Polytope of Type {5,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,6}*960
if this polytope has a name.
Group : SmallGroup(960,11358)
Rank : 3
Schlafli Type : {5,6}
Number of vertices, edges, etc : 80, 240, 96
Order of s₀s₁s₂ : 5
Order of s₀s₁s₂s₁ : 5
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {5,6,2} of size 1920
Vertex Figure Of :
   {2,5,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   16-fold quotients : {5,3}*60
Covers (Minimal Covers in Boldface) :
   2-fold covers : {5,6}*1920b, {5,6}*1920c, {5,6}*1920d, {10,6}*1920b, {10,6}*1920c, {10,6}*1920d, {10,6}*1920e, {5,12}*1920c, {5,12}*1920d, {5,12}*1920e, {5,12}*1920f
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      48 facets:
         48 of {5}*10
      40 vertex figures:
         40 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 2.
      48 facets:
         48 of {5}*10
      44 vertex figures:
         36 of {6}*12
         8 of {3}*6
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      32 facets:
         32 of {5}*10
      28 vertex figures:
         26 of {6}*12
         2 of {2}*4
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 4.
      24 facets:
         24 of {5}*10
      26 vertex figures:
         12 of {3}*6
         14 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2> of order 4.
      24 facets:
         24 of {5}*10
      22 vertex figures:
         18 of {6}*12
         4 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1> of order 4.
      24 facets:
         24 of {5}*10
      24 vertex figures:
         16 of {6}*12
         8 of {3}*6
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1> of order 8.
      12 facets:
         12 of {5}*10
      16 vertex figures:
         12 of {3}*6
         4 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2> of order 8.
      12 facets:
         12 of {5}*10
      14 vertex figures:
         8 of {3}*6
         6 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 12.
      8 facets:
         8 of {5}*10
      10 vertex figures:
         4 of {3}*6
         4 of {6}*12
         2 of {2}*4

Permutation Representation (GAP) :

s0 := (2,3)(4,5)(6,9)(7,8);;
s1 := ( 1, 2)( 3, 4)( 6, 8)( 9,10);;
s2 := (2,7)(3,8)(4,6)(5,9);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(10)!(2,3)(4,5)(6,9)(7,8);
s1 := Sym(10)!( 1, 2)( 3, 4)( 6, 8)( 9,10);
s2 := Sym(10)!(2,7)(3,8)(4,6)(5,9);
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*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope

Polytope of Type {5,6}

Twisty Puzzle