   0   2
-----------------+-----+------+----------+--------
o5/2o   .   x5*b ♦  12 |   30 |   12   0 | 120   *
.   o3/2o5/2x5*b ♦  20 |   60 |   12  12 |   * 120

o5/2o3/2o5/3x5/4*b

.   .   .   .      | 120 |   60 |   60  30 |  12  20
-------------------+-----+------+----------+--------
.   .   .   x      |   2 | 3600 |    2   1 |   1   2
-------------------+-----+------+----------+--------
.   o   .   x5/4*b |   5 |    5 | 1440   * |   1   1
.   .   o5/3x      |   5 |    5 |    * 720 |   0   2
-------------------+-----+------+----------+--------
o5/2o   .   x5/4*b ♦  12 |   30 |   12   0 | 120   *
.   o3/2o5/3x5/4*b ♦  20 |   60 |   12  12 |   * 120

o5/3o3o5/2x5/4*b

.   . .   .      | 120 |   60 |   60  30 |  12  20
-----------------+-----+------+----------+--------
.   . .   x      |   2 | 3600 |    2   1 |   1   2
-----------------+-----+------+----------+--------
.   o .   x5/4*b |   5 |    5 | 1440   * |   1   1
.   . o5/2x      |   5 |    5 |    * 720 |   0   2
-----------------+-----+------+----------+--------
o5/3o .   x5/4*b ♦  12 |   30 |   12   0 | 120   *
.   o3o5/2x5/4*b ♦  20 |   60 |   12  12 |   * 120

o5/3o3o5/3x5*b

.   . .   .    | 120 |   60 |   60  30 |  12  20
---------------+-----+------+----------+--------
.   . .   x    |   2 | 3600 |    2   1 |   1   2
---------------+-----+------+----------+--------
.   o .   x5*b |   5 |    5 | 1440   * |   1   1
.   . o5/3x    |   5 |    5 |    * 720 |   0   2
---------------+-----+------+----------+--------
o5/3o .   x5*b ♦  12 |   30 |   12   0 | 120   *
.   o3o5/3x5*b ♦  20 |   60 |   12  12 |   * 120

o5/3o3/2o5/2x5*b

.   .   .   .    | 120 |   60 |   60  30 |  12  20
-----------------+-----+------+----------+--------
.   .   .   x    |   2 | 3600 |    2   1 |   1   2
-----------------+-----+------+----------+--------
.   o   .   x5*b |   5 |    5 | 1440   * |   1   1
.   .   o5/2x    |   5 |    5 |    * 720 |   0   2
-----------------+-----+------+----------+--------
o5/3o   .   x5*b ♦  12 |   30 |   12   0 | 120   *
.   o3/2o5/2x5*b ♦  20 |   60 |   12  12 |   * 120

o5/3o3/2o5/3x5/4*b

.   .   .   .      | 120 |   60 |   60  30 |  12  20
-------------------+-----+------+----------+--------
.   .   .   x      |   2 | 3600 |    2   1 |   1   2
-------------------+-----+------+----------+--------
.   o   .   x5/4*b |   5 |    5 | 1440   * |   1   1
.   .   o5/3x      |   5 |    5 |    * 720 |   0   2
-------------------+-----+------+----------+--------
o5/3o   .   x5/4*b ♦  12 |   30 |   12   0 | 120   *
.   o3/2o5/3x5/4*b ♦  20 |   60 |   12  12 |   * 120