Acronym ... Name gahi+gohi (?) Circumradius (1+sqrt(5))/2 = 1.618034 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 gahi   gohi   fix+gahi+120id   2gahi   ex+gohi+120id   2gohi

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

As abstract polytope gahi+gohi is isomorphic to gishi+gashi, thereby just scaling the edge figure, a complete pentagon, resp. replacing doe by gissid, gad by sissid, and replacing gacid vertex figures by cid ones.

Just as gacid itself (with vertices being identified) comes in 2 flavors, type A) as in o5/3x3o5*a with vertex figure [(5/3,3)5] or type B) as in o5/3o5/2x3*a with vertex figure [(5/2,3)5]/3, accordingly either one can be used as vertex figure here too. Thus gahi+gohi comes alike in 2 corresponding flavors (types being marked accordingly).

Incidence matrix according to Dynkin symbol

```x5o3o5o5/3*b (type A)

. . . .      | 120 ♦  12 |   60 |  20  12
-------------+-----+-----+------+--------
x . . .      |   2 | 720 |   10 |   5   5
-------------+-----+-----+------+--------
x5o . .      |   5 |   5 | 1440 |   1   1
-------------+-----+-----+------+--------
x5o3o .      ♦  20 |  30 |   12 | 120   *
x5o . o5/3*b ♦  12 |  30 |   12 |   * 120
```

```x5o3o5/4o5/2*b (type A)

. . .   .      | 120 ♦  12 |   60 |  20  12
---------------+-----+-----+------+--------
x . .   .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5o .   .      |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5o3o   .      ♦  20 |  30 |   12 | 120   *
x5o .   o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5o3/2o5o5/2*b (type A)

. .   . .      | 120 ♦  12 |   60 |  20  12
---------------+-----+-----+------+--------
x .   . .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5o   . .      |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5o3/2o .      ♦  20 |  30 |   12 | 120   *
x5o   . o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5o3/2o5/4o5/3*b (type A)

. .   .   .      | 120 ♦  12 |   60 |  20  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5o   .   o5/3*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3o5o5/3*b (type A)

.   . . .      | 120 ♦  12 |   60 |  20  12
---------------+-----+-----+------+--------
x   . . .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/4o . .      |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/4o3o .      ♦  20 |  30 |   12 | 120   *
x5/4o . o5/3*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3o5/4o5/2*b (type A)

.   . .   .      | 120 ♦  12 |   60 |  20  12
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/4o .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/4o3o   .      ♦  20 |  30 |   12 | 120   *
x5/4o .   o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3/2o5o5/2*b (type A)

.   .   . .      | 120 ♦  12 |   60 |  20  12
-----------------+-----+-----+------+--------
x   .   . .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/4o   . .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/4o3/2o .      ♦  20 |  30 |   12 | 120   *
x5/4o   . o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3/2o5/4o5/3*b (type A)

.   .   .   .      | 120 ♦  12 |   60 |  20  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5/4o   .   o5/3*b ♦  12 |  30 |   12 |   * 120
```

```x5o3o5/2o5/3*b (type B)

. . .   .      | 120 ♦  12 |   60 |  20  12
---------------+-----+-----+------+--------
x . .   .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5o .   .      |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5o3o   .      ♦  20 |  30 |   12 | 120   *
x5o .   o5/3*b ♦  12 |  30 |   12 |   * 120
```

```x5o3o5/3o5/2*b (type B)

. . .   .      | 120 ♦  12 |   60 |  20  12
---------------+-----+-----+------+--------
x . .   .      |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5o .   .      |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5o3o   .      ♦  20 |  30 |   12 | 120   *
x5o .   o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5o3/2o5/2o5/2*b (type B)

. .   .   .      | 120 ♦  12 |   60 |  20  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5o   .   o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5o3/2o5/3o5/3*b (type B)

. .   .   .      | 120 ♦  12 |   60 |  20  12
-----------------+-----+-----+------+--------
x .   .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5o   .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5o   .   o5/3*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3o5/2o5/3*b (type B)

.   . .   .      | 120 ♦  12 |   60 |  20  12
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/4o .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/4o3o   .      ♦  20 |  30 |   12 | 120   *
x5/4o .   o5/3*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3o5/3o5/2*b (type B)

.   . .   .      | 120 ♦  12 |   60 |  20  12
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/4o .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/4o3o   .      ♦  20 |  30 |   12 | 120   *
x5/4o .   o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3/2o5/2o5/2*b (type B)

.   .   .   .      | 120 ♦  12 |   60 |  20  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5/4o   .   o5/2*b ♦  12 |  30 |   12 |   * 120
```

```x5/4o3/2o5/3o5/3*b (type B)

.   .   .   .      | 120 ♦  12 |   60 |  20  12
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/4o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/4o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5/4o   .   o5/3*b ♦  12 |  30 |   12 |   * 120
```