Acronym ...
Name 2rico (?)
Circumradius sqrt(3) = 1.732051
Coordinates (sqrt(2), 1/sqrt(2), 1/sqrt(2), 0)   & all permutations, all changes of sign
General of army rico
Colonel of regiment rico
Confer
non-Grünbaumian master:
rico  
Grünbaumian relatives:
2rico+64{6}+128{3}   2rico+64{6}+192{3}  

Looks like a compound of 2 coincident rectified icositetrachora (rico), and indeed edges, squars, and cubes (cube) all coincide by pairs.


Incidence matrix according to Dynkin symbol

β3x4o3o

both( . . . . ) | 192 |   3   3 |   3  3   3 |  1  3  1
----------------+-----+---------+------------+---------
both( . x . . ) |   2 | 288   * |   2  1   0 |  1  2  0
sefa( β3x . . ) |   2 |   * 288 |   0  1   2 |  0  2  1
----------------+-----+---------+------------+---------
both( . x4o . ) |   4 |   4   0 | 144  *   * |  1  1  0
      β3x . .      6 |   3   3 |   * 96   * |  0  2  0
sefa( β3x4o . ) |   4 |   0   4 |   *  * 144 |  0  1  1
----------------+-----+---------+------------+---------
both( . x4o3o )    8 |  12   0 |   6  0   0 | 24  *  *
      β3x4o .     24 |  24  24 |   6  8   6 |  * 24  *
sefa( β3x4o3o )    8 |   0  12 |   0  0   6 |  *  * 24

starting figure: x3x4o3o

x3β3x4o

both( . . . . ) | 192 |  1   2  1   2 |  2  1  1  2  2  1 |  1  2 1  1
----------------+-----+---------------+-------------------+-----------
both( x . . . ) |   2 | 96   *  *   * |  2  0  1  0  0  0 |  1  2 0  0
both( . . x . ) |   2 |  * 192  *   * |  1  1  0  1  0  0 |  1  1 1  0
sefa( x3β . . ) |   2 |  *   * 96   * |  0  0  1  0  2  0 |  0  2 0  1
sefa( . β3x . ) |   2 |  *   *  * 192 |  0  0  0  1  1  1 |  0  1 1  1
----------------+-----+---------------+-------------------+-----------
both( x . x . ) |   4 |  2   2  0   0 | 96  *  *  *  *  * |  1  1 0  0
both( . . x4o ) |   4 |  0   4  0   0 |  * 48  *  *  *  * |  1  0 1  0
      x3β . .      6 |  3   0  3   0 |  *  * 32  *  *  * |  0  2 0  0
      . β3x .      6 |  0   3  0   3 |  *  *  * 64  *  * |  0  1 1  0
sefa( x3β3x . ) |   4 |  0   0  2   2 |  *  *  *  * 96  * |  0  1 0  1
sefa( . β3x4o ) |   4 |  0   0  0   4 |  *  *  *  *  * 48 |  0  0 1  1
----------------+-----+---------------+-------------------+-----------
both( x . x4o )    8 |  4   8  0   0 |  4  2  0  0  0  0 | 24  * *  *
      x3β3x .     24 | 12  12 12  12 |  6  0  4  4  6  0 |  * 16 *  *
      . β3x4o     24 |  0  24  0  24 |  0  6  0  8  0  6 |  *  * 8  *
sefa( x3β3x4o )    8 |  0   0  4   8 |  0  0  0  0  4  2 |  *  * * 24

starting figure: x3x3x4o

β3β3x4o

both( . . . . ) | 192 |   2   2   2 |  1  1  2   4  1 |  2 1  2
----------------+-----+-------------+-----------------+--------
both( . . x . ) |   2 | 192   *   * |  1  0  1   1  0 |  1 1  1
sefa( s3s . . ) |   2 |   * 192   * |  0  1  0   2  0 |  2 0  1
sefa( . β3x . ) |   2 |   *   * 192 |  0  0  1   1  1 |  1 1  1
----------------+-----+-------------+-----------------+--------
both( . . x4o ) |   4 |   4   0   0 | 48  *  *   *  * |  0 1  1
both( s3s . . )    3 |   0   3   0 |  * 64  *   *  * |  2 0  0
      . β3x .      6 |   3   0   3 |  *  * 64   *  * |  1 1  0
sefa( β3β3x . ) |   4 |   1   2   1 |  *  *  * 192  * |  1 0  1
sefa( . β3x4o ) |   4 |   0   0   4 |  *  *  *   * 48 |  0 1  1
----------------+-----+-------------+-----------------+--------
      β3β3x .     24 |  12  24  12 |  0  8  4  12  0 | 16 *  *
      . β3x4o     24 |  24   0  24 |  6  0  8   0  6 |  * 8  *
sefa( β3β3x4o )    8 |   4   4   4 |  1  0  0   4  1 |  * * 48

