The relation to ex runs as follows: ex in axial icosahedral subsymmetry can be given as VFfxo2oxofo3oooox5ooxoo&#zx = oxofofoxo3ooooxoooo5ooxoooxoo&#xt (F = ff = f+x, V = uf = f+f). That will be transformed into VFfxo2oxofx3oooo(-x)5ooxof&#zx. And finally into VFfxo2oxofx3oofo(-x)5oo(-x)of&#zx. Then a Stott expansion wrt. the third and fourth nodes produces this polychoron.

Incidence matrix according to Dynkin symbol

oxofxfoxo3xxFxoxFxx5xxoxFxoxx&#xt   → height(1,2) = height(3,4) = height(6,7) = height(8,9) = (sqrt(5)-1)/4 = 0.309017
                                      height(2,3) = height(4,5) = height(5,6) = height(7,8) = 1/2
(tid || pseudo grid || pseudo F-id || pseudo (f,x,x)-grid || pseudo (x,F)-srid || pseudo (f,x,x)-grid || pseudo F-id || pseudo grid || tid)

o........3o........5o........     & | 120   *  *   *  * |   2  1   2   0   0   0   0   0   0   0   0   0   0  0 |  1  2   1   2   2  0  0  0   0   0   0   0   0   0   0  0   0  0  0   0  0 | 1  1  1  2  0  0  0   0  0  0  0
.o.......3.o.......5.o.......     & |   * 240  *   *  * |   0  0   1   1   1   1   1   1   0   0   0   0   0  0 |  0  0   1   1   1  1  1  1   1   1   1   1   1   0   0  0   0  0  0   0  0 | 0  1  1  1  1  1  1   1  0  0  0
..o......3..o......5..o......     & |   *   * 60   *  * |   0  0   0   0   0   0   4   0   4   0   0   0   0  0 |  0  0   0   0   0  0  0  0   2   2   0   0   4   2   2  0   0  0  0   0  0 | 0  0  0  0  2  1  0   2  1  0  0
...o.....3...o.....5...o.....     & |   *   *  * 240  * |   0  0   0   0   0   0   0   1   1   1   1   1   1  0 |  0  0   0   0   0  0  0  0   0   0   1   1   1   1   1  1   1  1  1   1  0 | 0  0  0  0  1  0  1   1  1  1  1
....o....3....o....5....o....       |   *   *  *   * 60 |   0  0   0   0   0   0   0   0   0   0   0   4   0  2 |  0  0   0   0   0  0  0  0   0   0   0   0   0   0   4  0   2  0  0   2  1 | 0  0  0  0  2  0  0   0  2  1  0
------------------------------------+-------------------+-------------------------------------------------------+----------------------------------------------------------------------------+---------------------------------
......... x........ .........     & |   2   0  0   0  0 | 120  *   *   *   *   *   *   *   *   *   *   *   *  * |  1  1   0   1   0  0  0  0   0   0   0   0   0   0   0  0   0  0  0   0  0 | 1  1  0  1  0  0  0   0  0  0  0
......... ......... x........     & |   2   0  0   0  0 |   * 60   *   *   *   *   *   *   *   *   *   *   *  * |  0  2   0   0   2  0  0  0   0   0   0   0   0   0   0  0   0  0  0   0  0 | 1  0  1  2  0  0  0   0  0  0  0
oo.......3oo.......5oo.......&#x  & |   1   1  0   0  0 |   *  * 240   *   *   *   *   *   *   *   *   *   *  * |  0  0   1   1   1  0  0  0   0   0   0   0   0   0   0  0   0  0  0   0  0 | 0  1  1  1  0  0  0   0  0  0  0
.x....... ......... .........     & |   0   2  0   0  0 |   *  *   * 120   *   *   *   *   *   *   *   *   *  * |  0  0   1   0   0  1  1  0   1   0   0   0   0   0   0  0   0  0  0   0  0 | 0  1  1  0  1  1  0   0  0  0  0
......... .x....... .........     & |   0   2  0   0  0 |   *  *   *   * 120   *   *   *   *   *   *   *   *  * |  0  0   0   1   0  1  0  1   0   0   1   0   0   0   0  0   0  0  0   0  0 | 0  1  0  1  1  0  1   0  0  0  0
