Is this shape possible?
Let’s get the faces (done by hand).
In[]:=
f={{27,20,11,19},{20,21,14,13},{20,13,5,12},{20,12,4,11},{19,11,4,10},{12,5,1,4},{13,14,8,5},{14,22,15,8},{5,8,2,1},{10,4,1,3},{19,10,18,26},{10,3,9,18},{1,2,6,3},{3,6,43,9},{6,44,24,43},{6,2,7,44},{8,15,7,2},{15,23,16,7},{7,16,24,44},{9,43,24,17},{18,9,17,25},{20,29,28,27},{20,30,32,29},{20,21,31,30},{31,22,39,34},{21,14,22,31},{30,31,34,32},{29,32,33,28},{27,28,26,19},{28,33,36,26},{32,34,35,33},{34,39,38,35},{35,38,40,37},{33,35,37,36},{26,36,25,18},{36,37,41,25},{37,40,24,41},{25,41,24,17},{38,42,24,40},{39,23,42,38},{22,15,23,39},{23,16,24,42}};

Init code for canonical forms


f42

Eventually we get this polyhedron:
In[]:=
f42=Polyhedron
Number of points: 44
Number of faces: 42
;
In[]:=
Graphics3D[{Opacity[.6],f42},Boxed->False]
Out[]=