Polytope of Type {6,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,46101)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 144, 432, 144
Order of s₀s₁s₂ : 6
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 : {6,6}*576e
   16-fold quotients : {3,6}*108
   48-fold quotients : {3,6}*36
   144-fold quotients : {3,2}*12
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.
      84 facets:
         24 of {3}*6
         60 of {6}*12
      72 vertex figures:
         72 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0> of order 2.
      72 facets:
         72 of {6}*12
      72 vertex figures:
         72 of {6}*12
   P/N, where N=<s0*s1*s0*s1> of order 3.
      52 facets:
         6 of {2}*4
         46 of {6}*12
      48 vertex figures:
         48 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
      54 facets:
         36 of {3}*6
         18 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0> of order 4.
      42 facets:
         12 of {3}*6
         30 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
      48 facets:
         24 of {6}*12
         24 of {3}*6
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1> of order 4.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 6.
      30 facets:
         8 of {3}*6
         19 of {6}*12
         3 of {2}*4
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
      30 facets:
         24 of {3}*6
         6 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1> of order 8.
      24 facets:
         12 of {3}*6
         12 of {6}*12
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0, s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 8.
      24 facets:
         12 of {6}*12
         12 of {3}*6
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0> of order 12.
      22 facets:
         12 of {3}*6
         4 of {6}*12
         6 of {2}*4
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0> of order 24.
      12 facets:
         8 of {3}*6
         1 of {6}*12
         3 of {2}*4
      6 vertex figures:
         6 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,6}

Twisty Puzzle