......... ......... .x.......     & |   0   2  0   0  0 |   *  *   *   *   * 120   *   *   *   *   *   *   *  * |  0  0   0   0   1  0  1  1   0   1   0   1   0   0   0  0   0  0  0   0  0 | 0  0  1  1  0  1  1   1  0  0  0
.oo......3.oo......5.oo......&#x  & |   0   1  1   0  0 |   *  *   *   *   *   * 240   *   *   *   *   *   *  * |  0  0   0   0   0  0  0  0   1   1   0   0   1   0   0  0   0  0  0   0  0 | 0  0  0  0  1  1  0   1  0  0  0
.o.o.....3.o.o.....5.o.o.....&#x  & |   0   1  0   1  0 |   *  *   *   *   *   *   * 240   *   *   *   *   *  * |  0  0   0   0   0  0  0  0   0   0   1   1   1   0   0  0   0  0  0   0  0 | 0  0  0  0  1  0  1   1  0  0  0
..oo.....3..oo.....5..oo.....&#x  & |   0   0  1   1  0 |   *  *   *   *   *   *   *   * 240   *   *   *   *  * |  0  0   0   0   0  0  0  0   0   0   0   0   1   1   1  0   0  0  0   0  0 | 0  0  0  0  1  0  0   1  1  0  0
......... ...x..... .........     & |   0   0  0   2  0 |   *  *   *   *   *   *   *   *   * 120   *   *   *  * |  0  0   0   0   0  0  0  0   0   0   1   0   0   0   0  1   1  1  0   0  0 | 0  0  0  0  1  0  1   0  0  1  1
......... ......... ...x.....     & |   0   0  0   2  0 |   *  *   *   *   *   *   *   *   *   * 120   *   *  * |  0  0   0   0   0  0  0  0   0   0   0   1   0   1   0  1   0  0  1   0  0 | 0  0  0  0  0  0  1   1  1  0  1
...oo....3...oo....5...oo....&#x  & |   0   0  0   1  1 |   *  *   *   *   *   *   *   *   *   *   * 240   *  * |  0  0   0   0   0  0  0  0   0   0   0   0   0   0   1  0   1  0  0   1  0 | 0  0  0  0  1  0  0   0  1  1  0
...o.o...3...o.o...5...o.o...&#x    |   0   0  0   2  0 |   *  *   *   *   *   *   *   *   *   *   *   * 120  * |  0  0   0   0   0  0  0  0   0   0   0   0   0   0   0  0   0  1  1   1  0 | 0  0  0  0  0  0  0   0  1  1  1
....x.... ......... .........       |   0   0  0   0  2 |   *  *   *   *   *   *   *   *   *   *   *   *   * 60 |  0  0   0   0   0  0  0  0   0   0   0   0   0   0   2  0   0  0  0   0  1 | 0  0  0  0  2  0  0   0  1  0  0
------------------------------------+-------------------+-------------------------------------------------------+----------------------------------------------------------------------------+---------------------------------
o........3x........ .........     & |   3   0  0   0  0 |   3  0   0   0   0   0   0   0   0   0   0   0   0  0 | 40  *   *   *   *  *  *  *   *   *   *   *   *   *   *  *   *  *  *   *  * | 1  1  0  0  0  0  0   0  0  0  0
......... x........5x........     & |  10   0  0   0  0 |   5  5   0   0   0   0   0   0   0   0   0   0   0  0 |  * 24   *   *   *  *  *  *   *   *   *   *   *   *   *  *   *  *  *   *  * | 1  0  0  1  0  0  0   0  0  0  0
ox....... ......... .........&#x  & |   1   2  0   0  0 |   0  0   2   1   0   0   0   0   0   0   0   0   0  0 |  *  * 120   *   *  *  *  *   *   *   *   *   *   *   *  *   *  *  *   *  * | 0  1  1  0  0  0  0   0  0  0  0
......... xx....... .........&#x  & |   2   2  0   0  0 |   1  0   2   0   1   0   0   0   0   0   0   0   0  0 |  *  *   * 120   *  *  *  *   *   *   *   *   *   *   *  *   *  *  *   *  * | 0  1  0  1  0  0  0   0  0  0  0
