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}*288
Also Known As : {4,4}_(6,0), {4,4|6}. if this polytope has another name.
Group : SmallGroup(288,889)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 36, 72, 36
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
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 576
   {4,4,4} of size 1152
   {4,4,6} of size 1728
Vertex Figure Of :
   {2,4,4} of size 576
   {4,4,4} of size 1152
   {6,4,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4}*144
   4-fold quotients : {4,4}*72
   9-fold quotients : {4,4}*32
   18-fold quotients : {2,4}*16, {4,2}*16
   36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4}*576, {4,8}*576a, {8,4}*576a, {4,8}*576b, {8,4}*576b
   3-fold covers : {4,12}*864a, {12,4}*864a, {4,12}*864d, {12,4}*864c
   4-fold covers : {4,8}*1152a, {8,4}*1152a, {8,8}*1152a, {8,8}*1152b, {8,8}*1152c, {8,8}*1152d, {4,16}*1152a, {16,4}*1152a, {4,16}*1152b, {16,4}*1152b, {4,8}*1152b, {8,4}*1152b, {4,4}*1152
   5-fold covers : {4,20}*1440, {20,4}*1440
   6-fold covers : {4,12}*1728b, {12,4}*1728a, {4,24}*1728b, {8,12}*1728b, {12,8}*1728b, {24,4}*1728b, {4,24}*1728d, {8,12}*1728c, {12,8}*1728c, {24,4}*1728d, {4,24}*1728f, {24,4}*1728e, {8,12}*1728e, {12,8}*1728e, {4,24}*1728h, {24,4}*1728h, {8,12}*1728f, {12,8}*1728f, {4,12}*1728d, {12,4}*1728d
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      18 facets:
         18 of {4}*8
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s1*s0*s1> of order 2.
      20 facets:
         4 of {2}*4
         16 of {4}*8
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {4}*8
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {4}*8
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      18 facets:
         18 of {4}*8
      20 vertex figures:
         16 of {4}*8
         4 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      12 facets:
         12 of {4}*8
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      12 facets:
         12 of {4}*8
      12 vertex figures:
         12 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      9 facets:
         9 of {4}*8
      9 vertex figures:
         9 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      10 facets:
         2 of {2}*4
         8 of {4}*8
      9 vertex figures:
         9 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      9 facets:
         9 of {4}*8
      10 vertex figures:
         8 of {4}*8
         2 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
      6 facets:
         6 of {4}*8
      6 vertex figures:
         6 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      8 facets:
         4 of {2}*4
         4 of {4}*8
      6 vertex figures:
         6 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 6.
      6 facets:
         6 of {4}*8
      6 vertex figures:
         6 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
      8 facets:
         4 of {2}*4
         4 of {4}*8
      6 vertex figures:
         6 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 6.
      6 facets:
         6 of {4}*8
      6 vertex figures:
         6 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      6 facets:
         6 of {4}*8
      6 vertex figures:
         6 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
      6 facets:
         6 of {4}*8
      8 vertex figures:
         4 of {4}*8
         4 of {2}*4
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 6.
      6 facets:
         6 of {4}*8
      8 vertex figures:
         4 of {2}*4
         4 of {4}*8

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {4,4}

Twisty Puzzle