Acronym ...
Name ico+gico+24co (?)
Circumradius 1
General of army ico
Colonel of regiment ico
Confer ico   gico   ico+gico+72{4}  

Looks like a compound of an icositetrachoron (ico) with the compound of 3 vertex-inscribed tesseracts (tes, i.e. the great icositetrachoron, gico). Cells are connected by further 24 diametral co. Its vertex figure is a variation of querco, in fact q3o4/3x or q3/2o4x (which degenerates to something what looks like a cube plus an inscribed stella octangula (so), but diametral rectangles will there be in addition). Accordingly edges come in coincident triples, all faces come in coincident pairs, and the co also come in coincident pairs.


Incidence matrix according to Dynkin symbol

x3o4o3o4/3*a

. . . .      | 24 |  24 |  24  24 |  6 12  8
-------------+----+-----+---------+---------
x . . .      |  2 | 288 |   2   2 |  1  2  1
-------------+----+-----+---------+---------
x3o . .      |  3 |   3 | 192   * |  1  1  0
x . . o4/3*a |  4 |   4 |   * 144 |  0  1  1
-------------+----+-----+---------+---------
x3o4o .        6 |  12 |   8   0 | 24  *  *
x3o . o4/3*a  12 |  24 |   8   6 |  * 24  *
x . o3o4/3*a   8 |  12 |   0   6 |  *  * 24

x3o4o3/2o4*a

. . .   .    | 24 |  24 |  24  24 |  6 12  8
-------------+----+-----+---------+---------
x . .   .    |  2 | 288 |   2   2 |  1  2  1
-------------+----+-----+---------+---------
x3o .   .    |  3 |   3 | 192   * |  1  1  0
x . .   o4*a |  4 |   4 |   * 144 |  0  1  1
-------------+----+-----+---------+---------
x3o4o   .      6 |  12 |   8   0 | 24  *  *
x3o .   o4*a  12 |  24 |   8   6 |  * 24  *
x . o3/2o4*a   8 |  12 |   0   6 |  *  * 24

x3o4/3o3o4*a

. .   . .    | 24 |  24 |  24  24 |  6 12  8
-------------+----+-----+---------+---------
x .   . .    |  2 | 288 |   2   2 |  1  2  1
-------------+----+-----+---------+---------
x3o   . .    |  3 |   3 | 192   * |  1  1  0
x .   . o4*a |  4 |   4 |   * 144 |  0  1  1
-------------+----+-----+---------+---------
x3o4/3o .      6 |  12 |   8   0 | 24  *  *
x3o   . o4*a  12 |  24 |   8   6 |  * 24  *
x .   o3o4*a   8 |  12 |   0   6 |  *  * 24

x3/2o4o3o4*a

.   . . .    | 24 |  24 |  24  24 |  6 12  8
-------------+----+-----+---------+---------
x   . . .    |  2 | 288 |   2   2 |  1  2  1
-------------+----+-----+---------+---------
x3/2o . .    |  3 |   3 | 192   * |  1  1  0
x   . . o4*a |  4 |   4 |   * 144 |  0  1  1
-------------+----+-----+---------+---------
x3/2o4o .      6 |  12 |   8   0 | 24  *  *
x3/2o . o4*a  12 |  24 |   8   6 |  * 24  *
x   . o3o4*a   8 |  12 |   0   6 |  *  * 24

x3o4/3o3/2o4/3*a

. .   .   .      | 24 |  24 |  24  24 |  6 12  8
-----------------+----+-----+---------+---------
x .   .   .      |  2 | 288 |   2   2 |  1  2  1
-----------------+----+-----+---------+---------
x3o   .   .      |  3 |   3 | 192   * |  1  1  0
x .   .   o4/3*a |  4 |   4 |   * 144 |  0  1  1
-----------------+----+-----+---------+---------
x3o4/3o   .        6 |  12 |   8   0 | 24  *  *
x3o   .   o4/3*a  12 |  24 |   8   6 |  * 24  *
x .   o3/2o4/3*a   8 |  12 |   0   6 |  *  * 24

x3/2o4o3/2o4/3*a

.   . .   .      | 24 |  24 |  24  24 |  6 12  8
-----------------+----+-----+---------+---------
x   . .   .      |  2 | 288 |   2   2 |  1  2  1
-----------------+----+-----+---------+---------
x3/2o .   .      |  3 |   3 | 192   * |  1  1  0
x   . .   o4/3*a |  4 |   4 |   * 144 |  0  1  1
-----------------+----+-----+---------+---------
x3/2o4o   .        6 |  12 |   8   0 | 24  *  *
x3/2o .   o4/3*a  12 |  24 |   8   6 |  * 24  *
x   . o3/2o4/3*a   8 |  12 |   0   6 |  *  * 24

x3/2o4/3o3o4/3*a

.   .   . .      | 24 |  24 |  24  24 |  6 12  8
-----------------+----+-----+---------+---------
x   .   . .      |  2 | 288 |   2   2 |  1  2  1
-----------------+----+-----+---------+---------
x3/2o   . .      |  3 |   3 | 192   * |  1  1  0
x   .   . o4/3*a |  4 |   4 |   * 144 |  0  1  1
-----------------+----+-----+---------+---------
x3/2o4/3o .        6 |  12 |   8   0 | 24  *  *
x3/2o   . o4/3*a  12 |  24 |   8   6 |  * 24  *
x   .   o3o4/3*a   8 |  12 |   0   6 |  *  * 24

x3/2o4/3o3/2o4*a

.   .   .   .    | 24 |  24 |  24  24 |  6 12  8
-----------------+----+-----+---------+---------
x   .   .   .    |  2 | 288 |   2   2 |  1  2  1
-----------------+----+-----+---------+---------
x3/2o   .   .    |  3 |   3 | 192   * |  1  1  0
x   .   .   o4*a |  4 |   4 |   * 144 |  0  1  1
-----------------+----+-----+---------+---------
x3/2o4/3o   .      6 |  12 |   8   0 | 24  *  *
x3/2o   .   o4*a  12 |  24 |   8   6 |  * 24  *
x   .   o3/2o4*a   8 |  12 |   0   6 |  *  * 24

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