Polytope of Type {4,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8}*1344f
if this polytope has a name.
Group : SmallGroup(1344,11684)
Rank : 3
Schlafli Type : {4,8}
Number of vertices, edges, etc : 84, 336, 168
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 8
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 : {4,8}*672d, {4,8}*672e, {4,8}*672f
   4-fold quotients : {4,8}*336b
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      88 facets:
         80 of {4}*8
         8 of {2}*4
      44 vertex figures:
         4 of {4}*8
         40 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      84 facets:
         84 of {4}*8
      42 vertex figures:
         42 of {8}*16
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 2.
      84 facets:
         84 of {4}*8
      42 vertex figures:
         42 of {8}*16
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
      84 facets:
         84 of {4}*8
      42 vertex figures:
         42 of {8}*16
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 4.
      48 facets:
         36 of {4}*8
         12 of {2}*4
      24 vertex figures:
         6 of {4}*8
         18 of {8}*16
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 4.
      44 facets:
         40 of {4}*8
         4 of {2}*4
      22 vertex figures:
         2 of {4}*8
         20 of {8}*16
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      44 facets:
         40 of {4}*8
         4 of {2}*4
      22 vertex figures:
         2 of {4}*8
         20 of {8}*16
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
      44 facets:
         40 of {4}*8
         4 of {2}*4
      22 vertex figures:
         20 of {8}*16
         2 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
      44 facets:
         40 of {4}*8
         4 of {2}*4
      22 vertex figures:
         20 of {8}*16
         2 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 7.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2> of order 8.
      26 facets:
         16 of {4}*8
         10 of {2}*4
      14 vertex figures:
         4 of {4}*8
         8 of {8}*16
         2 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
      22 facets:
         20 of {4}*8
         2 of {2}*4
      11 vertex figures:
         10 of {8}*16
         1 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 14.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 14.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {4,8}

Twisty Puzzle