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The base polychoron here is a uniform sidpixhi, then augmented by dodecahedral pyramids with lacing edges of size q = sqrt(2) = 1.414214. That lacing edge size thereby allows for a common circumradius.

Incidence matrix according to Dynkin symbol

((xV3oo3oo5xo))&#zq   → height = 0

o.3o.3o.5o.     | 2400   * |    3    3    1 |    3    6    3    3 |   1    3   3    3
.o3.o3.o5.o     |    * 120 ♦    0    0   20 |    0    0    0   30 |   0    0   0   12
----------------+----------+----------------+---------------------+------------------
x. .. .. ..     |    2   0 | 3600    *    * |    2    2    0    0 |   1    2   1    0  x
.. .. .. x.     |    2   0 |    * 3600    * |    0    2    2    1 |   0    1   2    2  x
oo3oo3oo5oo&#q  |    1   1 |    *    * 2400 |    0    0    0    3 |   0    0   0    3  q
----------------+----------+----------------+---------------------+------------------
x.3o. .. ..     |    3   0 |    3    0    0 | 2400    *    *    * |   1    1   0    0  verf(tet)
x. .. .. x.     |    4   0 |    2    2    0 |    * 3600    *    * |   0    1   1    0  verf(oct)
.. .. o.5x.     |    5   0 |    0    5    0 |    *    * 1440    * |   0    0   1    1  verf(ike)
.. .. .. xo&#q  |    2   1 |    0    1    2 |    *    *    * 3600 |   0    0   0    2  verf(trip)
----------------+----------+----------------+---------------------+------------------
x.3o.3o. ..     ♦    4   0 |    6    0    0 |    4    0    0    0 | 600    *   *    *  verf(pen)
x.3o. .. x.     ♦    6   0 |    6    3    0 |    2    3    0    0 |   * 1200   *    *  verf(rap)
x. .. o.5x.     ♦   10   0 |    5   10    0 |    0    5    2    0 |   *    * 720    *  verf(rox)
.. .. oo5xo&#q  |    5   1 |    0    5    5 |    0    0    1    5 |   *    *   * 1440  verf(ipe)