This polyteron can be obtained as a diminished version of the 2^nd segment of spat in spid-first orientation.

Incidence matrix according to Dynkin symbol

xx3ox3xx3xo&#x   → height = sqrt(2/5) = 0.632456
(prip || grip)

o.3o.3o.3o.    | 60  * |  2  2  1   2  0  0  0 |  1  2  2  1  2  2  1  2  2  0  0  0  0 | 1  1  2 1  1  2  2  1  1  2 0  0 0 | 1 1  1  2 1 0
.o3.o3.o3.o    |  * 60 |  0  0  0   2  1  1  2 |  0  0  0  0  0  2  2  2  1  1  2  2  1 | 0  0  0 0  2  2  1  2  1  1 2  1 1 | 0 2  1  1 1 1
---------------+-------+-----------------------+----------------------------------------+------------------------------------+--------------
x. .. .. ..    |  2  0 | 60  *  *   *  *  *  * |  1  1  1  0  0  1  0  0  0  0  0  0  0 | 1  1  1 0  1  1  1  0  0  0 0  0 0 | 1 1  1  1 0 0
.. .. x. ..    |  2  0 |  * 60  *   *  *  *  * |  0  1  0  1  1  0  0  1  0  0  0  0  0 | 1  0  1 1  0  1  0  1  0  1 0  0 0 | 1 1  0  1 1 0
.. .. .. x.    |  2  0 |  *  * 30   *  *  *  * |  0  0  2  0  2  0  0  0  2  0  0  0  0 | 0  1  2 1  0  0  2  0  1  2 0  0 0 | 1 0  1  2 1 0
oo3oo3oo3oo&#x |  1  1 |  *  *  * 120  *  *  * |  0  0  0  0  0  1  1  1  1  0  0  0  0 | 0  0  0 0  1  1  1  1  1  1 0  0 0 | 0 1  1  1 1 0
.x .. .. ..    |  0  2 |  *  *  *   * 30  *  * |  0  0  0  0  0  2  0  0  0  1  2  0  0 | 0  0  0 0  2  2  1  0  0  0 2  1 0 | 0 2  1  1 0 1
.. .x .. ..    |  0  2 |  *  *  *   *  * 30  * |  0  0  0  0  0  0  2  0  0  1  0  2  0 | 0  0  0 0  2  0  0  2  1  0 2  0 1 | 0 2  1  0 1 1
.. .. .x ..    |  0  2 |  *  *  *   *  *  * 60 |  0  0  0  0  0  0  0  1  0  0  1  1  1 | 0  0  0 0  0  1  0  1  0  1 1  1 1 | 0 1  0  1 1 1
---------------+-------+-----------------------+----------------------------------------+------------------------------------+--------------
x.3o. .. ..    |  3  0 |  3  0  0   0  0  0  0 | 20  *  *  *  *  *  *  *  *  *  *  *  * | 1  1  0 0  1  0  0  0  0  0 0  0 0 | 1 1  1  0 0 0
x. .. x. ..    |  4  0 |  2  2  0   0  0  0  0 |  * 30  *  *  *  *  *  *  *  *  *  *  * | 1  0  1 0  0  1  0  0  0  0 0  0 0 | 1 1  0  1 0 0
x. .. .. x.    |  4  0 |  2  0  2   0  0  0  0 |  *  * 30  *  *  *  *  *  *  *  *  *  * | 0  1  1 0  0  0  1  0  0  0 0  0 0 | 1 0  1  1 0 0
.. o.3x. ..    |  3  0 |  0  3  0   0  0  0  0 |  *  *  * 20  *  *  *  *  *  *  *  *  * | 1  0  0 1  0  0  0  1  0  0 0  0 0 | 1 1  0  0 1 0
.. .. x.3x.    |  6  0 |  0  3  3   0  0  0  0 |  *  *  *  * 20  *  *  *  *  *  *  *  * | 0  0  1 1  0  0  0  0  0  1 0  0 0 | 1 0  0  1 1 0
xx .. .. ..&#x |  2  2 |  1  0  0   2  1  0  0 |  *  *  *  *  * 60  *  *  *  *  *  *  * | 0  0  0 0  1  1  1  0  0  0 0  0 0 | 0 1  1  1 0 0
.. ox .. ..&#x |  1  2 |  0  0  0   2  0  1  0 |  *  *  *  *  *  * 60  *  *  *  *  *  * | 0  0  0 0  1  0  0  1  1  0 0  0 0 | 0 1  1  0 1 0
.. .. xx ..&#x |  2  2 |  0  1  0   2  0  0  1 |  *  *  *  *  *  *  * 60  *  *  *  *  * | 0  0  0 0  0  1  0  1  0  1 0  0 0 | 0 1  0  1 1 0
.. .. .. xo&#x |  2  1 |  0  0  1   2  0  0  0 |  *  *  *  *  *  *  *  * 60  *  *  *  * | 0  0  0 0  0  0  1  0  1  1 0  0 0 | 0 0  1  1 1 0
.x3.x .. ..    |  0  6 |  0  0  0   0  3  3  0 |  *  *  *  *  *  *  *  *  * 10  *  *  * | 0  0  0 0  2  0  0  0  0  0 2  0 0 | 0 2  1  0 0 1
.x .. .x ..    |  0  4 |  0  0  0   0  2  0  2 |  *  *  *  *  *  *  *  *  *  * 30  *  * | 0  0  0 0  0  1  0  0  0  0 1  1 0 | 0 1  0  1 0 1
