As abstract polytope gogishi || gax occurs itself in 2 isomorphic variants with exactly the same incidences: the taller one here, the antiprism, where the first type vertex figure in turn is gissid-gike antiprism, and the narrower one, the gogishi-gax retroprism, where that vertex figure acts as gissid-gike retroprism instead.

Incidence matrix according to Dynkin symbol

xo3oo3oo5/2ox&#x   → height = sqrt[6 sqrt(5)-13] = 0.645297
(taller gax || gogishi)

o.3o.3o.5/2o.    | 120   * ♦  12   20    0 |   30   60   30   0 |  20   60   60   12   0 | 1  20   30  12   1 0
.o3.o3.o5/2.o    |   * 600 |   0    4    4 |    0    6   12   6 |   0    4   12   12   4 | 0   1    4   6   4 1
-----------------+---------+---------------+--------------------+------------------------+---------------------
x. .. ..   ..    |   2   0 | 720    *    * |    5    5    0   0 |   5   10    5    0   0 | 1   5    5   1   0 0
oo3oo3oo5/2oo&#x |   1   1 |   * 2400    * |    0    3    3   0 |   0    3    6    3   0 | 0   1    3   3   1 0
.. .. ..   .x    |   0   2 |   *    * 1200 |    0    0    3   3 |   0    0    3    6   3 | 0   0    1   3   3 1
-----------------+---------+---------------+--------------------+------------------------+---------------------
x.3o. ..   ..    |   3   0 |   3    0    0 | 1200    *    *   * |   2    2    0    0   0 | 1   2    1   0   0 0
xo .. ..   ..&#x |   2   1 |   1    2    0 |    * 3600    *   * |   0    2    2    0   0 | 0   1    2   1   0 0
.. .. ..   ox&#x |   1   2 |   0    2    1 |    *    * 3600   * |   0    0    2    2   0 | 0   0    1   2   1 0
.. .. .o5/2.x    |   0   5 |   0    0    5 |    *    *    * 720 |   0    0    0    2   2 | 0   0    0   1   2 1
-----------------+---------+---------------+--------------------+------------------------+---------------------
x.3o.3o.   ..    ♦   4   0 |   6    0    0 |    4    0    0   0 | 600    *    *    *   * | 1   1    0   0   0 0
xo3oo ..   ..&#x ♦   3   1 |   3    3    0 |    1    3    0   0 |   * 2400    *    *   * | 0   1    1   0   0 0
xo .. ..   ox&#x ♦   2   2 |   1    4    1 |    0    2    2   0 |   *    * 3600    *   * | 0   0    1   1   0 0
.. .. oo5/2ox&#x ♦   1   5 |   0    5    5 |    0    0    5   1 |   *    *    * 1440   * | 0   0    0   1   1 0
.. .o3.o5/2.x    ♦   0  20 |   0    0   30 |    0    0    0  12 |   *    *    *    * 120 | 0   0    0   0   1 1
-----------------+---------+---------------+--------------------+------------------------+---------------------
x.3o.3o.5/2o.    ♦ 120   0 | 720    0    0 | 1200    0    0   0 | 600    0    0    0   0 | 1   *    *   *   * *
xo3oo3oo   ..&#x ♦   4   1 |   6    4    0 |    4    6    0   0 |   1    4    0    0   0 | * 600    *   *   * *
xo3oo ..   ox&#x ♦   3   2 |   3    6    1 |    1    6    3   0 |   0    2    3    0   0 | *   * 1200   *   * *
xo .. oo5/2ox&#x ♦   2   5 |   1   10    5 |    0    5   10   1 |   0    0    5    2   0 | *   *    * 720   * *
.. oo3oo5/2ox&#x ♦   1  20 |   0   20   30 |    0    0   30  12 |   0    0    0   12   1 | *   *    *   * 120 *
.o3.o3.o5/2.x    ♦   0 600 |   0    0 1200 |    0    0    0 720 |   0    0    0    0 120 | *   *    *   *   * 1