Acronym ...
Name gax+gashi+120gid (?)
Circumradius (sqrt(5)-1)/2 = 0.618034
General of army ex
Colonel of regiment gishi
Confer gax   2gax   gax+gofix   gashi   2gashi  

Looks like a compound of a gax with gashi. Cells are connected by further 120 diametral gid. Its vertex figure is a variation of gidtid, in fact x3o5/4v or x3/2o5v (which degenerates to something what looks like an gike plus an encasing v-gad, but diametral rectangles will there be in addition). Accordingly edges come coincident by 5, all faces come in coincident pairs, and the gid also come in coincident pairs.

As abstract polytope gax+gashi+120gid is isomorphic to ex+gohi+120id, thereby replacing pentagrams by pentagons resp. replacing sissid by gad and gid by id.


Incidence matrix according to Dynkin symbol

x3o3o5o5/3*a

. . . .      | 120 |   60 |   60   60 |  20  30  12
-------------+-----+------+-----------+------------
x . . .      |   2 | 3600 |    2    2 |   1   2   1
-------------+-----+------+-----------+------------
x3o . .      |   3 |    3 | 2400    * |   1   1   0
x . . o5/3*a |   5 |    5 |    * 1440 |   0   1   1
-------------+-----+------+-----------+------------
x3o3o .         4 |    6 |    4    0 | 600   *   *
x3o . o5/3*a   30 |   60 |   20   12 |   * 120   *
x . o5o5/3*a   12 |   30 |    0   12 |   *   * 120

x3o3o5/4o5/2*a

. . .   .      | 120 |   60 |   60   60 |  20  30  12
---------------+-----+------+-----------+------------
x . .   .      |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x3o .   .      |   3 |    3 | 2400    * |   1   1   0
x . .   o5/2*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x3o3o   .         4 |    6 |    4    0 | 600   *   *
x3o .   o5/2*a   30 |   60 |   20   12 |   * 120   *
x . o5/4o5/2*a   12 |   30 |    0   12 |   *   * 120

x3o3/2o5o5/2*a

. .   . .      | 120 |   60 |   60   60 |  20  30  12
---------------+-----+------+-----------+------------
x .   . .      |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x3o   . .      |   3 |    3 | 2400    * |   1   1   0
x .   . o5/2*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x3o3/2o .         4 |    6 |    4    0 | 600   *   *
x3o   . o5/2*a   30 |   60 |   20   12 |   * 120   *
x .   o5o5/2*a   12 |   30 |    0   12 |   *   * 120

x3/2o3o5o5/2*a

.   . . .      | 120 |   60 |   60   60 |  20  30  12
---------------+-----+------+-----------+------------
x   . . .      |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x3/2o . .      |   3 |    3 | 2400    * |   1   1   0
x   . . o5/2*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x3/2o3o .         4 |    6 |    4    0 | 600   *   *
x3/2o . o5/2*a   30 |   60 |   20   12 |   * 120   *
x   . o5o5/2*a   12 |   30 |    0   12 |   *   * 120

x3o3/2o5/4o5/3*a

. .   .   .      | 120 |   60 |   60   60 |  20  30  12
-----------------+-----+------+-----------+------------
x .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3o   .   .      |   3 |    3 | 2400    * |   1   1   0
x .   .   o5/3*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3o3/2o   .         4 |    6 |    4    0 | 600   *   *
x3o   .   o5/3*a   30 |   60 |   20   12 |   * 120   *
x .   o5/4o5/3*a   12 |   30 |    0   12 |   *   * 120

x3/2o3o5/4o5/3*a

.   . .   .      | 120 |   60 |   60   60 |  20  30  12
-----------------+-----+------+-----------+------------
x   . .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3/2o .   .      |   3 |    3 | 2400    * |   1   1   0
x   . .   o5/3*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3/2o3o   .         4 |    6 |    4    0 | 600   *   *
x3/2o .   o5/3*a   30 |   60 |   20   12 |   * 120   *
x   . o5/4o5/3*a   12 |   30 |    0   12 |   *   * 120

x3/2o3/2o5o5/3*a

.   .   . .      | 120 |   60 |   60   60 |  20  30  12
-----------------+-----+------+-----------+------------
x   .   . .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3/2o   . .      |   3 |    3 | 2400    * |   1   1   0
x   .   . o5/3*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3/2o3/2o .         4 |    6 |    4    0 | 600   *   *
x3/2o   . o5/3*a   30 |   60 |   20   12 |   * 120   *
x   .   o5o5/3*a   12 |   30 |    0   12 |   *   * 120

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

.   .   .   .      | 120 |   60 |   60   60 |  20  30  12
-------------------+-----+------+-----------+------------
x   .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-------------------+-----+------+-----------+------------
x3/2o   .   .      |   3 |    3 | 2400    * |   1   1   0
x   .   .   o5/2*a |   5 |    5 |    * 1440 |   0   1   1
-------------------+-----+------+-----------+------------
x3/2o3/2o   .         4 |    6 |    4    0 | 600   *   *
x3/2o   .   o5/2*a   30 |   60 |   20   12 |   * 120   *
x   .   o5/4o5/2*a   12 |   30 |    0   12 |   *   * 120

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