Polytope of Type {4,5,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,5,3}*1920b
if this polytope has a name.
Group : SmallGroup(1920,240995)
Rank : 4
Schlafli Type : {4,5,3}
Number of vertices, edges, etc : 32, 160, 120, 6
Order of s₀s₁s₂s₃ : 10
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Non-Orientable
   Flat
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 : {2,5,3}*120
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s3*s2*s1*s2*s1*s0*s1*s2*s1*s2*s3> of order 2.
      6 facets:
         1 of {4,5}*160
         5 of 2-fold non-regular quotient of {4,5}*320
      16 vertex figures:
         16 of {5,3}*60
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      6 facets:
         2 of 2-fold non-regular quotient of {4,5}*320
         2 of 2-fold non-regular quotient of {4,5}*320
         2 of 2-fold non-regular quotient of {4,5}*320
      16 vertex figures:
         16 of {5,3}*60
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      6 facets:
         6 of 4-fold non-regular quotient of {4,5}*320
      8 vertex figures:
         8 of {5,3}*60
   P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      6 facets:
         3 of 4-fold non-regular quotient of {4,5}*320
         3 of 4-fold non-regular quotient of {4,5}*320
      8 vertex figures:
         8 of {5,3}*60
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s3*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s3> of order 4.
      6 facets:
         3 of 4-fold non-regular quotient of {4,5}*320
         3 of 4-fold non-regular quotient of {4,5}*320
      8 vertex figures:
         8 of {5,3}*60
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      6 facets:
         3 of 4-fold non-regular quotient of {4,5}*320
         3 of 4-fold non-regular quotient of {4,5}*320
      8 vertex figures:
         8 of {5,3}*60
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      6 facets:
         2 of 4-fold non-regular quotient of {4,5}*320
         2 of 2-fold non-regular quotient of {4,5}*160
         2 of 4-fold non-regular quotient of {4,5}*320
      8 vertex figures:
         8 of {5,3}*60
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
      6 facets:
         2 of 8-fold non-regular quotient of {4,5}*320
         2 of 8-fold non-regular quotient of {4,5}*320
         2 of 8-fold non-regular quotient of {4,5}*320
      4 vertex figures:
         4 of {5,3}*60

Permutation Representation (GAP) :

s0 := ( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20);;
s1 := ( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,16)( 7,14)( 8,15)(17,18)(19,20);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20);;
s3 := ( 1, 5)( 2, 7)( 3, 8)( 4, 6)( 9,13)(10,14)(11,16)(12,15)(17,19)(18,20);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s0*s1*s2*s1*s3*s0*s1*s2*s3*s1*s0*s1*s2*s3*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(20)!( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20);
s1 := Sym(20)!( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,16)( 7,14)( 8,15)(17,18)(19,20);
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, 5)( 2, 7)( 3, 8)( 4, 6)( 9,13)(10,14)(11,16)(12,15)(17,19)(18,20);
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, 
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s0*s1*s2*s1*s3*s0*s1*s2*s3*s1*s0*s1*s2*s3*s0*s1 >;

References : None.
to this polytope