Polytope of Type {9,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1788)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 108, 324, 72
Order of s₀s₁s₂ : 12
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 : {3,6}*432
   4-fold quotients : {9,6}*324b
   9-fold quotients : {3,6}*144
   12-fold quotients : {3,6}*108
   27-fold quotients : {3,6}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {3,3}*24
   108-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=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      36 facets:
         36 of {9}*18
      60 vertex figures:
         48 of {6}*12
         12 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 2.
      36 facets:
         36 of {9}*18
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0> of order 3.
      24 facets:
         24 of {9}*18
      42 vertex figures:
         33 of {6}*12
         9 of {2}*4
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      18 facets:
         18 of {9}*18
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      18 facets:
         18 of {9}*18
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {9}*18
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {9}*18
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2> of order 6.
      12 facets:
         12 of {9}*18
      24 vertex figures:
         12 of {3}*6
         12 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s2*s1, s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {9}*18
      24 vertex figures:
         12 of {6}*12
         12 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 12.
      6 facets:
         6 of {9}*18
      9 vertex figures:
         9 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {9,6}

Twisty Puzzle