Acronym ...
Name ex+fix (?)
Circumradius (1+sqrt(5))/2 = 1.618034
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 masters:
ex   fix  
Grünbaumian relatives:
2ex   2ex+fix   3ex+fix   3ex+2fix+gaghi   2fix  

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

As abstract polytope ex+fix is isomorphic to gax+gofix, thereby just scaling the edge figure, a complete pentagon, resp. replacing ike by gike and replacing cid vertex figures by gacid ones.

Just as cid itself (with vertices being identified) comes in 2 flavors, type A) as in o5x3/2o5*a with vertex figure [(3/2,5)5] or type B) as in o5/3o3x5*a with vertex figure [(3,5)5]/3, accordingly either one can be used as vertex figure here too. Thus ex+fix comes alike in 2 corresponding flavors (types being marked accordingly).


Incidence matrix according to Dynkin symbol

x3o3o5o5/4*b (type A)

. . . .      | 120   12 |   60 |  20  12
-------------+-----+-----+------+--------
x . . .      |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x3o . .      |   3 |   3 | 2400 |   1   1
-------------+-----+-----+------+--------
x3o3o .         4 |   6 |    4 | 600   *
x3o . o5/4*b   12 |  30 |   20 |   * 120

x3o3o5/4o5*b (type A)

. . .   .    | 120   12 |   60 |  20  12
-------------+-----+-----+------+--------
x . .   .    |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x3o .   .    |   3 |   3 | 2400 |   1   1
-------------+-----+-----+------+--------
x3o3o   .       4 |   6 |    4 | 600   *
x3o .   o5*b   12 |  30 |   20 |   * 120

x3o3/2o5o5*b (type A)

. .   . .    | 120   12 |   60 |  20  12
-------------+-----+-----+------+--------
x .   . .    |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x3o   . .    |   3 |   3 | 2400 |   1   1
-------------+-----+-----+------+--------
x3o3/2o .       4 |   6 |    4 | 600   *
x3o   . o5*b   12 |  30 |   20 |   * 120

x3o3/2o5/4o5/4*b (type A)

. .   .   .      | 120   12 |   60 |  20  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x3o   .   .      |   3 |   3 | 2400 |   1   1
-----------------+-----+-----+------+--------
x3o3/2o   .         4 |   6 |    4 | 600   *
x3o   .   o5/4*b   12 |  30 |   20 |   * 120

x3/2o3o5o5/4*b (type A)

.   . . .      | 120   12 |   60 |  20  12
---------------+-----+-----+------+--------
x   . . .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3/2o . .      |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3/2o3o .         4 |   6 |    4 | 600   *
x3/2o . o5/4*b   12 |  30 |   20 |   * 120

x3/2o3o5/4o5*b (type A)

.   . .   .    | 120   12 |   60 |  20  12
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3/2o .   .    |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3/2o3o   .       4 |   6 |    4 | 600   *
x3/2o .   o5*b   12 |  30 |   20 |   * 120

x3/2o3/2o5o5*b (type A)

.   .   . .    | 120   12 |   60 |  20  12
---------------+-----+-----+------+--------
x   .   . .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3/2o   . .    |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3/2o3/2o .       4 |   6 |    4 | 600   *
x3/2o   . o5*b   12 |  30 |   20 |   * 120

x3/2o3/2o5/4o5/4*b (type A)

.   .   .   .      | 120   12 |   60 |  20  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x3/2o   .   .      |   3 |   3 | 2400 |   1   1
-------------------+-----+-----+------+--------
x3/2o3/2o   .         4 |   6 |    4 | 600   *
x3/2o   .   o5/4*b   12 |  30 |   20 |   * 120

x3o3o5/2o5/4*b (type B)

. . .   .      | 120   12 |   60 |  20  12
---------------+-----+-----+------+--------
x . .   .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3o .   .      |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3o3o   .         4 |   6 |    4 | 600   *
x3o .   o5/4*b   12 |  30 |   20 |   * 120

x3o3o5/3o5*b (type B)

. . .   .    | 120   12 |   60 |  20  12
-------------+-----+-----+------+--------
x . .   .    |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x3o .   .    |   3 |   3 | 2400 |   1   1
-------------+-----+-----+------+--------
x3o3o   .       4 |   6 |    4 | 600   *
x3o .   o5*b   12 |  30 |   20 |   * 120

x3o3/2o5/2o5*b (type B)

. .   .   .    | 120   12 |   60 |  20  12
---------------+-----+-----+------+--------
x .   .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3o   .   .    |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3o3/2o   .       4 |   6 |    4 | 600   *
x3o   .   o5*b   12 |  30 |   20 |   * 120

x3o3/2o5/3o5/4*b (type B)

. .   .   .      | 120   12 |   60 |  20  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x3o   .   .      |   3 |   3 | 2400 |   1   1
-----------------+-----+-----+------+--------
x3o3/2o   .         4 |   6 |    4 | 600   *
x3o   .   o5/4*b   12 |  30 |   20 |   * 120

x3/2o3o5/2o5/4*b (type B)

.   . .   .      | 120   12 |   60 |  20  12
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x3/2o .   .      |   3 |   3 | 2400 |   1   1
-----------------+-----+-----+------+--------
x3/2o3o   .         4 |   6 |    4 | 600   *
x3/2o .   o5/4*b   12 |  30 |   20 |   * 120

x3/2o3o5/3o5*b (type B)

.   . .   .    | 120   12 |   60 |  20  12
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x3/2o .   .    |   3 |   3 | 2400 |   1   1
---------------+-----+-----+------+--------
x3/2o3o   .       4 |   6 |    4 | 600   *
x3/2o .   o5*b   12 |  30 |   20 |   * 120

x3/2o3/2o5/2o5*b (type B)

.   .   .   .    | 120   12 |   60 |  20  12
-----------------+-----+-----+------+--------
x   .   .   .    |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x3/2o   .   .    |   3 |   3 | 2400 |   1   1
-----------------+-----+-----+------+--------
x3/2o3/2o   .       4 |   6 |    4 | 600   *
x3/2o   .   o5*b   12 |  30 |   20 |   * 120

x3/2o3/2o5/3o5/4*b (type B)

.   .   .   .      | 120   12 |   60 |  20  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x3/2o   .   .      |   3 |   3 | 2400 |   1   1
-------------------+-----+-----+------+--------
x3/2o3/2o   .         4 |   6 |    4 | 600   *
x3/2o   .   o5/4*b   12 |  30 |   20 |   * 120

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