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}*1944c
if this polytope has a name.
Group : SmallGroup(1944,2324)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 81, 486, 162
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   9-fold quotients : {6,12}*216c
   27-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 3.
      72 facets:
         45 of {6}*12
         27 of {2}*4
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      33 vertex figures:
         24 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 9.
      36 facets:
         9 of {6}*12
         27 of {2}*4
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 9.
      24 facets:
         15 of {6}*12
         9 of {2}*4
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 9.
      18 facets:
         18 of {6}*12
      15 vertex figures:
         6 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 9.
      30 facets:
         18 of {2}*4
         12 of {6}*12
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 9.
      18 facets:
         18 of {6}*12
      15 vertex figures:
         9 of {4}*8
         6 of {12}*24

Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);;
s1 := ( 2, 3)( 4, 5)( 7, 9)(10,26)(11,25)(12,27)(13,21)(14,20)(15,19)(16,22)(17,24)(18,23);;
s2 := ( 1,10)( 2,17)( 3,15)( 4,16)( 5,14)( 6,12)( 7,13)( 8,11)( 9,18)(20,26)(21,24)(22,25);;
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, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);
s1 := Sym(27)!( 2, 3)( 4, 5)( 7, 9)(10,26)(11,25)(12,27)(13,21)(14,20)(15,19)(16,22)(17,24)(18,23);
s2 := Sym(27)!( 1,10)( 2,17)( 3,15)( 4,16)( 5,14)( 6,12)( 7,13)( 8,11)( 9,18)(20,26)(21,24)(22,25);
poly := sub<Sym(27)|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, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope

Polytope of Type {6,12}

Twisty Puzzle