Acronym ...
Name 2quiphi (?)
Circumradius sqrt[5-2 sqrt(5)] = 0.726543
General of army rox
Colonel of regiment rigfix
Confer
non-Grünbaumian master:
quiphi  
general polytopal classes:
Wythoffian polychora  

Looks like a compound of 2 quasiprismatohecatonicosachora (quiphi), and indeed all elements coincide by pairs.

As abstract polytope quiphi is isomorphic to sophi, thereby replacing pentagrams by pentagons, resp. by replacing stip by pip and sissid by gad.


Incidence matrix according to Dynkin symbol

x5/2o5/4o5/2x

.   .   .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
--------------+------+-----------+----------------+----------------
x   .   .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
.   .   .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
--------------+------+-----------+----------------+----------------
x5/2o   .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x   .   .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
.   .   o5/2x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
--------------+------+-----------+----------------+----------------
x5/2o5/4o   .    12 |   30    0 |   12    0    0 | 120   *   *   *
x5/2o   .   x    10 |   10    5 |    2    5    0 |   * 720   *   *
x   .   o5/2x    10 |    5   10 |    0    5    2 |   *   * 720   *
.   o5/4o5/2x    12 |    0   30 |    0    0   12 |   *   *   * 120
or
.   .   .   .    | 1440 |   10 |   10   10 |   2   10
-----------------+------+------+-----------+---------
x   .   .   .  & |    2 | 7200 |    2    2 |   1    3
-----------------+------+------+-----------+---------
x5/2o   .   .  & |    5 |    5 | 2880    * |   1    1
x   .   .   x    |    4 |    4 |    * 3600 |   0    2
-----------------+------+------+-----------+---------
x5/2o5/4o   .  &    12 |   30 |   12    0 | 240    *
x5/2o   .   x  &    10 |   15 |    2    5 |   * 1440

x5/2o5o5/3x

.   . .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
------------+------+-----------+----------------+----------------
x   . .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
.   . .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
------------+------+-----------+----------------+----------------
x5/2o .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x   . .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
.   . o5/3x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
------------+------+-----------+----------------+----------------
x5/2o5o   .    12 |   30    0 |   12    0    0 | 120   *   *   *
x5/2o .   x    10 |   10    5 |    2    5    0 |   * 720   *   *
x   . o5/3x    10 |    5   10 |    0    5    2 |   *   * 720   *
.   o5o5/3x    12 |    0   30 |    0    0   12 |   *   *   * 120

x5/3o5/4o5/3x

.   .   .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
--------------+------+-----------+----------------+----------------
x   .   .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
.   .   .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
--------------+------+-----------+----------------+----------------
x5/3o   .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x   .   .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
.   .   o5/3x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
--------------+------+-----------+----------------+----------------
x5/3o5/4o   .    12 |   30    0 |   12    0    0 | 120   *   *   *
x5/3o   .   x    10 |   10    5 |    2    5    0 |   * 720   *   *
x   .   o5/3x    10 |    5   10 |    0    5    2 |   *   * 720   *
.   o5/4o5/3x    12 |    0   30 |    0    0   12 |   *   *   * 120
or
.   .   .   .    | 1440 |   10 |   10   10 |   2   10
-----------------+------+------+-----------+---------
x   .   .   .  & |    2 | 7200 |    2    2 |   1    3
-----------------+------+------+-----------+---------
x5/3o   .   .  & |    5 |    5 | 2880    * |   1    1
x   .   .   x    |    4 |    4 |    * 3600 |   0    2
-----------------+------+------+-----------+---------
x5/3o5/4o   .  &    12 |   30 |   12    0 | 240    *
x5/3o   .   x  &    10 |   15 |    2    5 |   * 1440

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