As abstract polytope gefirdit xethi is isomorphic to gefridit xethi, thereby interchanging the roles of pentagrams and pentagons, resp. replacing gissid by doe and gidtid by sidtid.

Looks like the compound of grad tathi and teppix. Indeed, vertices coincide by 4, edges by 3, and triangles and hexagons each coincide by pairs.

Incidence matrix according to Dynkin symbol

        _o_        
    _3-  |  5/2    
 o<     5/4     >x 
    -3_  |  _3-    
        -x-        

o3o3x3x5/2*a5/4*c

. . . .           | 2400 |    6    3 |    3    3    3    6 |   1   1   3   3
------------------+------+-----------+---------------------+----------------
. . x .           |    2 | 7200    * |    1    0    1    1 |   1   0   1   1
. . . x           |    2 |    * 3600 |    0    2    0    2 |   0   1   2   1
------------------+------+-----------+---------------------+----------------
o . x .   *a5/4*c |    5 |    5    0 | 1440    *    *    * |   1   0   1   0
o . . x5/2*a      |    5 |    0    5 |    * 1440    *    * |   0   1   1   0
. o3x .           |    3 |    3    0 |    *    * 2400    * |   1   0   0   1
. . x3x           |    6 |    3    3 |    *    *    * 2400 |   0   0   1   1
------------------+------+-----------+---------------------+----------------
o3o3x .   *a5/4*c ♦   20 |   60    0 |   12    0   20    0 | 120   *   *   *
o3o . x5/2*a      ♦   20 |    0   30 |    0   12    0    0 |   * 120   *   *
o . x3x5/2*a5/4*c ♦   60 |   60   60 |   12   12    0   20 |   *   * 120   *
. o3x3x           ♦   12 |   12    6 |    0    0    4    4 |   *   *   * 600

        _o_        
    _3-  |  5/3    
 o<      5      >x 
    3/2  |  _3-    
        -x-        

o3o3/2x3x5/3*a5*c

. .   . .         | 2400 |    6    3 |    3    3    3    6 |   1   1   3   3
------------------+------+-----------+---------------------+----------------
. .   x .         |    2 | 7200    * |    1    0    1    1 |   1   0   1   1
. .   . x         |    2 |    * 3600 |    0    2    0    2 |   0   1   2   1
------------------+------+-----------+---------------------+----------------
o .   x .   *a5*c |    5 |    5    0 | 1440    *    *    * |   1   0   1   0
o .   . x5/3*a    |    5 |    0    5 |    * 1440    *    * |   0   1   1   0
. o3/2x .         |    3 |    3    0 |    *    * 2400    * |   1   0   0   1
. .   x3x         |    6 |    3    3 |    *    *    * 2400 |   0   0   1   1
------------------+------+-----------+---------------------+----------------
o3o3/2x .   *a5*c ♦   20 |   60    0 |   12    0   20    0 | 120   *   *   *
o3o   . x5/3*a    ♦   20 |    0   30 |    0   12    0    0 |   * 120   *   *
o .   x3x5/3*a5*c ♦   60 |   60   60 |   12   12    0   20 |   *   * 120   *
. o3/2x3x         ♦   12 |   12    6 |    0    0    4    4 |   *   *   * 600

        _o_        
    3/2  |  5/3    
 o<      5      >x 
    -3_  |  _3-    
        -x-        

o3/2o3x3x5/3*a5*c

.   . . .         | 2400 |    6    3 |    3    3    3    6 |   1   1   3   3
------------------+------+-----------+---------------------+----------------
.   . x .         |    2 | 7200    * |    1    0    1    1 |   1   0   1   1
.   . . x         |    2 |    * 3600 |    0    2    0    2 |   0   1   2   1
------------------+------+-----------+---------------------+----------------
o   . x .   *a5*c |    5 |    5    0 | 1440    *    *    * |   1   0   1   0
o   . . x5/3*a    |    5 |    0    5 |    * 1440    *    * |   0   1   1   0
.   o3x .         |    3 |    3    0 |    *    * 2400    * |   1   0   0   1
.   . x3x         |    6 |    3    3 |    *    *    * 2400 |   0   0   1   1
------------------+------+-----------+---------------------+----------------
o3/2o3x .   *a5*c ♦   20 |   60    0 |   12    0   20    0 | 120   *   *   *
o3/2o . x5/3*a    ♦   20 |    0   30 |    0   12    0    0 |   * 120   *   *
o   . x3x5/3*a5*c ♦   60 |   60   60 |   12   12    0   20 |   *   * 120   *
.   o3x3x         ♦   12 |   12    6 |    0    0    4    4 |   *   *   * 600

        _o_        
    3/2  |  5/2    
 o<     5/4     >x 
    3/2  |  _3-    
        -x-        

o3/2o3/2x3x5/2*a5/4*c

.   .   . .           | 2400 |    6    3 |    3    3    3    6 |   1   1   3   3
----------------------+------+-----------+---------------------+----------------
.   .   x .           |    2 | 7200    * |    1    0    1    1 |   1   0   1   1
.   .   . x           |    2 |    * 3600 |    0    2    0    2 |   0   1   2   1
----------------------+------+-----------+---------------------+----------------
o   .   x .   *a5/4*c |    5 |    5    0 | 1440    *    *    * |   1   0   1   0
o   .   . x5/2*a      |    5 |    0    5 |    * 1440    *    * |   0   1   1   0
.   o3/2x .           |    3 |    3    0 |    *    * 2400    * |   1   0   0   1
.   .   x3x           |    6 |    3    3 |    *    *    * 2400 |   0   0   1   1
----------------------+------+-----------+---------------------+----------------
o3/2o3/2x .   *a5/4*c ♦   20 |   60    0 |   12    0   20    0 | 120   *   *   *
o3/2o   . x5/2*a      ♦   20 |    0   30 |    0   12    0    0 |   * 120   *   *
o   .   x3x5/2*a5/4*c ♦   60 |   60   60 |   12   12    0   20 |   *   * 120   *
.   o3/2x3x           ♦   12 |   12    6 |    0    0    4    4 |   *   *   * 600