Acronym ...
Name 2gohi (?)
Circumradius (1+sqrt(5))/2 = 1.618034
Inradius (3+sqrt(5))/4 = 1.309017
Coordinates
  1. (τ, 0, 0, 0)             & all permutations, all changes of sign
    (vertex inscribed f/q-hex)
  2. (τ/2, τ/2, τ/2, τ/2)   & all permutations, all changes of sign
    (vertex inscribed f-tes)
  3. 2/2, τ/2, 1/2, 0)   & even permutations, all changes of sign
    (vertex inscribed sadi)
where τ = (1+sqrt(5))/2 (a. and b. together define a vertex inscribed f-ico)
General of army ex
Colonel of regiment ex
Confer
non-Grünbaumian master:
gohi  
Grünbaumian relatives:
ex+gohi+120id  

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

As abstract polytope 2gohi is isomorphic to 2gashi, thereby replacing gad by sissid, resp. replacing pentagonal faces by pentagrammal ones, resp. replacing 2sissid vertex figures by 2gad ones.


Incidence matrix according to Dynkin symbol

x5o5/2o5/2o5/2*b

. .   .   .      | 120   12 |   60 |  12  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o5/2o   .        12 |  30 |   12 | 120   *
x5o   .   o5/2*b   12 |  30 |   12 |   * 120
or
. .   .   .         | 120   12 |   60 |  24
--------------------+-----+-----+------+----
x .   .   .         |   2 | 720 |   10 |  10
--------------------+-----+-----+------+----
x5o   .   .         |   5 |   5 | 1440 |   2
--------------------+-----+-----+------+----
x5o5/2o   .       &   12 |  30 |   12 | 240

x5o5/2o5/3o5/3*b

. .   .   .      | 120   12 |   60 |  12  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o5/2o   .        12 |  30 |   12 | 120   *
x5o   .   o5/3*b   12 |  30 |   12 |   * 120

x5o5/3o5/2o5/3*b

. .   .   .      | 120   12 |   60 |  12  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o5/3o   .        12 |  30 |   12 | 120   *
x5o   .   o5/3*b   12 |  30 |   12 |   * 120
or
. .   .   .         | 120   12 |   60 |  24
--------------------+-----+-----+------+----
x .   .   .         |   2 | 720 |   10 |  10
--------------------+-----+-----+------+----
x5o   .   .         |   5 |   5 | 1440 |   2
--------------------+-----+-----+------+----
x5o5/3o   .       &   12 |  30 |   12 | 240

x5/4o5/2o5/2o5/2*b

.   .   .   .      | 120   12 |   60 |  12  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o5/2o   .        12 |  30 |   12 | 120   *
x5/4o   .   o5/2*b   12 |  30 |   12 |   * 120
or
.   .   .   .         | 120   12 |   60 |  24
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/4o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/4o5/2o   .       &   12 |  30 |   12 | 240

x5/4o5/2o5/3o5/3*b

.   .   .   .      | 120   12 |   60 |  12  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o5/2o   .        12 |  30 |   12 | 120   *
x5/4o   .   o5/3*b   12 |  30 |   12 |   * 120

x5/4o5/3o5/2o5/3*b

.   .   .   .      | 120   12 |   60 |  12  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o5/3o   .        12 |  30 |   12 | 120   *
x5/4o   .   o5/3*b   12 |  30 |   12 |   * 120
or
.   .   .   .         | 120   12 |   60 |  24
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/4o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/4o5/3o   .       &   12 |  30 |   12 | 240

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