Acronym ...
Name shafipto+32tet (?)
Circumradius sqrt[(3+sqrt(2))/2] = 1.485633
Coordinates ((1+sqrt(2))/2, 1/2, 1/2, 1/2)   & all permutations, all changes of sign
General of army sidpith
Colonel of regiment sidpith

Looks like a compound of a small hexadecafaceted prismatotesseractioctachoron (shafipto) plus 32 additional tetrahedra (tet) in hexadecachoral positions coincident by 2 each.


Incidence matrix according to Dynkin symbol

β3o3o4x

both( . . . . ) | 64 |  3   6 |  3  3  3  6 | 1  1  3 3
----------------+----+--------+-------------+----------
both( . . . x ) |  2 | 96   * |  2  0  0  2 | 1  0  1 2
sefa( β3o . . ) |  2 |  * 192 |  0  1  1  1 | 0  1  1 1
----------------+----+--------+-------------+----------
both( . . o4x ) |  4 |  4   0 | 48  *  *  * | 1  0  0 1
      β3o . .     3 |  0   3 |  * 64  *  * | 0  1  1 0
sefa( β3o3o . ) |  3 |  0   3 |  *  * 64  * | 0  1  0 1
sefa( β3o 2 x ) |  4 |  2   2 |  *  *  * 96 | 0  0  1 1
----------------+----+--------+-------------+----------
both( . o3o4x )   8 | 12   0 |  6  0  0  0 | 8  *  * *
      β3o3o .     4 |  0  12 |  0  4  4  0 | * 16  * *
      β3o 2 x     6 |  3   6 |  0  2  0  3 | *  * 32 *
sefa( β3o3o4x )  24 | 24  24 |  6  0  8 12 | *  *  * 8

starting figure: x3o3o4x

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