starting figure: x3x3x4o

x3β3x *b3x

both( . . .    . ) | 192 |  1  1  1  1  1  1 |  1  1  1  1  1  1  1  1  1 |  1 1 1 1  1
-------------------+-----+-------------------+----------------------------+------------
both( x . .    . ) |   2 | 96  *  *  *  *  * |  1  1  0  1  0  0  0  0  0 |  1 1 1 0  0
both( . . x    . ) |   2 |  * 96  *  *  *  * |  1  0  1  0  1  0  0  0  0 |  1 1 0 1  0
both( . . .    x ) |   2 |  *  * 96  *  *  * |  0  1  1  0  0  1  0  0  0 |  1 0 1 1  0
sefa( x3β .    . ) |   2 |  *  *  * 96  *  * |  0  0  0  1  0  0  1  1  0 |  0 1 1 0  1
sefa( . β3x    . ) |   2 |  *  *  *  * 96  * |  0  0  0  0  1  0  1  0  1 |  0 1 0 1  1
sefa( . β . *b3x ) |   2 |  *  *  *  *  * 96 |  0  0  0  0  0  1  0  1  1 |  0 0 1 1  1
-------------------+-----+-------------------+----------------------------+------------
both( x . x    . ) |   4 |  2  2  0  0  0  0 | 48  *  *  *  *  *  *  *  * |  1 1 0 0  0
both( x . .    x ) |   4 |  2  0  2  0  0  0 |  * 48  *  *  *  *  *  *  * |  1 0 1 0  0
both( . . x    x ) |   4 |  0  2  2  0  0  0 |  *  * 48  *  *  *  *  *  * |  1 0 0 1  0
      x3β .    .      6 |  3  0  0  3  0  0 |  *  *  * 32  *  *  *  *  * |  0 1 1 0  0
      . β3x    .      6 |  0  3  0  0  3  0 |  *  *  *  * 32  *  *  *  * |  0 1 0 1  0
      . β . *b3x      6 |  0  0  3  0  0  3 |  *  *  *  *  * 32  *  *  * |  0 0 1 1  0
sefa( x3β3x    . ) |   4 |  0  0  0  2  2  0 |  *  *  *  *  *  * 48  *  * |  0 1 0 0  1
sefa( x3β . *b3x ) |   4 |  0  0  0  2  0  2 |  *  *  *  *  *  *  * 48  * |  0 0 1 0  1
sefa( . β3x *b3x ) |   4 |  0  0  0  0  2  2 |  *  *  *  *  *  *  *  * 48 |  0 0 0 1  1
-------------------+-----+-------------------+----------------------------+------------
both( x . x    x )    8 |  4  4  4  0  0  0 |  2  2  2  0  0  0  0  0  0 | 24 * * *  *
      x3β3x    .     24 | 12 12  0 12 12  0 |  6  0  0  4  4  0  6  0  0 |  * 8 * *  *
      x3β . *b3x     24 | 12  0 12 12  0 12 |  0  6  0  4  0  4  0  6  0 |  * * 8 *  *
      . β3x *b3x     24 |  0 12 12  0 12 12 |  0  0  6  0  4  4  0  0  6 |  * * * 8  *
sefa( x3β3x *b3x )    8 |  0  0  0  4  4  4 |  0  0  0  0  0  0  2  2  2 |  * * * * 24

starting figure: x3x3x *b3x

β3β3x *b3x

both( . . .    . ) | 192 |  1  1   2  1  1 |  1  1  1  1  2  2  1 | 1 1 1  2
-------------------+-----+-----------------+----------------------+---------
both( . . x    . ) |   2 | 96  *   *  *  * |  1  0  1  0  1  0  0 | 1 0 1  1
both( . . .    x ) |   2 |  * 96   *  *  * |  1  0  0  1  0  1  0 | 0 1 1  1
sefa( s3s .    . ) |   2 |  *  * 192  *  * |  0  1  0  0  1  1  0 | 1 1 0  1
sefa( . β3x    . ) |   2 |  *  *   * 96  * |  0  0  1  0  1  0  1 | 1 0 1  1
sefa( . β . *b3x ) |   2 |  *  *   *  * 96 |  0  0  0  1  0  1  1 | 0 1 1  1
-------------------+-----+-----------------+----------------------+---------
both( . . x    x ) |   4 |  2  2   0  0  0 | 48  *  *  *  *  *  * | 0 0 1  1
both( s3s .    . )    3 |  0  0   3  0  0 |  * 64  *  *  *  *  * | 1 1 0  0
      . β3x    .      6 |  3  0   0  3  0 |  *  * 32  *  *  *  * | 1 0 1  0
      . β . *b3x      6 |  3  0   0  0  3 |  *  *  * 32  *  *  * | 0 1 1  0
sefa( 3β3x    . ) |   4 |  1  0   2  1  0 |  *  *  *  * 96  *  * | 1 0 0  1
sefa( 3β . *b3x ) |   4 |  0  1   2  0  1 |  *  *  *  *  * 96  * | 0 1 0  1
sefa( . β3x *b3x ) |   4 |  0  0   0  2  2 |  *  *  *  *  *  * 48 | 0 0 1  1
-------------------+-----+-----------------+----------------------+---------
      β3β3x    .     24 | 12  0  24 12  0 |  0  8  4  0 12  0  0 | 8 * *  *
      β3β .    x     24 |  0 12  24  0 12 |  0  8  0  4  0 12  0 | * 8 *  *
      . β3x *b3x     24 | 12 12   0 12 12 |  6  0  4  4  0  0  6 | * * 8  *
sefa( β3β3x *b3x )    8 |  2  2   4  2  2 |  1  0  0  0  2  2  1 | * * * 48

starting figure: x3x3x *b3x

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