Polytope of Type {12,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1296f
if this polytope has a name.
Group : SmallGroup(1296,3528)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 54, 324, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,12}*432b, {12,4}*432b
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72, {6,6}*72a
   54-fold quotients : {2,6}*24, {6,2}*24
   108-fold quotients : {2,3}*12, {3,2}*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*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2> of order 2.
      27 facets:
         27 of {12}*24
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      27 facets:
         27 of {12}*24
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      30 facets:
         6 of {6}*12
         24 of {12}*24
      30 vertex figures:
         24 of {12}*24
         6 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      15 facets:
         3 of {6}*12
         12 of {12}*24
      15 vertex figures:
         12 of {12}*24
         3 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      12 vertex figures:
         6 of {12}*24
         6 of {6}*12

Permutation Representation (GAP) : 
s0 := ( 4, 7)( 5, 8)( 6, 9)(10,28)(11,29)(12,30)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,55)(20,56)(21,57)(22,61)(23,62)(24,63)(25,58)(26,59)(27,60)(40,43)(41,44)(42,45)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69)(76,79)(77,80)(78,81);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)(36,62)(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)(47,78)(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);;
s2 := ( 1,38)( 2,37)( 3,39)( 4,41)( 5,40)( 6,42)( 7,44)( 8,43)( 9,45)(10,11)(13,14)(16,17)(19,65)(20,64)(21,66)(22,68)(23,67)(24,69)(25,71)(26,70)(27,72)(28,29)(31,32)(34,35)(46,56)(47,55)(48,57)(49,59)(50,58)(51,60)(52,62)(53,61)(54,63)(73,74)(76,77)(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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(10,28)(11,29)(12,30)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,55)(20,56)(21,57)(22,61)(23,62)(24,63)(25,58)(26,59)(27,60)(40,43)(41,44)(42,45)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69)(76,79)(77,80)(78,81);
s1 := Sym(81)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)(36,62)(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)(47,78)(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);
s2 := Sym(81)!( 1,38)( 2,37)( 3,39)( 4,41)( 5,40)( 6,42)( 7,44)( 8,43)( 9,45)(10,11)(13,14)(16,17)(19,65)(20,64)(21,66)(22,68)(23,67)(24,69)(25,71)(26,70)(27,72)(28,29)(31,32)(34,35)(46,56)(47,55)(48,57)(49,59)(50,58)(51,60)(52,62)(53,61)(54,63)(73,74)(76,77)(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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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