......... ......... xx.......&#x  & |   2   2  0   0  0 |   0  1   2   0   0   1   0   0   0   0   0   0   0  0 |  *  *   *   * 120  *  *  *   *   *   *   *   *   *   *  *   *  *  *   *  * | 0  0  1  1  0  0  0   0  0  0  0
.x.......3.x....... .........     & |   0   6  0   0  0 |   0  0   0   3   3   0   0   0   0   0   0   0   0  0 |  *  *   *   *   * 40  *  *   *   *   *   *   *   *   *  *   *  *  *   *  * | 0  1  0  0  1  0  0   0  0  0  0
.x....... ......... .x.......     & |   0   4  0   0  0 |   0  0   0   2   0   2   0   0   0   0   0   0   0  0 |  *  *   *   *   *  * 60  *   *   *   *   *   *   *   *  *   *  *  *   *  * | 0  0  1  0  0  1  0   0  0  0  0
......... .x.......5.x.......     & |   0  10  0   0  0 |   0  0   0   0   5   5   0   0   0   0   0   0   0  0 |  *  *   *   *   *  *  * 24   *   *   *   *   *   *   *  *   *  *  *   *  * | 0  0  0  1  0  0  1   0  0  0  0
.xo...... ......... .........&#x  & |   0   2  1   0  0 |   0  0   0   1   0   0   2   0   0   0   0   0   0  0 |  *  *   *   *   *  *  *  * 120   *   *   *   *   *   *  *   *  *  *   *  * | 0  0  0  0  1  1  0   0  0  0  0
......... ......... .xo......&#x  & |   0   2  1   0  0 |   0  0   0   0   0   1   2   0   0   0   0   0   0  0 |  *  *   *   *   *  *  *  *   * 120   *   *   *   *   *  *   *  *  *   *  * | 0  0  0  0  0  1  0   1  0  0  0
......... .x.x..... .........&#x  & |   0   2  0   2  0 |   0  0   0   0   1   0   0   2   0   1   0   0   0  0 |  *  *   *   *   *  *  *  *   *   * 120   *   *   *   *  *   *  *  *   *  * | 0  0  0  0  1  0  1   0  0  0  0
......... ......... .x.x.....&#x  & |   0   2  0   2  0 |   0  0   0   0   0   1   0   2   0   0   1   0   0  0 |  *  *   *   *   *  *  *  *   *   *   * 120   *   *   *  *   *  *  *   *  * | 0  0  0  0  0  0  1   1  0  0  0
.ooo.....3.ooo.....5.ooo.....&#x  & |   0   1  1   1  0 |   0  0   0   0   0   0   1   1   1   0   0   0   0  0 |  *  *   *   *   *  *  *  *   *   *   *   * 240   *   *  *   *  *  *   *  * | 0  0  0  0  1  0  0   1  0  0  0
......... ......... ..ox.....&#x  & |   0   0  1   2  0 |   0  0   0   0   0   0   0   0   2   0   1   0   0  0 |  *  *   *   *   *  *  *  *   *   *   *   *   * 120   *  *   *  *  *   *  * | 0  0  0  0  0  0  0   1  1  0  0
..ofx.... ......... .........&#x  & |   0   0  1   2  2 |   0  0   0   0   0   0   0   0   2   0   0   2   0  1 |  *  *   *   *   *  *  *  *   *   *   *   *   *   * 120  *   *  *  *   *  * | 0  0  0  0  1  0  0   0  1  0  0
......... ...x.....5...x.....&#x  & |   0   0  0  10  0 |   0  0   0   0   0   0   0   0   0   5   5   0   0  0 |  *  *   *   *   *  *  *  *   *   *   *   *   *   *   * 24   *  *  *   *  * | 0  0  0  0  0  0  1   0  0  0  1
......... ...xo.... .........&#x  & |   0   0  0   2  1 |   0  0   0   0   0   0   0   0   0   1   0   2   0  0 |  *  *   *   *   *  *  *  *   *   *   *   *   *   *   *  * 120  *  *   *  * | 0  0  0  0  1  0  0   0  0  1  0
......... ...x.x... .........&#x    |   0   0  0   4  0 |   0  0   0   0   0   0   0   0   0   2   0   0   2  0 |  *  *   *   *   *  *  *  *   *   *   *   *   *   *   *  *   * 60  *   *  * | 0  0  0  0  0  0  0   0  0  1  1
......... ......... ...x.x...&#x    |   0   0  0   4  0 |   0  0   0   0   0   0   0   0   0   0   2   0   2  0 |  *  *   *   *   *  *  *  *   *   *   *   *   *   *   *  *   *  * 60   *  * | 0  0  0  0  0  0  0   0  1  0  1
