Polytope of Type {8,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4}*1440f
Also Known As : {8,4}₅. if this polytope has another name.
Group : SmallGroup(1440,5843)
Rank : 3
Schlafli Type : {8,4}
Number of vertices, edges, etc : 180, 360, 90
Order of s₀s₁s₂ : 5
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,4}*720
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1> of order 2.
      46 facets:
         44 of {8}*16
         2 of {4}*8
      92 vertex figures:
         88 of {4}*8
         4 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 3.
      30 facets:
         30 of {8}*16
      60 vertex figures:
         60 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1> of order 4.
      24 facets:
         21 of {8}*16
         3 of {4}*8
      48 vertex figures:
         42 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0> of order 6.
      16 facets:
         14 of {8}*16
         2 of {4}*8
      32 vertex figures:
         28 of {4}*8
         4 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 9.
      10 facets:
         10 of {8}*16
      20 vertex figures:
         20 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
      8 facets:
         7 of {8}*16
         1 of {4}*8
      16 vertex figures:
         14 of {4}*8
         2 of {2}*4
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1> of order 18.
      6 facets:
         4 of {8}*16
         2 of {4}*8
      12 vertex figures:
         8 of {4}*8
         4 of {2}*4

Permutation Representation (GAP) :

s0 := ( 2, 8)( 3, 6)( 4, 9)( 7,10)(11,12);;
s1 := ( 1, 2)( 3, 6)( 4, 7)( 5, 9)( 8,10);;
s2 := ( 1, 5)( 2, 3)( 4, 7)( 6, 8)( 9,10)(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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {8,4}

Twisty Puzzle