Polytope of Type {6,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*432
if this polytope has a name.
Group : SmallGroup(432,521)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 24, 108, 36
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,9,2} of size 864
   {6,9,4} of size 1728
Vertex Figure Of :
   {2,6,9} of size 864
   {4,6,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3}*144
   4-fold quotients : {6,9}*108
   9-fold quotients : {6,3}*48
   12-fold quotients : {2,9}*36, {6,3}*36
   18-fold quotients : {3,3}*24
   36-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,9}*864, {6,18}*864
   3-fold covers : {6,27}*1296, {18,9}*1296a, {6,9}*1296b
   4-fold covers : {6,9}*1728, {6,36}*1728a, {12,18}*1728a, {6,18}*1728a, {6,36}*1728c, {12,18}*1728b, {12,9}*1728
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {6}*12
      12 vertex figures:
         12 of {9}*18
   P/N, where N=<s0*s1*s0*s1> of order 3.
      18 facets:
         9 of {2}*4
         9 of {6}*12
      8 vertex figures:
         8 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 4.
      9 facets:
         9 of {6}*12
      6 vertex figures:
         6 of {9}*18

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,9}

Twisty Puzzle