Acronym ...
Name 2fix (?)
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:
fix  
Grünbaumian relatives:
fix+gahi+120id  

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

As abstract polytope 2fix is isomorphic to 2gofix, thereby replacing ike by gike, resp. replacing pentagrammal edge figures by pentagonal ones, resp. replacing 2gad vertex figures by 2sissid ones.


Incidence matrix according to Dynkin symbol

x3o5o5o5/4*b

. . . .      | 120   12 |   60 |  12  12
-------------+-----+-----+------+--------
x . . .      |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x3o . .      |   3 |   3 | 2400 |   1   1
-------------+-----+-----+------+--------
x3o5o .        12 |  30 |   20 | 120   *
x3o . o5/4*b   12 |  30 |   20 |   * 120

x3o5o5/4o5*b

. . .   .    | 120   12 |   60 |  12  12
-------------+-----+-----+------+--------
x . .   .    |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x3o .   .    |   3 |   3 | 2400 |   1   1
-------------+-----+-----+------+--------
x3o5o   .      12 |  30 |   20 | 120   *
x3o .   o5*b   12 |  30 |   20 |   * 120
or
. . .   .       | 120   12 |   60 |  24
----------------+-----+-----+------+----
x . .   .       |   2 | 720 |   10 |  10
----------------+-----+-----+------+----
x3o .   .       |   3 |   3 | 2400 |   2
----------------+-----+-----+------+----
x3o5o   .     &   12 |  30 |   20 | 240

x3o5/4o5/4o5/4*b

. .   .   .      | 120   12 |   60 |  12  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x3o   .   .      |   3 |   3 | 2400 |   1   1
-----------------+-----+-----+------+--------
x3o5/4o   .        12 |  30 |   20 | 120   *
x3o   .   o5/4*b   12 |  30 |   20 |   * 120
or
. .   .   .         | 120   12 |   60 |  24
--------------------+-----+-----+------+----
x .   .   .         |   2 | 720 |   10 |  10
--------------------+-----+-----+------+----
x3o   .   .         |   3 |   3 | 2400 |   2
--------------------+-----+-----+------+----
x3o5/4o   .       &   12 |  30 |   20 | 240

x3/2o5o5o5/4*b

.   . . .      | 120   12 |   60 |  12  12
---------------+-----+-----+------+--------
x   . . .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3/2o . .      |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3/2o5o .        12 |  30 |   20 | 120   *
x3/2o . o5/4*b   12 |  30 |   20 |   * 120

x3/2o5o5/4o5*b

.   . .   .    | 120   12 |   60 |  12  12
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3/2o .   .    |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3/2o5o   .      12 |  30 |   20 | 120   *
x3/2o .   o5*b   12 |  30 |   20 |   * 120
or
.   . .   .       | 120   12 |   60 |  24
------------------+-----+-----+------+----
x   . .   .       |   2 | 720 |   10 |  10
------------------+-----+-----+------+----
x3/2o .   .       |   3 |   3 | 2400 |   2
------------------+-----+-----+------+----
x3/2o5o   .     &   12 |  30 |   20 | 240

x3/2o5/4o5/4o5/4*b

.   .   .   .      | 120   12 |   60 |  12  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x3/2o   .   .      |   3 |   3 | 2400 |   1   1
-------------------+-----+-----+------+--------
x3/2o5/4o   .        12 |  30 |   20 | 120   *
x3/2o   .   o5/4*b   12 |  30 |   20 |   * 120
or
.   .   .   .         | 120   12 |   60 |  24
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x3/2o   .   .         |   3 |   3 | 2400 |   2
----------------------+-----+-----+------+----
x3/2o5/4o   .       &   12 |  30 |   20 | 240

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