o o3x    
             
x x3o   x x3x

This polyteron can be obtained as a wedge of nit. In fact use the oct-first segment of nit, the octacope, and then diminish that one by chopping off exactly half of the lower base, diminishing it into a tricupe, while the other base gets rasped down to a single face only.

Incidence matrix according to Dynkin symbol

oxx oxx3xox&#x   → height(1,2) = height(1,3) = sqrt(5/12) = 0.645497
                   height(2,3) = sqrt(2/3) = 0.816497
                   height(1,23) = 1/2 ({3} || tricupe)

o.. o..3o..    | 3 *  * | 2  4  4 0 0  0 0 0 0 | 1 2 2 4  4 2 2 4 0 0 0 0 0 0 0 0 | 1 2 2 2 2 4 1 2 2 0 0 0 0 0 | 1 1 2 2 1 0
.o. .o.3.o.    | * 6  * | 0  2  0 1 2  2 0 0 0 | 0 2 2 1  2 0 0 0 2 1 2 2 1 0 0 0 | 2 1 1 2 2 1 0 0 0 1 2 1 1 0 | 1 2 1 1 0 1
..o ..o3..o    | * * 12 | 0  0  1 0 0  1 1 1 1 | 0 0 0 0  1 1 1 1 0 0 1 1 1 1 1 1 | 0 0 0 1 1 1 1 1 1 0 1 1 1 1 | 0 1 1 1 1 1
---------------+--------+----------------------+----------------------------------+-----------------------------+------------
... ... x..    | 2 0  0 | 3  *  * * *  * * * * | 1 0 0 2  0 0 0 2 0 0 0 0 0 0 0 0 | 0 1 2 0 0 2 0 1 2 0 0 0 0 0 | 1 0 1 2 1 0
oo. oo.3oo.&#x | 1 1  0 | * 12  * * *  * * * * | 0 1 1 1  1 0 0 0 0 0 0 0 0 0 0 0 | 1 1 1 1 1 1 0 0 0 0 0 0 0 0 | 1 1 1 1 0 0
o.o o.o3o.o&#x | 1 0  1 | *  * 12 * *  * * * * | 0 0 0 0  1 1 1 1 0 0 0 0 0 0 0 0 | 0 0 0 1 1 1 1 1 1 0 0 0 0 0 | 0 1 1 1 1 0
.x. ... ...    | 0 2  0 | *  *  * 3 *  * * * * | 0 2 0 0  0 0 0 0 2 0 2 0 0 0 0 0 | 2 1 0 2 0 0 0 0 0 1 2 1 0 0 | 1 2 1 0 0 1
... .x. ...    | 0 2  0 | *  *  * * 6  * * * * | 0 0 1 0  0 0 0 0 1 1 0 1 0 0 0 0 | 1 0 1 0 1 0 0 0 0 1 1 0 1 0 | 1 1 0 1 0 1
.oo .oo3.oo&#x | 0 1  1 | *  *  * * * 12 * * * | 0 0 0 0  1 0 0 0 0 0 1 1 1 0 0 0 | 0 0 0 1 1 1 0 0 0 0 1 1 1 0 | 0 1 1 1 0 1
..x ... ...    | 0 0  2 | *  *  * * *  * 6 * * | 0 0 0 0  0 1 0 0 0 0 1 0 0 1 1 0 | 0 0 0 1 0 0 1 1 0 0 1 1 0 1 | 0 1 1 0 1 1
... ..x ...    | 0 0  2 | *  *  * * *  * * 6 * | 0 0 0 0  0 0 1 0 0 0 0 1 0 1 0 1 | 0 0 0 0 1 0 1 0 1 0 1 0 1 1 | 0 1 0 1 1 1
... ... ..x    | 0 0  2 | *  *  * * *  * * * 6 | 0 0 0 0  0 0 0 1 0 0 0 0 1 0 1 1 | 0 0 0 0 0 1 0 1 1 0 0 1 1 1 | 0 0 1 1 1 1
---------------+--------+----------------------+----------------------------------+-----------------------------+------------
