Polytope of Type {4,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*256
Also Known As : {4,4}_(4,4), {4,4}₈. if this polytope has another name.
Group : SmallGroup(256,6661)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 32, 64, 32
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   {4,4,2} of size 512
Vertex Figure Of :
   {2,4,4} of size 512
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*128
   4-fold quotients : {4,4}*64
   8-fold quotients : {4,4}*32
   16-fold quotients : {2,4}*16, {4,2}*16
   32-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,8}*512b, {8,4}*512b, {4,4}*512, {4,8}*512d, {8,4}*512d
   3-fold covers : {4,12}*768a, {12,4}*768a
   5-fold covers : {4,20}*1280a, {20,4}*1280a
   7-fold covers : {4,28}*1792, {28,4}*1792
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
      16 facets:
         16 of {4}*8
      16 vertex figures:
         16 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 2.
      18 facets:
         14 of {4}*8
         4 of {2}*4
      16 vertex figures:
         16 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 2.
      16 facets:
         16 of {4}*8
      16 vertex figures:
         16 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      16 facets:
         16 of {4}*8
      18 vertex figures:
         14 of {4}*8
         4 of {2}*4
   P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      8 facets:
         8 of {4}*8
      10 vertex figures:
         6 of {4}*8
         4 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 4.
      10 facets:
         4 of {2}*4
         6 of {4}*8
      8 vertex figures:
         8 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*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      8 facets:
         8 of {4}*8
      8 vertex figures:
         8 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      9 facets:
         7 of {4}*8
         2 of {2}*4
      8 vertex figures:
         8 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      8 facets:
         8 of {4}*8
      9 vertex figures:
         7 of {4}*8
         2 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      8 facets:
         8 of {4}*8
      8 vertex figures:
         8 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 8.
      5 facets:
         2 of {2}*4
         3 of {4}*8
      5 vertex figures:
         3 of {4}*8
         2 of {2}*4

Permutation Representation (GAP) :

s0 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);;
s1 := ( 5, 7)( 6, 8)(11,12)(13,14)(15,16);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15);;
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, 
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(16)!( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);
s1 := Sym(16)!( 5, 7)( 6, 8)(11,12)(13,14)(15,16);
s2 := Sym(16)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15);
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, 
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 {4,4}

Twisty Puzzle