Polytope of Type {6,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1296m
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 54, 324, 108
Order of s₀s₁s₂ : 12
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 : {6,4}*432a, {6,12}*432h
   6-fold quotients : {6,4}*216
   9-fold quotients : {6,4}*144
   18-fold quotients : {6,4}*72
   81-fold quotients : {2,4}*16
   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*s2> of order 2.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 2.
      57 facets:
         51 of {6}*12
         6 of {3}*6
      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.
      54 facets:
         54 of {6}*12
      30 vertex figures:
         24 of {12}*24
         6 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 3.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 3.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
      30 facets:
         6 of {3}*6
         24 of {6}*12
      15 vertex figures:
         12 of {12}*24
         3 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      27 facets:
         27 of {6}*12
      15 vertex figures:
         12 of {12}*24
         3 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1> of order 6.
      18 facets:
         18 of {6}*12
      12 vertex figures:
         6 of {6}*12
         6 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 9.
      12 facets:
         12 of {6}*12
      6 vertex figures:
         6 of {12}*24

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,12}

Twisty Puzzle