Polytope of Type {4,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*1296
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 54, 324, 162
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 6
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) :
   3-fold quotients : {4,12}*432a, {4,12}*432b
   6-fold quotients : {4,12}*216
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72
   54-fold quotients : {2,6}*24
   108-fold quotients : {2,3}*12
   162-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=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      81 facets:
         81 of {4}*8
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      81 facets:
         81 of {4}*8
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s1> of order 2.
      90 facets:
         18 of {2}*4
         72 of {4}*8
      30 vertex figures:
         24 of {12}*24
         6 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      54 facets:
         54 of {4}*8
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 3.
      54 facets:
         54 of {4}*8
      36 vertex figures:
         9 of {12}*24
         27 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0> of order 3.
      54 facets:
         54 of {4}*8
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      45 facets:
         9 of {2}*4
         36 of {4}*8
      15 vertex figures:
         12 of {12}*24
         3 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      45 facets:
         9 of {2}*4
         36 of {4}*8
      18 vertex figures:
         12 of {12}*24
         6 of {3}*6
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 6.
      36 facets:
         18 of {2}*4
         18 of {4}*8
      12 vertex figures:
         6 of {6}*12
         6 of {12}*24
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 6.
      36 facets:
         18 of {2}*4
         18 of {4}*8
      12 vertex figures:
         6 of {12}*24
         6 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 6.
      30 facets:
         6 of {2}*4
         24 of {4}*8
      20 vertex figures:
         4 of {12}*24
         12 of {4}*8
         1 of {6}*12
         3 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 9.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         3 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 18.
      12 facets:
         6 of {2}*4
         6 of {4}*8
      8 vertex figures:
         1 of {12}*24
         3 of {4}*8
         1 of {6}*12
         3 of {2}*4

Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)(27,62)(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,74)(76,77)(79,80);;
s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);;
s2 := ( 1,42)( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,45)( 8,44)( 9,43)(10,15)(11,14)(12,13)(16,18)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,72)(26,71)(27,70)(28,32)(29,31)(30,33)(34,35)(46,58)(47,60)(48,59)(49,55)(50,57)(51,56)(52,61)(53,63)(54,62)(73,77)(74,76)(75,78)(79,80);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,55)(20,57)(21,56)(22,58)(23,60)(24,59)(25,61)(26,63)(27,62)(37,38)(40,41)(43,44)(46,66)(47,65)(48,64)(49,69)(50,68)(51,67)(52,72)(53,71)(54,70)(73,74)(76,77)(79,80);
s1 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);
s2 := Sym(81)!( 1,42)( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,45)( 8,44)( 9,43)(10,15)(11,14)(12,13)(16,18)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,72)(26,71)(27,70)(28,32)(29,31)(30,33)(34,35)(46,58)(47,60)(48,59)(49,55)(50,57)(51,56)(52,61)(53,63)(54,62)(73,77)(74,76)(75,78)(79,80);
poly := sub<Sym(81)|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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0 >;

References : None.
to this polytope

Polytope of Type {4,12}

Twisty Puzzle