Polytope of Type {3,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,9}*648
if this polytope has a name.
Group : SmallGroup(648,703)
Rank : 3
Schlafli Type : {3,9}
Number of vertices, edges, etc : 36, 162, 108
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 9
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,9,2} of size 1296
Vertex Figure Of :
   {2,3,9} of size 1296
Quotients (Maximal Quotients in Boldface) :
   27-fold quotients : {3,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,18}*1296b, {6,9}*1296f
   3-fold covers : {9,9}*1944a
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
      36 facets:
         36 of {3}*6
      12 vertex figures:
         12 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      36 facets:
         36 of {3}*6
      12 vertex figures:
         12 of {9}*18
   P/N, where N=<s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      36 facets:
         36 of {3}*6
      18 vertex figures:
         9 of {9}*18
         9 of {3}*6
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 9.
      12 facets:
         12 of {3}*6
      6 vertex figures:
         3 of {9}*18
         3 of {3}*6
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 9.
      12 facets:
         12 of {3}*6
      8 vertex figures:
         2 of {9}*18
         6 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
      12 facets:
         12 of {3}*6
      4 vertex figures:
         4 of {9}*18

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {3,9}

Twisty Puzzle