Polytope of Type {4,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*672a
if this polytope has a name.
Group : SmallGroup(672,1254)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 56, 168, 84
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   {4,6,2} of size 1344
Vertex Figure Of :
   {2,4,6} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6}*336
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12}*1344a, {4,12}*1344b, {4,6}*1344
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 2.
      42 facets:
         42 of {4}*8
      29 vertex figures:
         27 of {6}*12
         2 of {3}*6
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      28 facets:
         28 of {4}*8
      20 vertex figures:
         18 of {6}*12
         2 of {2}*4
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
      14 facets:
         14 of {4}*8
      11 vertex figures:
         8 of {6}*12
         2 of {3}*6
         1 of {2}*4

Permutation Representation (GAP) :

s0 := (1,2)(3,6)(4,8)(5,7);;
s1 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,10);;
s2 := (1,2)(3,5)(6,7);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0, 
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(10)!(1,2)(3,6)(4,8)(5,7);
s1 := Sym(10)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,10);
s2 := Sym(10)!(1,2)(3,5)(6,7);
poly := sub<Sym(10)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0, 
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 {4,6}

Twisty Puzzle