Acronym ...
Name a3b4c,
general variation of great rhombicuboctahedron

Vertex layers
 Layer Symmetry Subsymmetries o3o4o o3o . . o4o 1 a3b4c a3b .(a,b)-{6} first . b4c(b,c)-{8} first 2 a3B . . C4c 3 C3B . . a4A 4 b3X .(layers 4 & 5 interchange for c > aq) . a4A 5 X3b . . C4c 6 B3C . . b4copposite (b,c)-{8} 7 B3a . 8 b3a .opposite (b,a)-{6}
(A=c+bq, B=b+cq, C=a+b, X=a+b+cq)
Lace tower
and approx. ASCII-art
```          b   a   b
o---o---o---o           - a3b
c / B c | a | c B \ c           height = c/sqrt(3)
o-------o---o-------o       - a3B
b /  B  b /  C  \ b  B  \ b        height = bq/sqrt(3)
o-------o-------o-------o     - C3B
c | b  c/  \a b a/  \c  b | c       height = c/sqrt(3) resp. height = aq/sqrt(3)
o---o     o---o     o---o     - b3X + X3b [comp. only if c=aq, else 2 layers: heigth = |c-aq|/sqrt(3)]
a | C a\  /c  B  c\  /a C | a       height = aq/sqrt(3) resp. height = c/sqrt(3)
o-----o-----------o-----o     - B3C
b \  a |b    B    b| a  / b        height = bq/sqrt(3)
o---o-----------o---o       - B3a
c \   \c     c/   / c           height = c/sqrt(3)
o---o---o---o           - b3a
a   b   a
```
```      c b   c   b c
o---o---o---o          - b4c
a  / C a | c | a C \  a          height = a/sqrt(2)
c o-------o---o-------o c    - C4c
b | a b /   A   \ b a | b         height = b/sqrt(2)
A o---o-----------o---o A    - a4A
c | a | c   A   c | a | c         height = c
A o---o-----------o---o A    - a4A
b | C b \   c   / b C | b         height = b/sqrt(2)
c o-------o---o-------o c    - C4c
a  \   a |   | a   /  a          height = a/sqrt(2)
c  o---o---o---o  c       - b4c
b   c   b
```
Coordinates (c/2, (bq+c)/2, (aq+bq+c)/2)     & all permutations & all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Especially
 girco (a=b=c=x) x3x4q q3x4x w3x4x toe (a=b=x, c=o) x3f4o x3u4o x3w4o u3x4o (-x)3x4o sirco (a=c=x, b=o) q3o4x f3o4x u3o4x x3o4q tic (a=o, b=c=x) o3x4q o3x4u o3q4x
Confer
general polytopal classes:
isogonal

Incidence matrix according to Dynkin symbol

```a3b4c   (a ≠ 0, b ≠ 0, c ≠ 0 : general girco-variant)

. . . | 48 |  1  1  1 | 1  1 1
------+----+----------+-------
a . . |  2 | 24  *  * | 1  1 0  a
. b . |  2 |  * 24  * | 1  0 1  b
. . c |  2 |  *  * 24 | 0  1 1  c
------+----+----------+-------
a3b . |  6 |  3  3  0 | 8  * *
a . c |  4 |  2  0  2 | * 12 *
. b4c |  8 |  0  4  4 | *  * 6
```
```Bb3aa3bB&#zc   → height = 0,
where: B = b+cq = b+c sqrt(2) (pseudo),
same as general girco-variant a3b4c)

o.3o.3o.     & | 48 |  1  1  1 | 1 1  1
---------------+----+----------+-------
.. a. ..     & |  2 | 24  *  * | 1 0  1  a
.. .. b.     & |  2 |  * 24  * | 1 1  0  b
oo3oo3oo&#c    |  2 |  *  * 24 | 0 1  1  c
---------------+----+----------+-------
.. a.3b.     & |  6 |  3  3  0 | 8 *  *
Bb .. bB&#zc   |  8 |  0  4  4 | * 6  *
.. aa ..&#c    |  4 |  2  0  2 | * * 12
```

```a3b4o   (a ≠ 0, b ≠ 0, c = 0 : general toe-variant)

. . . | 24 |  1  2 | 2 1
------+----+-------+----
a . . |  2 | 12  * | 2 0  a
. b . |  2 |  * 24 | 1 1  b
------+----+-------+----
a3b . |  6 |  3  3 | 8 *
. b4o |  4 |  0  4 | * 6
```
```b3a3b   (same as general toe-variant a3b4o)

. . .    | 24 |  2  1 | 2 1
---------+----+-------+----
b . .  & |  2 | 24  * | 1 1  b
. a .    |  2 |  * 12 | 2 0  a
---------+----+-------+----
b3a .  & |  6 |  3  3 | 8 *
b . b    |  4 |  4  0 | * 6
```

```a3o4c   (a ≠ 0, b = 0, c ≠ 0 : general sirco-variant)

. . . | 24 |  2  2 | 1  2 1
------+----+-------+-------
a . . |  2 | 24  * | 1  1 0  a
. . c |  2 |  * 24 | 0  1 1  c
------+----+-------+-------
a3o . |  3 |  3  0 | 8  * *
a . c |  4 |  2  2 | * 12 *
. o4c |  4 |  0  4 | *  * 6
```
```Co3aa3oC&#zc   → height = 0,
where: C = cq = c sqrt(2) (pseudo),
same as general sirco-variant a3o4c)

o.3o.3o.     & | 24 |  2  2 | 1 1  2
---------------+----+-------+-------
.. a. ..     & |  2 | 24  * | 1 0  1  a
oo3oo3oo&#c    |  2 |  * 24 | 0 1  1  c
---------------+----+-------+-------
.. a.3o.     & |  3 |  3  0 | 8 *  *
Co .. oC&#zc   |  4 |  0  4 | * 6  *
.. aa ..&#c    |  4 |  2  2 | * * 12
```

```o3b4c   (a = 0, b ≠ 0, c ≠ 0 : general tic-variant)

. . . | 24 |  2  1 | 1 2
------+----+-------+----
. b . |  2 | 24  * | 1 1  b
. . c |  2 |  * 12 | 0 2  c
------+----+-------+----
o3b . |  3 |  3  0 | 8 *
. b4c |  8 |  4  4 | * 6
```
```Bb3oo3bB&#zc   → height = 0,
where: B = b+cq = b+c sqrt(2) (pseudo),
same as general tic-variant o3b4c)

o.3o.3o.     & | 24 |  2  1 | 1 2
---------------+----+-------+----
.. .. b.     & |  2 | 24  * | 1 1  b
oo3oo3oo&#c    |  2 |  * 12 | 0 2  c
---------------+----+-------+----
.. o.3b.     & |  3 |  3  0 | 8 *
Bb .. bB&#zc   |  8 |  4  4 | * 6
```

```o3o4c   (a = b = 0, c ≠ 0 : c-scaled cube o3o4c)

. . . | 8 |  3 | 3
------+---+----+--
. . c | 2 | 12 | 2  c
------+---+----+--
. o4c | 4 |  4 | 6
```
```Co3oo3oC&#zc   → height = 0,
where: C = cq = c sqrt(2) (pseudo),
same as c-scaled cube o3o4c)

o.3o.3o.     & | 8 |  3 | 3
---------------+---+----+--
oo3oo3oo&#c    | 2 | 12 | 2  c
---------------+---+----+--
Co .. oC&#zc   | 4 |  4 | 6
```