Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*720a
if this polytope has a name.
Group : SmallGroup(720,767)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 60, 180, 30
Order of s₀s₁s₂ : 15
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 1440
Vertex Figure Of :
   {2,12,6} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,6}*240c
   6-fold quotients : {4,6}*120
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,6}*1440a, {24,6}*1440b, {12,6}*1440c
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      15 facets:
         15 of {12}*24
      30 vertex figures:
         30 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      16 facets:
         2 of {6}*12
         14 of {12}*24
      30 vertex figures:
         30 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      9 facets:
         3 of {6}*12
         6 of {12}*24
      15 vertex figures:
         15 of {6}*12

Permutation Representation (GAP) :

s0 := (2,3)(7,8);;
s1 := (1,2)(5,7)(6,8);;
s2 := (4,5);;
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*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(8)!(2,3)(7,8);
s1 := Sym(8)!(1,2)(5,7)(6,8);
s2 := Sym(8)!(4,5);
poly := sub<Sym(8)|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*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1 >;

References : None.
to this polytope

Polytope of Type {12,6}

Twisty Puzzle