Polytope of Type {4,5}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,5}*1440
Also Known As : {4,5}₈. if this polytope has another name.
Group : SmallGroup(1440,5843)
Rank : 3
Schlafli Type : {4,5}
Number of vertices, edges, etc : 144, 360, 180
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,5}*720
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*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      92 facets:
         88 of {4}*8
         4 of {2}*4
      72 vertex figures:
         72 of {5}*10
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      60 facets:
         60 of {4}*8
      48 vertex figures:
         48 of {5}*10
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      48 facets:
         42 of {4}*8
         6 of {2}*4
      36 vertex figures:
         36 of {5}*10
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 6.
      32 facets:
         4 of {2}*4
         28 of {4}*8
      24 vertex figures:
         24 of {5}*10
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 9.
      20 facets:
         20 of {4}*8
      16 vertex figures:
         16 of {5}*10
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 12.
      16 facets:
         14 of {4}*8
         2 of {2}*4
      12 vertex figures:
         12 of {5}*10
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 18.
      12 facets:
         4 of {2}*4
         8 of {4}*8
      8 vertex figures:
         8 of {5}*10

Permutation Representation (GAP) :

s0 := ( 1, 3)( 2, 6)( 4, 9)( 5, 7)( 8,10);;
s1 := ( 1, 3)( 2, 8)( 4, 6)( 5,10)( 7, 9)(11,12);;
s2 := ( 1, 2)( 3, 6)( 4, 7)( 5, 9)( 8,10)(11,12);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(12)!( 1, 3)( 2, 6)( 4, 9)( 5, 7)( 8,10);
s1 := Sym(12)!( 1, 3)( 2, 8)( 4, 6)( 5,10)( 7, 9)(11,12);
s2 := Sym(12)!( 1, 2)( 3, 6)( 4, 7)( 5, 9)( 8,10)(11,12);
poly := sub<Sym(12)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope

Polytope of Type {4,5}

Twisty Puzzle