...ooo...3...ooo...5...ooo...&#x    |   0   0  0   2  1 |   0  0   0   0   0   0   0   0   0   0   0   2   1  0 |  *  *   *   *   *  *  *  *   *   *   *   *   *   *   *  *   *  *  * 120  * | 0  0  0  0  0  0  0   0  1  1  0
....x....3....o.... .........       |   0   0  0   0  3 |   0  0   0   0   0   0   0   0   0   0   0   0   0  3 |  *  *   *   *   *  *  *  *   *   *   *   *   *   *   *  *   *  *  *   * 20 | 0  0  0  0  2  0  0   0  0  0  0
------------------------------------+-------------------+-------------------------------------------------------+----------------------------------------------------------------------------+---------------------------------
o........3x........5x........     & ♦  60   0  0   0  0 |  60 30   0   0   0   0   0   0   0   0   0   0   0  0 | 20 12   0   0   0  0  0  0   0   0   0   0   0   0   0  0   0  0  0   0  0 | 2  *  *  *  *  *  *   *  *  *  *
ox.......3xx....... .........&#x  & ♦   3   6  0   0  0 |   3  0   6   3   3   0   0   0   0   0   0   0   0  0 |  1  0   3   3   0  1  0  0   0   0   0   0   0   0   0  0   0  0  0   0  0 | * 40  *  *  *  *  *   *  *  *  *
ox....... ......... xx.......&#x  & ♦   2   4  0   0  0 |   0  1   4   2   0   2   0   0   0   0   0   0   0  0 |  0  0   2   0   2  0  1  0   0   0   0   0   0   0   0  0   0  0  0   0  0 | *  * 60  *  *  *  *   *  *  *  *
......... xx.......5xx.......&#x  & ♦  10  10  0   0  0 |   5  5  10   0   5   5   0   0   0   0   0   0   0  0 |  0  1   0   5   5  0  0  1   0   0   0   0   0   0   0  0   0  0  0   0  0 | *  *  * 24  *  *  *   *  *  *  *
.xofx....3.xFxo.... .........&#xt & ♦   0   6  3   6  3 |   0  0   0   3   3   0   6   6   6   3   0   6   0  3 |  0  0   0   0   0  1  0  0   3   0   3   0   6   0   3  0   3  0  0   0  1 | *  *  *  * 40  *  *   *  *  *  *
.xo...... ......... .xo......&#x  & ♦   0   4  1   0  0 |   0  0   0   2   0   2   4   0   0   0   0   0   0  0 |  0  0   0   0   0  0  1  0   2   2   0   0   0   0   0  0   0  0  0   0  0 | *  *  *  *  * 60  *   *  *  *  *
......... .x.x.....5.x.x.....&#x  & ♦   0  10  0  10  0 |   0  0   0   0   5   5   0  10   0   5   5   0   0  0 |  0  0   0   0   0  0  0  1   0   0   5   5   0   0   0  1   0  0  0   0  0 | *  *  *  *  *  * 24   *  *  *  *
......... ......... .xox.....&#x  & ♦   0   2  1   2  0 |   0  0   0   0   0   1   2   2   2   0   1   0   0  0 |  0  0   0   0   0  0  0  0   0   1   0   1   2   1   0  0   0  0  0   0  0 | *  *  *  *  *  *  * 120  *  *  *
..ofxfo.. ......... ..oxFxo..&#xt   ♦   0   0  2   8  4 |   0  0   0   0   0   0   0   0   8   0   4   8   4  2 |  0  0   0   0   0  0  0  0   0   0   0   0   0   4   4  0   0  0  2   4  0 | *  *  *  *  *  *  *   * 30  *  *
......... ...xox... .........&#x    ♦   0   0  0   4  1 |   0  0   0   0   0   0   0   0   0   2   0   4   2  0 |  0  0   0   0   0  0  0  0   0   0   0   0   0   0   0  0   2  1  0   2  0 | *  *  *  *  *  *  *   *  * 60  *
......... ...x.x...5...x.x...&#x    ♦   0   0  0  20  0 |   0  0   0   0   0   0   0   0   0  10  10   0  10  0 |  0  0   0   0   0  0  0  0   0   0   0   0   0   0   0  2   0  5  5   0  0 | *  *  *  *  *  *  *   *  *  * 12