Acronym ... Name 2gahi (?) Circumradius (1+sqrt(5))/2 = 1.618034 Inradius (1+sqrt(5))/4 = 0.809017 Coordinates (τ, 0, 0, 0)             & all permutations, all changes of sign (vertex inscribed f/q-hex) (τ/2, τ/2, τ/2, τ/2)   & all permutations, all changes of sign (vertex inscribed f-tes) (τ2/2, 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: gahi   Grünbaumian relatives: fix+gahi+120id   gahi+gohi

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

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

Incidence matrix according to Dynkin symbol

```x5o3o5o3/2*b

. . . .      | 120 ♦  12 |   60 |  20  20
-------------+-----+-----+------+--------
x . . .      |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x5o . .      |   5 |   5 | 1440 |   1   1
-------------+-----+-----+------+--------
x5o3o .      ♦  20 |  30 |   12 | 120   *
x5o . o3/2*b ♦  20 |  30 |   12 |   * 120
```

```x5o3o5/4o3*b

. . .   .    | 120 ♦  12 |   60 |  20  20
-------------+-----+-----+------+--------
x . .   .    |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x5o .   .    |   5 |   5 | 1440 |   1   1
-------------+-----+-----+------+--------
x5o3o   .    ♦  20 |  30 |   12 | 120   *
x5o .   o3*b ♦  20 |  30 |   12 |   * 120
```
```or
. . .   .       | 120 ♦  12 |   60 |  40
----------------+-----+-----+------+----
x . .   .       |   2 | 720 |   10 |  10
----------------+-----+-----+------+----
x5o .   .       |   5 |   5 | 1440 |   2
----------------+-----+-----+------+----
x5o3o   .     & ♦  20 |  30 |   12 | 240
```

```x5o3/2o5/4o3/2*b

. .   .   .      | 120 ♦  12 |   60 |  20  20
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5o   .   o3/2*b ♦  20 |  30 |   12 |   * 120
```
```or
. .   .   .         | 120 ♦  12 |   60 |  40
--------------------+-----+-----+------+----
x .   .   .         |   2 | 720 |   10 |  10
--------------------+-----+-----+------+----
x5o   .   .         |   5 |   5 | 1440 |   2
--------------------+-----+-----+------+----
x5o3/2o   .       & ♦  20 |  30 |   12 | 240
```

```x5/4o3o5o3/2*b

.   . . .      | 120 ♦  12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . . .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/4o . .      |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/4o3o .      ♦  20 |  30 |   12 | 120   *
x5/4o . o3/2*b ♦  20 |  30 |   12 |   * 120
```

```x5/4o3o5/4o3*b

.   . .   .    | 120 ♦  12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/4o .   .    |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/4o3o   .    ♦  20 |  30 |   12 | 120   *
x5/4o .   o3*b ♦  20 |  30 |   12 |   * 120
```
```or
.   . .   .       | 120 ♦  12 |   60 |  40
------------------+-----+-----+------+----
x   . .   .       |   2 | 720 |   10 |  10
------------------+-----+-----+------+----
x5/4o .   .       |   5 |   5 | 1440 |   2
------------------+-----+-----+------+----
x5/4o3o   .     & ♦  20 |  30 |   12 | 240
```

```x5/4o3/2o5/4o3/2*b

.   .   .   .      | 120 ♦  12 |   60 |  20  20
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5/4o   .   o3/2*b ♦  20 |  30 |   12 |   * 120
```
```or
.   .   .   .         | 120 ♦  12 |   60 |  40
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/4o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/4o3/2o   .       & ♦  20 |  30 |   12 | 240
```