Acronym ...
Name o3o3q4x,
variation of tat
Circumradius sqrt(7) = 2.645751
Lace city
in approx. ASCII-art
q4x Q4o Q4o q4x		-- o3q4x
               
Q4o         Q4o		-- o3o4Q
               
Q4o         Q4o		-- o3o4Q
               
q4x Q4o Q4o q4x		-- o3q4x


where:
Q = 2q = sqrt(8)
Coordinates (3/2, 3/2, 3/2, 1/2)   & all permutations, all changes of sign
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Dihedral angles
Face vector 64, 128, 88, 24
Confer
uniform variant:
tat  

Incidence matrix according to Dynkin symbol

o3o3q4x

. . . . | 64 |  3  1 |  3  3 |  1 3
--------+----+-------+-------+-----
. . q . |  2 | 96  * |  2  1 |  1 2  q
. . . x |  2 |  * 32 |  0  3 |  0 3  x
--------+----+-------+-------+-----
. o3q . |  3 |  3  0 | 64  * |  1 1
. . q4x |  8 |  4  4 |  * 24 |  0 2
--------+----+-------+-------+-----
o3o3q .   4 |  6  0 |  4  0 | 16 *
. o3q4x  24 | 24 12 |  8  6 |  * 8

oo3oo3qQ *b3Qq&#zx   → height = 0
                       where  Q = sqrt(8) = 2.828427 (pseudo)

o.3o.3o. *b3o.     | 32  * |  3  1  0 |  3  3  0 | 1 3 0
.o3.o3.o *b3.o     |  * 32 |  0  1  3 |  0  3  3 | 0 3 1
-------------------+-------+----------+----------+------
.. .. q.   ..      |  2  0 | 48  *  * |  2  1  0 | 1 2 0  q
oo3oo3oo *b3oo&#x  |  1  1 |  * 32  * |  0  3  0 | 0 3 0  x
.. .. ..    .q     |  0  2 |  *  * 48 |  0  1  2 | 0 2 1  q
-------------------+-------+----------+----------+------
.. o.3q.   ..      |  3  0 |  3  0  0 | 32  *  * | 1 1 0
.. .. qQ   Qq&#zx  |  4  4 |  2  4  2 |  * 24  * | 0 2 0  q4x
.. .o .. *b3.q     |  0  3 |  0  0  3 |  *  * 32 | 0 1 1
-------------------+-------+----------+----------+------
o.3o.3q.   ..        4  0 |  6  0  0 |  4  0  0 | 8 * *
.. oo3qQ *b3Qq&#zx  12 12 | 12 12 12 |  4  6  4 | * 8 *  o3q4x
.o3.o .. *b3.q       0  4 |  0  0  6 |  0  0  4 | * * 8

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