Polytope of Type {12,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*1296
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 162, 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
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,4}*432a, {12,4}*432b
   6-fold quotients : {12,4}*216
   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
      81 vertex figures:
         81 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 2.
      27 facets:
         27 of {12}*24
      81 vertex figures:
         81 of {4}*8
   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
      90 vertex figures:
         72 of {4}*8
         18 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      36 facets:
         27 of {4}*8
         9 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      54 vertex figures:
         54 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      15 facets:
         3 of {6}*12
         12 of {12}*24
      45 vertex figures:
         36 of {4}*8
         9 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 4.
      18 facets:
         6 of {3}*6
         12 of {12}*24
      45 vertex figures:
         36 of {4}*8
         9 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      36 vertex figures:
         18 of {4}*8
         18 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      36 vertex figures:
         18 of {4}*8
         18 of {2}*4
   P/N, where N=<s0*s1*s0*s1> of order 6.
      20 facets:
         3 of {2}*4
         1 of {6}*12
         12 of {4}*8
         4 of {12}*24
      30 vertex figures:
         24 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s1*s2*s1*s0*s1*s2> of order 9.
      12 facets:
         9 of {4}*8
         3 of {12}*24
      18 vertex figures:
         18 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 18.
      8 facets:
         3 of {2}*4
         1 of {6}*12
         3 of {4}*8
         1 of {12}*24
      12 vertex figures:
         6 of {4}*8
         6 of {2}*4

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

References : None.
to this polytope

Polytope of Type {12,4}

Twisty Puzzle