Polytope of Type {4,3,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,5}*1920
Also Known As : hemi-2{3,5}/2. if this polytope has another name.
Group : SmallGroup(1920,240995)
Rank : 4
Schlafli Type : {4,3,5}
Number of vertices, edges, etc : 32, 96, 120, 40
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 4
Special Properties :
   Locally Projective
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   16-fold quotients : {2,3,5}*120
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2*s1> of order 2.
      22 facets:
         18 of {4,3}*48
         4 of {4,3}*24
      16 vertex figures:
         16 of {3,5}*60
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3> of order 2.
      24 facets:
         16 of {4,3}*48
         8 of 2-fold non-regular quotient of {4,3}*48
      16 vertex figures:
         16 of {3,5}*60
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3> of order 4.
      16 facets:
         12 of 2-fold non-regular quotient of {4,3}*48
         4 of {4,3}*48
      8 vertex figures:
         8 of {3,5}*60
   P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s1*s3*s2*s1*s0*s1*s2*s3> of order 4.
      16 facets:
         12 of 2-fold non-regular quotient of {4,3}*48
         4 of {4,3}*48
      8 vertex figures:
         8 of {3,5}*60
   P/N, where N=<s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2, s1*s0*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1> of order 4.
      16 facets:
         6 of {4,3}*48
         4 of {2,3}*12
         6 of 2-fold non-regular quotient of {4,3}*48
      8 vertex figures:
         8 of {3,5}*60
   P/N, where N=<s3*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2*s3, s0*s1*s0*s1*s2*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3> of order 4.
      16 facets:
         4 of {4,3}*48
         12 of 2-fold non-regular quotient of {4,3}*48
      8 vertex figures:
         8 of {3,5}*60
   P/N, where N=<s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2, s0*s2*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3> of order 4.
      14 facets:
         6 of {4,3}*48
         4 of {4,3}*24
         4 of 2-fold non-regular quotient of {4,3}*48
      8 vertex figures:
         8 of {3,5}*60
   P/N, where N=<s0*s3*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2*s3, s1*s0*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2*s1> of order 4.
      14 facets:
         6 of {4,3}*48
         4 of 2-fold non-regular quotient of {4,3}*48
         4 of {4,3}*24
      8 vertex figures:
         8 of {3,5}*60
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3, s0*s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2> of order 8.
      10 facets:
         6 of 2-fold non-regular quotient of {4,3}*48
         4 of {4,3}*24
      4 vertex figures:
         4 of {3,5}*60
   P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s1*s3*s2*s1*s0*s1*s2*s3, s0*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2> of order 8.
      10 facets:
         6 of 2-fold non-regular quotient of {4,3}*48
         4 of {4,3}*24
      4 vertex figures:
         4 of {3,5}*60
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s1*s2*s3*s2*s1*s0*s1*s2*s3, s2*s1*s0*s1*s3*s2*s1*s0*s1*s2*s3*s2> of order 8.
      12 facets:
         8 of 2-fold non-regular quotient of {4,3}*48
         4 of {2,3}*12
      4 vertex figures:
         4 of {3,5}*60

Permutation Representation (GAP) : 
s0 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,20)(18,19);;
s1 := ( 1, 2)( 3, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,18)(14,19)(15,17)(16,20);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20);;
s3 := ( 1, 3)( 2, 4)( 5,17)( 6,20)( 7,18)( 8,19)( 9,15)(10,14)(11,13)(12,16);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3, 
s0*s2*s1*s0*s2*s3*s2*s1*s2*s0*s3*s1*s2*s3*s1*s0*s1*s2*s3*s1*s0*s1*s2*s3*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(20)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,20)(18,19);
s1 := Sym(20)!( 1, 2)( 3, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,18)(14,19)(15,17)(16,20);
s2 := Sym(20)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20);
s3 := Sym(20)!( 1, 3)( 2, 4)( 5,17)( 6,20)( 7,18)( 8,19)( 9,15)(10,14)(11,13)(12,16);
poly := sub<Sym(20)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s2*s1*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3, 
s0*s2*s1*s0*s2*s3*s2*s1*s2*s0*s3*s1*s2*s3*s1*s0*s1*s2*s3*s1*s0*s1*s2*s3*s0*s1 >;
References :
  1. Hartley, M. I.; Quotients of Some Finite Universal Locally Projective Polytopes, Discrete and Computational Geometry 29 pp435-443 (2003)

to this polytope