Acronym ...
Name q x3o3o,
variation of tetrahedron prism,
vertex figure of rectified penteract
Circumradius sqrt(7/8) = 0.935414
Confer
uniform variant:
tepe  

using edge sizes x = 1 and q = sqrt(2) = 1.414214


Incidence matrix according to Dynkin symbol

q x3o3o

. . . . | 8 | 1  3 | 3 3 | 3 1
--------+---+------+-----+----
q . . . | 2 | 4  * | 3 0 | 3 0
. x . . | 2 | * 12 | 1 2 | 2 1
--------+---+------+-----+----
q x . . | 4 | 2  2 | 6 * | 2 0
. x3o . | 3 | 0  3 | * 8 | 1 1
--------+---+------+-----+----
q x3o .  6 | 3  6 | 3 2 | 4 *
. x3o3o  4 | 0  6 | 0 4 | * 2

xx3oo3oo&#q   → height = sqrt(2) = 1.414214
(tet || tet)

o.3o.3o.    | 4 * | 3 1 0 | 3 3 0 | 1 3 0
.o3.o3.o    | * 4 | 0 1 3 | 0 3 3 | 0 3 1
------------+-----+-------+-------+------
x. .. ..    | 2 0 | 6 * * | 2 1 0 | 1 2 0
oo3oo3oo&#q | 1 1 | * 4 * | 0 3 0 | 0 3 0
.x .. ..    | 0 2 | * * 6 | 0 1 2 | 0 2 1
------------+-----+-------+-------+------
x.3o. ..    | 3 0 | 3 0 0 | 4 * * | 1 1 0
xx .. ..&#q | 2 2 | 1 2 1 | * 6 * | 0 2 0
.x3.o ..    | 0 3 | 0 0 3 | * * 4 | 0 1 1
------------+-----+-------+-------+------
x.3o.3o.     4 0 | 6 0 0 | 4 0 0 | 1 * *
xx3oo ..&#q  3 3 | 3 3 3 | 1 3 1 | * 4 *
.x3.o3.o     0 4 | 0 0 6 | 0 0 4 | * * 1

qq ox3oo&#x   → height = sqrt(2/3) = 0.816497
(line || q x3o)

o. o.3o.    | 2 * | 1 3 0 0 | 3 3 0 0 | 3 1 0
.o .o3.o    | * 6 | 0 1 1 2 | 1 2 2 1 | 2 1 1
------------+-----+---------+---------+------
q. .. ..    | 2 0 | 1 * * * | 3 0 0 0 | 3 0 0
oo oo3oo&#x | 1 1 | * 6 * * | 1 2 0 0 | 2 1 0
.q .. ..    | 0 2 | * * 3 * | 1 0 2 0 | 2 0 1
.. .x ..    | 0 2 | * * * 6 | 0 1 1 1 | 1 1 1
------------+-----+---------+---------+------
qq .. ..&#x | 2 2 | 1 2 1 0 | 3 * * * | 2 0 0
.. ox ..&#x | 1 2 | 0 2 0 1 | * 6 * * | 1 1 0
.q .x ..    | 0 4 | 0 0 2 2 | * * 3 * | 1 0 1
.. .x3.o    | 0 3 | 0 0 0 3 | * * * 2 | 0 1 1
------------+-----+---------+---------+------
qq ox ..&#x  2 4 | 1 4 2 2 | 2 2 1 0 | 3 * *
.. ox3oo&#x  1 3 | 0 3 0 3 | 0 3 0 1 | * 2 *
.q .x3.o     0 6 | 0 0 3 6 | 0 0 3 2 | * * 1

qq xo ox&#x   → height = 1/sqrt(2) = 0.707107
((q,x)-{4} || ortho (q,x)-{4})

o. o. o.    | 4 * | 1 1 2 0 0 | 1 2 2 1 0 | 2 1 1
.o .o .o    | * 4 | 0 0 2 1 1 | 0 2 1 2 1 | 1 2 1
------------+-----+-----------+-----------+------
q. .. ..    | 2 0 | 2 * * * * | 1 2 0 0 0 | 2 1 0
.. x. ..    | 2 0 | * 2 * * * | 1 0 2 0 0 | 2 0 1
oo oo oo&#x | 1 1 | * * 8 * * | 0 1 1 1 0 | 1 1 1
.q .. ..    | 0 2 | * * * 2 * | 0 2 0 0 1 | 1 2 0
.. .. .x    | 0 2 | * * * * 2 | 0 0 0 2 1 | 0 2 1
------------+-----+-----------+-----------+------
q. x. ..    | 4 0 | 2 2 0 0 0 | 1 * * * * | 2 0 0
qq .. ..&#x | 2 2 | 1 0 2 1 0 | * 4 * * * | 1 1 0
.. xo ..&#x | 2 1 | 0 1 2 0 0 | * * 4 * * | 1 0 1
.. .. ox&#x | 1 2 | 0 0 2 0 1 | * * * 4 * | 0 1 1
.q .. .x    | 0 4 | 0 0 0 2 2 | * * * * 1 | 0 2 0
------------+-----+-----------+-----------+------
qq xo ..&#x  4 2 | 2 2 4 1 0 | 1 2 2 0 0 | 2 * *
qq .. ox&#x  2 4 | 1 0 4 2 2 | 0 2 0 2 1 | * 2 *
.. xo ox&#x  2 2 | 0 1 4 0 1 | 0 0 2 2 0 | * * 2
or
o. o. o.    & | 8 | 1 1 1 | 1 2 3 | 3 1
--------------+---+-------+-------+----
q. .. ..    & | 2 | 4 * * | 1 2 0 | 3 0
.. x. ..    & | 2 | * 4 * | 1 0 2 | 2 1
oo oo oo&#x   | 2 | * * 8 | 0 1 2 | 2 1
--------------+---+-------+-------+----
q. x. ..    & | 4 | 2 2 0 | 2 * * | 2 0
qq .. ..&#x   | 4 | 2 0 2 | * 4 * | 2 0
.. xo ..&#x & | 3 | 0 1 2 | * * 8 | 1 1
--------------+---+-------+-------+----
qq xo ..&#x &  6 | 3 2 4 | 1 2 2 | 4 *
.. xo ox&#x    4 | 0 2 4 | 0 0 4 | * 2

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