Looks like a compound of a gaghi with didhi. The vertex figure is f3x5/3o5/2*a, a facetting of doe. Edges coincide by 3 and pentagons come in coincident pairs.

As abstract polytope gaghi+didhi is isomorphic to sishi+ofiddady, thereby replacing pentagons by pentagrams, resp. gidtid by sidtid, gad by sissid and gike by ike.

Incidence matrix according to Dynkin symbol

o3x5o5/3o5/3*a3/2*c

. . .   .           | 120 |   60 |   60   60 |  20  12  12
--------------------+-----+------+-----------+------------
. x .   .           |   2 | 3600 |    2    2 |   2   1   1
--------------------+-----+------+-----------+------------
o3x .   .           |   3 |    3 | 2400    * |   1   1   0
. x5o   .           |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3x5o   .   *a3/2*c ♦  20 |   60 |   20   12 | 120   *   *
o3x .   o5/3*a      ♦  12 |   30 |   20    0 |   * 120   *
. x5o5/3o           ♦  12 |   30 |    0   12 |   *   * 120

o3x5/4o5/3o5/2*a3*c

. .   .   .         | 120 |   60 |   60   60 |  20  12  12
--------------------+-----+------+-----------+------------
. x   .   .         |   2 | 3600 |    2    2 |   2   1   1
--------------------+-----+------+-----------+------------
o3x   .   .         |   3 |    3 | 2400    * |   1   1   0
. x5/4o   .         |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3x5/4o   .   *a3*c ♦  20 |   60 |   20   12 | 120   *   *
o3x   .   o5/2*a    ♦  12 |   30 |   20    0 |   * 120   *
. x5/4o5/3o         ♦  12 |   30 |    0   12 |   *   * 120

o3x5/4o5/2o5/3*a3*c

. .   .   .         | 120 |   60 |   60   60 |  20  12  12
--------------------+-----+------+-----------+------------
. x   .   .         |   2 | 3600 |    2    2 |   2   1   1
--------------------+-----+------+-----------+------------
o3x   .   .         |   3 |    3 | 2400    * |   1   1   0
. x5/4o   .         |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3x5/4o   .   *a3*c ♦  20 |   60 |   20   12 | 120   *   *
o3x   .   o5/3*a    ♦  12 |   30 |   20    0 |   * 120   *
. x5/4o5/2o         ♦  12 |   30 |    0   12 |   *   * 120

o3x5o5/2o5/2*a3/2*c

. . .   .           | 120 |   60 |   60   60 |  20  12  12
--------------------+-----+------+-----------+------------
. x .   .           |   2 | 3600 |    2    2 |   2   1   1
--------------------+-----+------+-----------+------------
o3x .   .           |   3 |    3 | 2400    * |   1   1   0
. x5o   .           |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3x5o   .   *a3/2*c ♦  20 |   60 |   20   12 | 120   *   *
o3x .   o5/2*a      ♦  12 |   30 |   20    0 |   * 120   *
. x5o5/2o           ♦  12 |   30 |    0   12 |   *   * 120

o3/2x5/4o5/3o5/3*a3/2*c

.   .   .   .           | 120 |   60 |   60   60 |  20  12  12
------------------------+-----+------+-----------+------------
.   x   .   .           |   2 | 3600 |    2    2 |   2   1   1
------------------------+-----+------+-----------+------------
o3/2x   .   .           |   3 |    3 | 2400    * |   1   1   0
.   x5/4o   .           |   5 |    5 |    * 1440 |   1   0   1
------------------------+-----+------+-----------+------------
o3/2x5/4o   .   *a3/2*c ♦  20 |   60 |   20   12 | 120   *   *
o3/2x   .   o5/3*a      ♦  12 |   30 |   20    0 |   * 120   *
.   x5/4o5/3o           ♦  12 |   30 |    0   12 |   *   * 120

o3/2x5o5/3o5/2*a3*c

.   . .   .         | 120 |   60 |   60   60 |  20  12  12
--------------------+-----+------+-----------+------------
.   x .   .         |   2 | 3600 |    2    2 |   2   1   1
--------------------+-----+------+-----------+------------
o3/2x .   .         |   3 |    3 | 2400    * |   1   1   0
.   x5o   .         |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3/2x5o   .   *a3*c ♦  20 |   60 |   20   12 | 120   *   *
o3/2x .   o5/2*a    ♦  12 |   30 |   20    0 |   * 120   *
.   x5o5/3o         ♦  12 |   30 |    0   12 |   *   * 120

o3/2x5o5/2o5/3*a3*c

.   . .   .         | 120 |   60 |   60   60 |  20  12  12
--------------------+-----+------+-----------+------------
.   x .   .         |   2 | 3600 |    2    2 |   2   1   1
--------------------+-----+------+-----------+------------
o3/2x .   .         |   3 |    3 | 2400    * |   1   1   0
.   x5o   .         |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3/2x5o   .   *a3*c ♦  20 |   60 |   20   12 | 120   *   *
o3/2x .   o5/3*a    ♦  12 |   30 |   20    0 |   * 120   *
.   x5o5/2o         ♦  12 |   30 |    0   12 |   *   * 120

o3/2x5/4o5/2o5/2*a3/2*c

.   .   .   .           | 120 |   60 |   60   60 |  20  12  12
------------------------+-----+------+-----------+------------
.   x   .   .           |   2 | 3600 |    2    2 |   2   1   1
------------------------+-----+------+-----------+------------
o3/2x   .   .           |   3 |    3 | 2400    * |   1   1   0
.   x5/4o   .           |   5 |    5 |    * 1440 |   1   0   1
------------------------+-----+------+-----------+------------
o3/2x5/4o   .   *a3/2*c ♦  20 |   60 |   20   12 | 120   *   *
o3/2x   .   o5/2*a      ♦  12 |   30 |   20    0 |   * 120   *
.   x5/4o5/2o           ♦  12 |   30 |    0   12 |   *   * 120