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}*1296e
if this polytope has a name.
Group : SmallGroup(1296,1788)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 54, 324, 108
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 12
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) :
   3-fold quotients : {6,12}*432d
   4-fold quotients : {6,6}*324a
   9-fold quotients : {6,12}*144d
   12-fold quotients : {6,6}*108
   27-fold quotients : {6,4}*48b
   54-fold quotients : {3,4}*24
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*s1*s0*s1> of order 2.
      60 facets:
         12 of {3}*6
         48 of {6}*12
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      54 facets:
         54 of {6}*12
      36 vertex figures:
         18 of {12}*24
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
      36 facets:
         36 of {6}*12
      22 vertex figures:
         16 of {12}*24
         6 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1> 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*s2*s1*s0*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*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 6.
      24 facets:
         12 of {3}*6
         12 of {6}*12
      11 vertex figures:
         8 of {12}*24
         3 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 6.
      24 facets:
         12 of {3}*6
         12 of {6}*12
      9 vertex figures:
         9 of {12}*24

Permutation Representation (GAP) :

s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,21)(18,22)(19,24)(20,23)(27,28)(29,33)(30,34)(31,36)(32,35);;
s1 := ( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(21,33)(22,36)(23,35)(24,34);;
s2 := ( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,24)(12,23)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36);;
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*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,21)(18,22)(19,24)(20,23)(27,28)(29,33)(30,34)(31,36)(32,35);
s1 := Sym(36)!( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(21,33)(22,36)(23,35)(24,34);
s2 := Sym(36)!( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,24)(12,23)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36);
poly := sub<Sym(36)|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*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope

Polytope of Type {6,12}

Twisty Puzzle