Polytope of Type {3,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*972
Also Known As : {3,6}_(9,0), {3,6}₁₈. if this polytope has another name.
Group : SmallGroup(972,115)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 81, 243, 162
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,6,2} of size 1944
Vertex Figure Of :
   {2,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*324
   9-fold quotients : {3,6}*108
   27-fold quotients : {3,6}*36
   81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6}*1944b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
      54 facets:
         54 of {3}*6
      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*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {3}*6
      29 vertex figures:
         26 of {6}*12
         3 of {2}*4
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 9.
      18 facets:
         18 of {3}*6
      9 vertex figures:
         9 of {6}*12

Permutation Representation (GAP) :

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

Twisty Puzzle