sishi+paphicki+gridaphi (?) clearly is an edge-faceting of the small stellated hecatonicosachoron (sishi). Thus the vertices coincide by 12 each.

As abstract polytope sishi+paphicki+gridaphi is isomorphic to gaghi+paphacki+sridaphi, thereby replacing pentagrams by pentagons, resp. replacing ike by gike, sissid by gad, and stip by pip.

Incidence matrix according to Dynkin symbol

x3o5o5/3x

. . .   . | 1440 |    5    5 |    5   10    5 |   1    5   5   1
----------+------+-----------+----------------+-----------------
x . .   . |    2 | 7200    * |    2    2    0 |   1    2   1   0
. . .   x |    2 |    * 7200 |    0    2    2 |   0    1   2   1
----------+------+-----------+----------------+-----------------
x3o .   . |    3 |    3    0 | 2400    *    * |   1    1   0   0
x . .   x |    4 |    2    2 |    * 3600    * |   0    1   1   0
. . o5/3x |    5 |    0    5 |    *    * 1440 |   0    0   1   1
----------+------+-----------+----------------+-----------------
x3o5o   . ♦   12 |   30    0 |   20    0    0 | 120    *   *   *
x3o .   x ♦    6 |    6    3 |    2    3    0 |   * 1200   *   *
x . o5/3x ♦   10 |    5   10 |    0    5    2 |   *    * 720   *
. o5o5/3x ♦   12 |    0   30 |    0    0   12 |   *    *   * 120

x3o5/4o5/2x

. .   .   . | 1440 |    5    5 |    5   10    5 |   1    5   5   1
------------+------+-----------+----------------+-----------------
x .   .   . |    2 | 7200    * |    2    2    0 |   1    2   1   0
. .   .   x |    2 |    * 7200 |    0    2    2 |   0    1   2   1
------------+------+-----------+----------------+-----------------
x3o   .   . |    3 |    3    0 | 2400    *    * |   1    1   0   0
x .   .   x |    4 |    2    2 |    * 3600    * |   0    1   1   0
. .   o5/2x |    5 |    0    5 |    *    * 1440 |   0    0   1   1
------------+------+-----------+----------------+-----------------
x3o5/4o   . ♦   12 |   30    0 |   20    0    0 | 120    *   *   *
x3o   .   x ♦    6 |    6    3 |    2    3    0 |   * 1200   *   *
x .   o5/2x ♦   10 |    5   10 |    0    5    2 |   *    * 720   *
. o5/4o5/2x ♦   12 |    0   30 |    0    0   12 |   *    *   * 120

x3/2o5o5/2x

.   . .   . | 1440 |    5    5 |    5   10    5 |   1    5   5   1
------------+------+-----------+----------------+-----------------
x   . .   . |    2 | 7200    * |    2    2    0 |   1    2   1   0
.   . .   x |    2 |    * 7200 |    0    2    2 |   0    1   2   1
------------+------+-----------+----------------+-----------------
x3/2o .   . |    3 |    3    0 | 2400    *    * |   1    1   0   0
x   . .   x |    4 |    2    2 |    * 3600    * |   0    1   1   0
.   . o5/2x |    5 |    0    5 |    *    * 1440 |   0    0   1   1
------------+------+-----------+----------------+-----------------
x3/2o5o   . ♦   12 |   30    0 |   20    0    0 | 120    *   *   *
x3/2o .   x ♦    6 |    6    3 |    2    3    0 |   * 1200   *   *
x   . o5/2x ♦   10 |    5   10 |    0    5    2 |   *    * 720   *
.   o5o5/2x ♦   12 |    0   30 |    0    0   12 |   *    *   * 120

x3/2o5/4o5/3x

.   .   .   . | 1440 |    5    5 |    5   10    5 |   1    5   5   1
--------------+------+-----------+----------------+-----------------
x   .   .   . |    2 | 7200    * |    2    2    0 |   1    2   1   0
.   .   .   x |    2 |    * 7200 |    0    2    2 |   0    1   2   1
--------------+------+-----------+----------------+-----------------
x3/2o   .   . |    3 |    3    0 | 2400    *    * |   1    1   0   0
x   .   .   x |    4 |    2    2 |    * 3600    * |   0    1   1   0
.   .   o5/3x |    5 |    0    5 |    *    * 1440 |   0    0   1   1
--------------+------+-----------+----------------+-----------------
x3/2o5/4o   . ♦   12 |   30    0 |   20    0    0 | 120    *   *   *
x3/2o   .   x ♦    6 |    6    3 |    2    3    0 |   * 1200   *   *
x   .   o5/3x ♦   10 |    5   10 |    0    5    2 |   *    * 720   *
.   o5/4o5/3x ♦   12 |    0   30 |    0    0   12 |   *    *   * 120