... o..3x..    | 3 0  0 | 3  0  0 0 0  0 0 0 0 | 1 * * *  * * * * * * * * * * * * | 0 0 2 0 0 0 0 0 2 0 0 0 0 0 | 1 0 0 1 1 0
ox. ... ...&#x | 1 2  0 | 0  2  0 1 0  0 0 0 0 | * 6 * *  * * * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
... ox. ...&#x | 1 2  0 | 0  2  0 0 1  0 0 0 0 | * * 6 *  * * * * * * * * * * * * | 1 0 1 0 1 0 0 0 0 0 0 0 0 0 | 1 1 0 1 0 0
... ... xo.&#x | 2 1  0 | 1  2  0 0 0  0 0 0 0 | * * * 6  * * * * * * * * * * * * | 0 1 1 0 0 1 0 0 0 0 0 0 0 0 | 1 0 1 1 0 0
ooo ooo3ooo&#x | 1 1  1 | 0  1  1 0 0  1 0 0 0 | * * * * 12 * * * * * * * * * * * | 0 0 0 1 1 1 0 0 0 0 0 0 0 0 | 0 1 1 1 0 0
o.x ... ...&#x | 1 0  2 | 0  0  2 0 0  0 1 0 0 | * * * *  * 6 * * * * * * * * * * | 0 0 0 1 0 0 1 1 0 0 0 0 0 0 | 0 1 1 0 1 0
... o.x ...&#x | 1 0  2 | 0  0  2 0 0  0 0 1 0 | * * * *  * * 6 * * * * * * * * * | 0 0 0 0 1 0 1 0 1 0 0 0 0 0 | 0 1 0 1 1 0
... ... x.x&#x | 2 0  2 | 1  0  2 0 0  0 0 0 1 | * * * *  * * * 6 * * * * * * * * | 0 0 0 0 0 1 0 1 1 0 0 0 0 0 | 0 0 1 1 1 0
.x. .x. ...    | 0 4  0 | 0  0  0 2 2  0 0 0 0 | * * * *  * * * * 3 * * * * * * * | 1 0 0 0 0 0 0 0 0 1 1 0 0 0 | 1 1 0 0 0 1
... .x.3.o.    | 0 3  0 | 0  0  0 0 3  0 0 0 0 | * * * *  * * * * * 2 * * * * * * | 0 0 1 0 0 0 0 0 0 1 0 0 1 0 | 1 0 0 1 0 1
.xx ... ...&#x | 0 2  2 | 0  0  0 1 0  2 1 0 0 | * * * *  * * * * * * 6 * * * * * | 0 0 0 1 0 0 0 0 0 0 1 1 0 0 | 0 1 1 0 0 1
... .xx ...&#x | 0 2  2 | 0  0  0 0 1  2 0 1 0 | * * * *  * * * * * * * 6 * * * * | 0 0 0 0 1 0 0 0 0 0 1 0 1 0 | 0 1 0 1 0 1
... ... .ox&#x | 0 1  2 | 0  0  0 0 0  2 0 0 1 | * * * *  * * * * * * * * 6 * * * | 0 0 0 0 0 1 0 0 0 0 0 1 1 0 | 0 0 1 1 0 1
..x ..x ...    | 0 0  4 | 0  0  0 0 0  0 2 2 0 | * * * *  * * * * * * * * * 3 * * | 0 0 0 0 0 0 1 0 0 0 1 0 0 1 | 0 1 0 0 1 1
..x ... ..x    | 0 0  4 | 0  0  0 0 0  0 2 0 2 | * * * *  * * * * * * * * * * 3 * | 0 0 0 0 0 0 0 1 0 0 0 1 0 1 | 0 0 1 0 1 1
... ..x3..x    | 0 0  6 | 0  0  0 0 0  0 0 3 3 | * * * *  * * * * * * * * * * * 2 | 0 0 0 0 0 0 0 0 1 0 0 0 1 1 | 0 0 0 1 1 1
---------------+--------+----------------------+----------------------------------+-----------------------------+------------
ox. ox. ...&#x ♦ 1 4  0 | 0  4  0 2 2  0 0 0 0 | 0 2 2 0  0 0 0 0 1 0 0 0 0 0 0 0 | 3 * * * * * * * * * * * * * | 1 1 0 0 0 0
ox. ... xo.&#x ♦ 2 2  0 | 1  4  0 1 0  0 0 0 0 | 0 2 0 2  0 0 0 0 0 0 0 0 0 0 0 0 | * 3 * * * * * * * * * * * * | 1 0 1 0 0 0
