Acronym ...
Name 2gishi (?)
Circumradius (sqrt(5)-1)/2 = 0.618034
Inradius (sqrt(5)-1)/4 = 0.309017
General of army ex
Colonel of regiment gishi
Confer
non-Grünbaumian master:
gishi  
Grünbaumian relatives:
gishi+gashi  

Looks like a compound of 2 gishi with vertices and edges identified, while faces and cells coincide by pairs.

As abstract polytope 2gishi is isomorphic to 2gahi, thereby replacing gissid by doe, resp. replacing pentagrammal faces by pentagonal ones, resp. replacing 2ike vertex figures by 2gike ones.


Incidence matrix according to Dynkin symbol

x5/2o3o5/2o3*b

.   . .   .    | 120   12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/2o .   .    |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/2o3o   .      20 |  30 |   12 | 120   *
x5/2o .   o3*b   20 |  30 |   12 |   * 120
or
.   . .   .       | 120   12 |   60 |  40
------------------+-----+-----+------+----
x   . .   .       |   2 | 720 |   10 |  10
------------------+-----+-----+------+----
x5/2o .   .       |   5 |   5 | 1440 |   2
------------------+-----+-----+------+----
x5/2o3o   .     &   20 |  30 |   12 | 240

x5/2o3o5/3o3/2*b

.   . .   .      | 120   12 |   60 |  20  20
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/2o .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/2o3o   .        20 |  30 |   12 | 120   *
x5/2o .   o3/2*b   20 |  30 |   12 |   * 120

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

.   .   .   .      | 120   12 |   60 |  20  20
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/2o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/2o3/2o   .        20 |  30 |   12 | 120   *
x5/2o   .   o3/2*b   20 |  30 |   12 |   * 120
or
.   .   .   .         | 120   12 |   60 |  40
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/2o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/2o3/2o   .       &   20 |  30 |   12 | 240

x5/3o3o5/2o3*b

.   . .   .    | 120   12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/3o .   .    |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/3o3o   .      20 |  30 |   12 | 120   *
x5/3o .   o3*b   20 |  30 |   12 |   * 120
or
.   . .   .       | 120   12 |   60 |  40
------------------+-----+-----+------+----
x   . .   .       |   2 | 720 |   10 |  10
------------------+-----+-----+------+----
x5/3o .   .       |   5 |   5 | 1440 |   2
------------------+-----+-----+------+----
x5/3o3o   .     &   20 |  30 |   12 | 240

x5/3o3o5/3o3/2*b

.   . .   .      | 120   12 |   60 |  20  20
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/3o .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/3o3o   .        20 |  30 |   12 | 120   *
x5/3o .   o3/2*b   20 |  30 |   12 |   * 120

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

.   .   .   .      | 120   12 |   60 |  20  20
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/3o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/3o3/2o   .        20 |  30 |   12 | 120   *
x5/3o   .   o3/2*b   20 |  30 |   12 |   * 120
or
.   .   .   .         | 120   12 |   60 |  40
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/3o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/3o3/2o   .       &   20 |  30 |   12 | 240

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