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}*1296d
if this polytope has a name.
Group : SmallGroup(1296,2909)
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
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 : {12,4}*432b
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72
   54-fold quotients : {6,2}*24
   108-fold quotients : {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*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      27 facets:
         27 of {12}*24
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      27 facets:
         27 of {12}*24
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      30 facets:
         24 of {12}*24
         6 of {6}*12
      30 vertex figures:
         24 of {12}*24
         6 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      15 facets:
         12 of {12}*24
         3 of {6}*12
      15 vertex figures:
         12 of {12}*24
         3 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*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 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,55)(20,57)(21,56)(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(37,38)(40,44)(41,43)(42,45)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(73,74)(76,80)(77,79)(78,81);;
s1 := ( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,41)(11,40)(12,42)(13,38)(14,37)(15,39)(16,44)(17,43)(18,45)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,54)(26,53)(27,52)(55,58)(56,60)(57,59)(62,63)(64,68)(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(10,55)(11,57)(12,56)(13,58)(14,60)(15,59)(16,61)(17,63)(18,62)(19,28)(20,30)(21,29)(22,31)(23,33)(24,32)(25,34)(26,36)(27,35)(37,74)(38,73)(39,75)(40,77)(41,76)(42,78)(43,80)(44,79)(45,81)(46,48)(49,51)(52,54)(64,66)(67,69)(70,72);;
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*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*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(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,55)(20,57)(21,56)(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(37,38)(40,44)(41,43)(42,45)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(73,74)(76,80)(77,79)(78,81);
s1 := Sym(81)!( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,41)(11,40)(12,42)(13,38)(14,37)(15,39)(16,44)(17,43)(18,45)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,54)(26,53)(27,52)(55,58)(56,60)(57,59)(62,63)(64,68)(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);
s2 := Sym(81)!( 2, 3)( 5, 6)( 8, 9)(10,55)(11,57)(12,56)(13,58)(14,60)(15,59)(16,61)(17,63)(18,62)(19,28)(20,30)(21,29)(22,31)(23,33)(24,32)(25,34)(26,36)(27,35)(37,74)(38,73)(39,75)(40,77)(41,76)(42,78)(43,80)(44,79)(45,81)(46,48)(49,51)(52,54)(64,66)(67,69)(70,72);
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*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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