... ox.3xo.&#x ♦ 3 3  0 | 3  6  0 0 3  0 0 0 0 | 1 0 3 3  0 0 0 0 0 1 0 0 0 0 0 0 | * * 2 * * * * * * * * * * * | 1 0 0 1 0 0
oxx ... ...&#x ♦ 1 2  2 | 0  2  2 1 0  2 1 0 0 | 0 1 0 0  2 1 0 0 0 0 1 0 0 0 0 0 | * * * 6 * * * * * * * * * * | 0 1 1 0 0 0
... oxx ...&#x ♦ 1 2  2 | 0  2  2 0 1  2 0 1 0 | 0 0 1 0  2 0 1 0 0 0 0 1 0 0 0 0 | * * * * 6 * * * * * * * * * | 0 1 0 1 0 0
... ... xox&#x ♦ 2 1  2 | 1  2  2 0 0  2 0 0 1 | 0 0 0 1  2 0 0 1 0 0 0 0 1 0 0 0 | * * * * * 6 * * * * * * * * | 0 0 1 1 0 0
o.x o.x ...&#x ♦ 1 0  4 | 0  0  4 0 0  0 2 2 0 | 0 0 0 0  0 2 2 0 0 0 0 0 0 1 0 0 | * * * * * * 3 * * * * * * * | 0 1 0 0 1 0
o.x ... x.x&#x ♦ 2 0  4 | 1  0  4 0 0  0 2 0 2 | 0 0 0 0  0 2 0 2 0 0 0 0 0 0 1 0 | * * * * * * * 3 * * * * * * | 0 0 1 0 1 0
... o.x3x.x&#x ♦ 3 0  6 | 3  0  6 0 0  0 0 3 3 | 1 0 0 0  0 0 3 3 0 0 0 0 0 0 0 1 | * * * * * * * * 2 * * * * * | 0 0 0 1 1 0
.x. .x.3.o.    ♦ 0 6  0 | 0  0  0 3 6  0 0 0 0 | 0 0 0 0  0 0 0 0 3 2 0 0 0 0 0 0 | * * * * * * * * * 1 * * * * | 1 0 0 0 0 1
.xx .xx ...&#x ♦ 0 4  4 | 0  0  0 2 2  4 2 2 0 | 0 0 0 0  0 0 0 0 1 0 2 2 0 1 0 0 | * * * * * * * * * * 3 * * * | 0 1 0 0 0 1
.xx ... .ox&#x ♦ 0 2  4 | 0  0  0 1 0  4 2 0 2 | 0 0 0 0  0 0 0 0 0 0 2 0 2 0 1 0 | * * * * * * * * * * * 3 * * | 0 0 1 0 0 1
... .xx3.ox&#x ♦ 0 3  6 | 0  0  0 0 3  6 0 3 3 | 0 0 0 0  0 0 0 0 0 1 0 3 3 0 0 1 | * * * * * * * * * * * * 2 * | 0 0 0 1 0 1
..x ..x3..x    ♦ 0 0 12 | 0  0  0 0 0  0 6 6 6 | 0 0 0 0  0 0 0 0 0 0 0 0 0 3 3 2 | * * * * * * * * * * * * * 1 | 0 0 0 0 1 1
---------------+--------+----------------------+----------------------------------+-----------------------------+------------
ox. ox.3xo.&#x ♦ 3 6  0 | 3 12  0 3 6  0 0 0 0 | 1 6 6 6  0 0 0 0 3 2 0 0 0 0 0 0 | 3 3 2 0 0 0 0 0 0 1 0 0 0 0 | 1 * * * * *
oxx oxx ...&#x ♦ 1 4  4 | 0  4  4 2 2  4 2 2 0 | 0 2 2 0  4 2 2 0 1 0 2 2 0 1 0 0 | 1 0 0 2 2 0 1 0 0 0 1 0 0 0 | * 3 * * * *
oxx ... xox&#x ♦ 2 2  4 | 1  4  4 1 0  4 2 0 2 | 0 2 0 2  4 2 0 2 0 0 2 0 2 0 1 0 | 0 1 0 2 0 2 0 1 0 0 0 1 0 0 | * * 3 * * *
... oxx3xox&#x ♦ 3 3  6 | 3  6  6 0 3  6 0 3 3 | 1 0 3 3  6 0 3 3 0 1 0 3 3 0 0 1 | 0 0 1 0 3 3 0 0 1 0 0 0 1 0 | * * * 2 * *
o.x o.x3x.x&#x ♦ 3 0 12 | 3  0 12 0 0  0 6 6 6 | 1 0 0 0  0 6 6 6 0 0 0 0 0 3 3 2 | 0 0 0 0 0 0 3 3 2 0 0 0 0 1 | * * * * 1 *
.xx .xx3.ox&#x ♦ 0 6 12 | 0  0  0 3 6 12 6 6 6 | 0 0 0 0  0 0 0 0 3 2 6 6 6 3 3 2 | 0 0 0 0 0 0 0 0 0 1 3 3 2 1 | * * * * * 1