Acronym ...
Name sishi+gaghi+120did (?)
Circumradius 1
General of army ex
Colonel of regiment sishi
Confer gaghi   gaghi+paphacki+sridaphi   2gaghi   2gaghi+2paphacki   sishi   sishi+idhi   sishi+paphicki+gridaphi   2sishi   2sishi+2paphicki  

Looks like a compound of a sishi with gaghi. Cells are connected by further 120 diametral did. Its vertex figure is a variation of cadditradid, in fact v5o5/3f or f5/2o5/4v. Edges coincide by 3, all faces come in coincident pairs, and the did also come in coincident pairs.

As abstract polytope sishi+gaghi+120did is automorph, thereby interchanging pentagrams and pentagons resp. sissid and gad.


Incidence matrix according to Dynkin symbol

x5o5/2o5o5/3*a

. .   . .      | 120 |   60 |   60   60 |  12  30  12
---------------+-----+------+-----------+------------
x .   . .      |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x5o   . .      |   5 |    5 | 1440    * |   1   1   0
x .   . o5/3*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x5o5/2o .        12 |   30 |   12    0 | 120   *   *
x5o   . o5/3*a   30 |   60 |   12   12 |   * 120   *
x .   o5o5/3*a   12 |   30 |    0   12 |   *   * 120

x5o5/2o5/4o5/2*a

. .   .   .      | 120 |   60 |   60   60 |  12  30  12
-----------------+-----+------+-----------+------------
x .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x5o   .   .      |   5 |    5 | 1440    * |   1   1   0
x .   .   o5/2*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x5o5/2o   .        12 |   30 |   12    0 | 120   *   *
x5o   .   o5/2*a   30 |   60 |   12   12 |   * 120   *
x .   o5/4o5/2*a   12 |   30 |    0   12 |   *   * 120

x5o5/3o5o5/2*a

. .   . .      | 120 |   60 |   60   60 |  12  30  12
---------------+-----+------+-----------+------------
x .   . .      |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x5o   . .      |   5 |    5 | 1440    * |   1   1   0
x .   . o5/2*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x5o5/3o .        12 |   30 |   12    0 | 120   *   *
x5o   . o5/2*a   30 |   60 |   12   12 |   * 120   *
x .   o5o5/2*a   12 |   30 |    0   12 |   *   * 120

x5/4o5/2o5o5/2*a

.   .   . .      | 120 |   60 |   60   60 |  12  30  12
-----------------+-----+------+-----------+------------
x   .   . .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x5/4o   . .      |   5 |    5 | 1440    * |   1   1   0
x   .   . o5/2*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x5/4o5/2o .        12 |   30 |   12    0 | 120   *   *
x5/4o   . o5/2*a   30 |   60 |   12   12 |   * 120   *
x   .   o5o5/2*a   12 |   30 |    0   12 |   *   * 120

x5o5/3o5/4o5/3*a

. .   .   .      | 120 |   60 |   60   60 |  12  30  12
-----------------+-----+------+-----------+------------
x .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x5o   .   .      |   5 |    5 | 1440    * |   1   1   0
x .   .   o5/3*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x5o5/3o   .        12 |   30 |   12    0 | 120   *   *
x5o   .   o5/3*a   30 |   60 |   12   12 |   * 120   *
x .   o5/4o5/3*a   12 |   30 |    0   12 |   *   * 120

x5/4o5/2o5/4o5/3*a

.   .   .   .      | 120 |   60 |   60   60 |  12  30  12
-------------------+-----+------+-----------+------------
x   .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-------------------+-----+------+-----------+------------
x5/4o   .   .      |   5 |    5 | 1440    * |   1   1   0
x   .   .   o5/3*a |   5 |    5 |    * 1440 |   0   1   1
-------------------+-----+------+-----------+------------
x5/4o5/2o   .        12 |   30 |   12    0 | 120   *   *
x5/4o   .   o5/3*a   30 |   60 |   12   12 |   * 120   *
x   .   o5/4o5/3*a   12 |   30 |    0   12 |   *   * 120

x5/4o5/3o5o5/3*a

.   .   . .      | 120 |   60 |   60   60 |  12  30  12
-----------------+-----+------+-----------+------------
x   .   . .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x5/4o   . .      |   5 |    5 | 1440    * |   1   1   0
x   .   . o5/3*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x5/4o5/3o .        12 |   30 |   12    0 | 120   *   *
x5/4o   . o5/3*a   30 |   60 |   12   12 |   * 120   *
x   .   o5o5/3*a   12 |   30 |    0   12 |   *   * 120

x5/4o5/3o5/4o5/2*a

.   .   .   .      | 120 |   60 |   60   60 |  12  30  12
-------------------+-----+------+-----------+------------
x   .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-------------------+-----+------+-----------+------------
x5/4o   .   .      |   5 |    5 | 1440    * |   1   1   0
x   .   .   o5/2*a |   5 |    5 |    * 1440 |   0   1   1
-------------------+-----+------+-----------+------------
x5/4o5/3o   .        12 |   30 |   12    0 | 120   *   *
x5/4o   .   o5/2*a   30 |   60 |   12   12 |   * 120   *
x   .   o5/4o5/2*a   12 |   30 |    0   12 |   *   * 120

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