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}*1152d
if this polytope has a name.
Group : SmallGroup(1152,157849)
Rank : 3
Schlafli Type : {4,8}
Number of vertices, edges, etc : 72, 288, 144
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 3
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) :
   16-fold quotients : {4,4}*72
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*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      72 facets:
         72 of {4}*8
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      72 facets:
         72 of {4}*8
      40 vertex figures:
         32 of {8}*16
         8 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      72 facets:
         72 of {4}*8
      36 vertex figures:
         36 of {8}*16
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      48 facets:
         48 of {4}*8
      24 vertex figures:
         24 of {8}*16
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      36 facets:
         36 of {4}*8
      24 vertex figures:
         12 of {8}*16
         12 of {4}*8
   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.
      36 facets:
         36 of {4}*8
      22 vertex figures:
         8 of {4}*8
         14 of {8}*16
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      36 facets:
         36 of {4}*8
      18 vertex figures:
         18 of {8}*16
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      36 facets:
         36 of {4}*8
      20 vertex figures:
         16 of {8}*16
         4 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
      24 facets:
         24 of {4}*8
      12 vertex figures:
         12 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 8.
      18 facets:
         18 of {4}*8
      14 vertex figures:
         4 of {8}*16
         10 of {4}*8
   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, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 8.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         6 of {4}*8
         6 of {8}*16
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 8.
      18 facets:
         18 of {4}*8
      11 vertex figures:
         7 of {8}*16
         4 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 8.
      18 facets:
         18 of {4}*8
      10 vertex figures:
         8 of {8}*16
         2 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 12.
      12 facets:
         12 of {4}*8
      6 vertex figures:
         6 of {8}*16

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

s0 := Sym(16)!( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15);
s1 := Sym(16)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,16)(12,15);
s2 := Sym(16)!( 2,13)( 3, 9)( 4, 5)( 6,16)( 7,12)(10,15);
poly := sub<Sym(16)|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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope

Polytope of Type {4,8}

Twisty Puzzle