Acronym hint
Name demipenteractic (line-)tegum,
demipenteractic dipyramid
Circumradius ...
Lace city
in approx. ASCII-art 
o    		where:
    E		E = o3o3x *b3o (hex)
X    		X = x3o3o *b3o
    o		o = o3o3o *b3o (point)
Coordinates
  1. (0, 0, 0, 0, 0, sqrt(3/8))   & change of sign
    tips
  2. (1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), 0)   & even permutations and even changes of sign in one but last coord.
    equator
Dihedral angles
(at margins)
Face vector 18, 112, 320, 440, 266, 52
Confer
mono-pyramid:
hinpy  

Incidence matrix according to Dynkin symbol

oxo3ooo3ooo *b3ooo3ooo&#xt   → both heights = sqrt(3/8) = 0.612372
(pt || pseudo hin || pt)

o..3o..3o.. *b3o..3o..     & | 2  * ♦ 16  0 |  80   0 | 160  0  0 | 40  80  0  0 | 10 16
.o.3.o.3.o. *b3.o.3.o.       | * 16 ♦  2 10 |  20  30 |  60 10 20 | 20  40  5  5 | 10 10
-----------------------------+------+-------+---------+-----------+--------------+------
oo.3oo.3oo. *b3oo.3oo.&#x  & | 1  1 | 32  * ♦  10   0 |  30  0  0 | 10  20  0  0 |  5  5
.x. ... ...    ... ...       | 0  2 |  * 80 ♦   2   6 |  12  3  6 |  6  12  3  2 |  6  4
-----------------------------+------+-------+---------+-----------+--------------+------
ox. ... ...    ... ...&#x  & | 1  2 |  2  1 | 160   * ♦   6  0  0 |  3   6  0  0 |  3  2
.x.3.o. ...    ... ...       | 0  3 |  0  3 |   * 160 |   2  1  2 |  2   4  2  1 |  4  2
-----------------------------+------+-------+---------+-----------+--------------+------
ox.3oo. ...    ... ...&#x  & ♦ 1  3 |  3  3 |   3   1 | 320  *  * |  1   2  0  0 |  2  1
.x.3.o.3.o.    ... ...       ♦ 0  4 |  0  6 |   0   4 |   * 40  * |  2   0  2  0 |  4  0
.x.3.o. ... *b3.o. ...       ♦ 0  4 |  0  6 |   0   4 |   *  * 80 |  0   2  1  1 |  2  2
-----------------------------+------+-------+---------+-----------+--------------+------
ox.3oo.3oo.    ... ...&#x  & ♦ 1  4 |  4  6 |   6   4 |   4  1  0 | 80   *  *  * |  2  0
ox.3oo. ... *b3oo. ...&#x  & ♦ 1  4 |  4  6 |   6   4 |   4  0  1 |  * 160  *  * |  1  1
.x.3.o.3.o. *b3.o. ...       ♦ 0  8 |  0 24 |   0  32 |   0  8  8 |  *   * 10  * |  2  0
.x.3.o. ... *b3.o.3.o.       ♦ 0  5 |  0 10 |   0  10 |   0  0  5 |  *   *  * 16 |  0  2
-----------------------------+------+-------+---------+-----------+--------------+------
ox.3oo.3oo. *b3oo. ...&#x  & ♦ 1  8 |  8 24 |  24  32 |  32  8  8 |  8   8  1  0 | 20  *
ox.3oo. ... *b3oo.3oo.&#x  & ♦ 1  5 |  5 10 |  10  10 |  10  0  5 |  0   5  0  1 |  * 32

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