Looks like a compound of a sishi with ofiddady. The vertex figure is v3x5o5/4*a, a facetting of doe. Edges coincide by 3 and pentagrams come in coincident pairs.

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

Incidence matrix according to Dynkin symbol

o3x5/3o5o5*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
. x5/3o .         |   5 |    5 |    * 1440 |   1   0   1
------------------+-----+------+-----------+------------
o3x5/3o . *a3/2*c ♦  20 |   60 |   20   12 | 120   *   *
o3x   . o5*a      ♦  12 |   30 |   20    0 |   * 120   *
. x5/3o5o         ♦  12 |   30 |    0   12 |   *   * 120

o3x5/2o5o5/4*a3*c

. .   . .         | 120 |   60 |   60   60 |  20  12  12
------------------+-----+------+-----------+------------
. x   . .         |   2 | 3600 |    2    2 |   2   1   1
------------------+-----+------+-----------+------------
o3x   . .         |   3 |    3 | 2400    * |   1   1   0
. x5/2o .         |   5 |    5 |    * 1440 |   1   0   1
------------------+-----+------+-----------+------------
o3x5/2o .   *a3*c ♦  20 |   60 |   20   12 | 120   *   *
o3x   . o5/4*a    ♦  12 |   30 |   20    0 |   * 120   *
. x5/2o5o         ♦  12 |   30 |    0   12 |   *   * 120

o3x5/2o5/4o5*a3*c

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

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

o3/2x5/2o5o5*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
.   x5/2o .         |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3/2x5/2o . *a3/2*c ♦  20 |   60 |   20   12 | 120   *   *
o3/2x   . o5*a      ♦  12 |   30 |   20    0 |   * 120   *
.   x5/2o5o         ♦  12 |   30 |    0   12 |   *   * 120

o3/2x5/3o5o5/4*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
.   x5/3o .         |   5 |    5 |    * 1440 |   1   0   1
--------------------+-----+------+-----------+------------
o3/2x5/3o .   *a3*c ♦  20 |   60 |   20   12 | 120   *   *
o3/2x   . o5/4*a    ♦  12 |   30 |   20    0 |   * 120   *
.   x5/3o5o         ♦  12 |   30 |    0   12 |   *   * 120

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

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