Polytope of Type {3,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*576
Also Known As : {3,6}_(4,4). if this polytope has another name.
Group : SmallGroup(576,5053)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 48, 144, 96
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,6,2} of size 1152
Vertex Figure Of :
   {2,3,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*192
   4-fold quotients : {3,6}*144
   12-fold quotients : {3,6}*48
   16-fold quotients : {3,6}*36
   24-fold quotients : {3,3}*24
   48-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,12}*1152a, {6,6}*1152a
   3-fold covers : {9,6}*1728, {3,6}*1728
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      48 facets:
         48 of {3}*6
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      48 facets:
         48 of {3}*6
      26 vertex figures:
         22 of {6}*12
         4 of {3}*6
   P/N, where N=<s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      32 facets:
         32 of {3}*6
      18 vertex figures:
         15 of {6}*12
         3 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {3}*6
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      24 facets:
         24 of {3}*6
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 4.
      24 facets:
         24 of {3}*6
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {3}*6
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1, s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      24 facets:
         24 of {3}*6
      14 vertex figures:
         10 of {6}*12
         4 of {3}*6
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1, s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {3}*6
      7 vertex figures:
         5 of {6}*12
         2 of {3}*6
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 8.
      12 facets:
         12 of {3}*6
      6 vertex figures:
         6 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 8.
      12 facets:
         12 of {3}*6
      6 vertex figures:
         6 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 8.
      12 facets:
         12 of {3}*6
      6 vertex figures:
         6 of {6}*12

Permutation Representation (GAP) :

s0 := ( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,33)(18,34)(19,36)(20,35)(21,44)(22,43)(23,41)(24,42)(25,39)(26,40)(27,38)(28,37)(29,46)(30,45)(31,47)(32,48);;
s1 := ( 1,17)( 2,19)( 3,18)( 4,20)( 5,24)( 6,22)( 7,23)( 8,21)( 9,32)(10,30)(11,31)(12,29)(13,28)(14,26)(15,27)(16,25)(34,35)(37,40)(41,48)(42,46)(43,47)(44,45);;
s2 := ( 1,15)( 2,16)( 3,14)( 4,13)( 5, 6)(11,12)(17,31)(18,32)(19,30)(20,29)(21,22)(27,28)(33,47)(34,48)(35,46)(36,45)(37,38)(43,44);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(48)!( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,33)(18,34)(19,36)(20,35)(21,44)(22,43)(23,41)(24,42)(25,39)(26,40)(27,38)(28,37)(29,46)(30,45)(31,47)(32,48);
s1 := Sym(48)!( 1,17)( 2,19)( 3,18)( 4,20)( 5,24)( 6,22)( 7,23)( 8,21)( 9,32)(10,30)(11,31)(12,29)(13,28)(14,26)(15,27)(16,25)(34,35)(37,40)(41,48)(42,46)(43,47)(44,45);
s2 := Sym(48)!( 1,15)( 2,16)( 3,14)( 4,13)( 5, 6)(11,12)(17,31)(18,32)(19,30)(20,29)(21,22)(27,28)(33,47)(34,48)(35,46)(36,45)(37,38)(43,44);
poly := sub<Sym(48)|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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

References : None.
to this polytope

Polytope of Type {3,6}

Twisty Puzzle