.. .x3.x ..    |  0  6 |  0  0  0   0  0  3  3 |  *  *  *  *  *  *  *  *  *  *  * 20  * | 0  0  0 0  0  0  0  1  0  0 1  0 1 | 0 1  0  0 1 1
.. .. .x3.o    |  0  3 |  0  0  0   0  0  0  3 |  *  *  *  *  *  *  *  *  *  *  *  * 20 | 0  0  0 0  0  0  0  0  0  1 0  1 1 | 0 0  0  1 1 1
---------------+-------+-----------------------+----------------------------------------+------------------------------------+--------------
x.3o.3x. ..    ♦ 12  0 | 12 12  0   0  0  0  0 |  4  6  0  4  0  0  0  0  0  0  0  0  0 | 5  *  * *  *  *  *  *  *  * *  * * | 1 1  0  0 0 0
x.3o. .. x.    ♦  6  0 |  6  0  3   0  0  0  0 |  2  0  3  0  0  0  0  0  0  0  0  0  0 | * 10  * *  *  *  *  *  *  * *  * * | 1 0  1  0 0 0
x. .. x.3x.    ♦ 12  0 |  6  6  6   0  0  0  0 |  0  3  3  0  2  0  0  0  0  0  0  0  0 | *  * 10 *  *  *  *  *  *  * *  * * | 1 0  0  1 0 0
.. o.3x.3x.    ♦ 12  0 |  0 12  6   0  0  0  0 |  0  0  0  4  4  0  0  0  0  0  0  0  0 | *  *  * 5  *  *  *  *  *  * *  * * | 1 0  0  0 1 0
xx3ox .. ..&#x ♦  3  6 |  3  0  0   6  3  3  0 |  1  0  0  0  0  3  3  0  0  1  0  0  0 | *  *  * * 20  *  *  *  *  * *  * * | 0 1  1  0 0 0
xx .. xx ..&#x ♦  4  4 |  2  2  0   4  2  0  2 |  0  1  0  0  0  2  0  2  0  0  1  0  0 | *  *  * *  * 30  *  *  *  * *  * * | 0 1  0  1 0 0
xx .. .. xo&#x ♦  4  2 |  2  0  2   4  1  0  0 |  0  0  1  0  0  2  0  0  2  0  0  0  0 | *  *  * *  *  * 30  *  *  * *  * * | 0 0  1  1 0 0
.. ox3xx ..&#x ♦  3  6 |  0  3  0   6  0  3  3 |  0  0  0  1  0  0  3  3  0  0  0  1  0 | *  *  * *  *  *  * 20  *  * *  * * | 0 1  0  0 1 0
.. ox .. xo&#x ♦  2  2 |  0  0  1   4  0  1  0 |  0  0  0  0  0  0  2  0  2  0  0  0  0 | *  *  * *  *  *  *  * 30  * *  * * | 0 0  1  0 1 0
.. .. xx3xo&#x ♦  6  3 |  0  3  3   6  0  0  3 |  0  0  0  0  1  0  0  3  3  0  0  0  1 | *  *  * *  *  *  *  *  * 20 *  * * | 0 0  0  1 1 0
.x3.x3.x ..    ♦  0 24 |  0  0  0   0 12 12 12 |  0  0  0  0  0  0  0  0  0  4  6  4  0 | *  *  * *  *  *  *  *  *  * 5  * * | 0 1  0  0 0 1
.x .. .x3.o    ♦  0  6 |  0  0  0   0  3  0  6 |  0  0  0  0  0  0  0  0  0  0  3  0  2 | *  *  * *  *  *  *  *  *  * * 10 * | 0 0  0  1 0 1
.. .x3x.3.o    ♦  0 12 |  0  0  0   0  0  6 12 |  0  0  0  0  0  0  0  0  0  0  0  4  4 | *  *  * *  *  *  *  *  *  * *  * 5 | 0 0  0  0 1 1
---------------+-------+-----------------------+----------------------------------------+------------------------------------+--------------
x.3o.3x.3x.    ♦ 60  0 | 60 60 30   0  0  0  0 | 20 30 30 20 20  0  0  0  0  0  0  0  0 | 5 10 10 5  0  0  0  0  0  0 0  0 0 | 1 *  *  * * *
xx3ox3xx ..&#x ♦ 12 24 | 12 12  0  24 12 12 12 |  4  6  0  4  0 12 12 12  0  4  6  4  0 | 1  0  0 0  4  6  0  4  0  0 1  0 0 | * 5  *  * * *
xx3ox .. xo&#x ♦  6  6 |  6  0  3  12  3  3  0 |  2  0  3  0  0  6  6  0  6  1  0  0  0 | 0  1  0 0  2  0  3  0  3  0 0  0 0 | * * 10  * * *
xx .. xx3xo&#x ♦ 12  6 |  6  6  6  12  3  0  6 |  0  3  3  0  2  6  0  6  6  0  3  0  2 | 0  0  1 0  0  3  3  0  0  2 0  1 0 | * *  * 10 * *
.. ox3xx3xo&#x ♦ 12 12 |  0 12  6  24  0  6 12 |  0  0  0  4  4  0 12 12 12  0  0  4  4 | 0  0  0 1  0  0  0  4  6  4 0  0 1 | * *  *  * 5 *
.x3.x3.x3.o    ♦  0 60 |  0  0  0   0 30 30 60 |  0  0  0  0  0  0  0  0  0 10 30 20 20 | 0  0  0 0  0  0  0  0  0  0 5 10 5 | * *  *  * * 1