ico+gico+72{4}

Acronym	...
Name	ico+gico+72{4} (?)
Circumradius	1
Coordinates	(1, 0, 0, 0) & all permutations, all changes of sign (vertex inscribed q-hex) (±1/2, ±1/2, ±1/2, ±1/2) & all permutations, even number of minus signs (vertex inscribed q-hex) (±1/2, ±1/2, ±1/2, ±1/2) & all permutations, odd number of minus signs (vertex inscribed q-hex) The hull of any pair of those sets describes a tesseracts.
General of army	ico
Colonel of regiment	ico
Confer	ico gico ico+gico+24co 2ico+2gico

Looks like a compound of an icositetrachoron (ico) with the compound of 3 vertex-inscribed tesseracts (tes, i.e. the great icositetrachoron, gico). But the squares all are doubled up to completely coincident pairs. Even though, the cubes are not doubled up.

Incidence matrix according to Dynkin symbol

β3o4o3o

both( . . . . ) | 24 |  24 | 12  24 |  6  8
----------------+----+-----+--------+------
sefa( β3o . . ) |  2 | 288 |  1   2 |  2  1
----------------+----+-----+--------+------
      β3o . .   ♦  3 |   3 | 96   * |  2  0
sefa( β3o4o . ) |  4 |   4 |  * 144 |  1  1
----------------+----+-----+--------+------
      β3o4o .   ♦  6 |  24 |  8   6 | 24  *
sefa( β3o4o3o ) ♦  8 |  12 |  0   6 |  * 24

starting figure: x3o4o3o

o3β3o4o

both( . . . . ) | 24 |  8  16 |  4  8 16  8 |  4 2  8
----------------+----+--------+-------------+--------
sefa( o3β . . ) |  2 | 96   * |  1  0  2  0 |  2 0  1
sefa( . β3o . ) |  2 |  * 192 |  0  1  1  1 |  1 1  1
----------------+----+--------+-------------+--------
      o3β . .   ♦  3 |  3   0 | 32  *  *  * |  2 0  0
      . β3o .   ♦  3 |  0   3 |  * 64  *  * |  1 1  0
sefa( o3β3o . ) |  4 |  2   2 |  *  * 96  * |  1 0  1
sefa( . β3o4o ) |  4 |  0   4 |  *  *  * 48 |  0 1  1
----------------+----+--------+-------------+--------
      o3β3o .   ♦  6 | 12  12 |  4  4  6  0 | 16 *  *
      . β3o4o   ♦  6 |  0  24 |  0  8  0  6 |  * 8  *
sefa( o3β3o4o ) ♦  8 |  4   8 |  0  0  4  2 |  * * 24

starting figure: o3x3o4o

o3β3o *b3o

both( . . .    . ) | 24 |  8  8  8 |  4  4  4  8  8  8 | 2 2 2  8
-------------------+----+----------+-------------------+---------
sefa( o3β .    . ) |  2 | 96  *  * |  1  0  0  1  1  0 | 1 1 0  1
sefa( . β3o    . ) |  2 |  * 96  * |  0  1  0  1  0  1 | 1 0 1  1
sefa( . β . *b3o ) |  2 |  *  * 96 |  0  0  1  0  1  1 | 0 1 1  1
-------------------+----+----------+-------------------+---------
      o3β .    .   ♦  3 |  3  0  0 | 32  *  *  *  *  * | 1 1 0  0
      . β3o    .   ♦  3 |  0  3  0 |  * 32  *  *  *  * | 1 0 1  0
      . β . *b3o   ♦  3 |  0  0  3 |  *  * 32  *  *  * | 0 1 1  0
sefa( o3β3o    . ) |  4 |  2  2  0 |  *  *  * 48  *  * | 1 0 0  1
sefa( o3β . *b3o ) |  4 |  2  0  2 |  *  *  *  * 48  * | 0 1 0  1
sefa( . β3o *b3o ) |  4 |  0  2  2 |  *  *  *  *  * 48 | 0 0 1  1
-------------------+----+----------+-------------------+---------
      o3β3o    .   ♦  6 | 12 12  0 |  4  4  0  6  0  0 | 8 * *  *
      o3β . *b3o   ♦  6 | 12  0 12 |  4  0  4  0  6  0 | * 8 *  *
      . β3o *b3o   ♦  6 |  0 12 12 |  0  4  4  0  0  6 | * * 8  *
sefa( o3β3o *b3o ) ♦  8 |  4  4  4 |  0  0  0  2  2  2 | * * * 24

starting figure: o3x3o *b3o

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