Polytope of Type {9,3}

Play with this polytope as a twisty puzzle

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

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

References : None.
to this polytope

Polytope of Type {9,3}

Twisty Puzzle