Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296j
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
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 : {12,6}*432f, {4,6}*432b
   6-fold quotients : {12,6}*216a
   9-fold quotients : {4,6}*144
   18-fold quotients : {4,6}*72
   27-fold quotients : {4,6}*48a
   54-fold quotients : {2,6}*24
   81-fold quotients : {4,2}*16
   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*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 2.
      27 facets:
         27 of {12}*24
      54 vertex figures:
         54 of {6}*12
   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
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      18 vertex figures:
         18 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
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s2*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 9.
      6 facets:
         6 of {12}*24
      12 vertex figures:
         12 of {6}*12

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

References : None.
to this polytope

Polytope of Type {12,6}

Twisty Puzzle