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}*480
if this polytope has a name.
Group : SmallGroup(480,1186)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 40, 120, 60
Order of s₀s₁s₂ : 10
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   {4,6,2} of size 960
   {4,6,4} of size 1920
Vertex Figure Of :
   {2,4,6} of size 960
   {4,4,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6}*240a, {4,6}*240b, {4,6}*240c
   4-fold quotients : {4,6}*120
   60-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,6}*960a, {4,6}*960, {4,12}*960a, {8,6}*960b, {4,12}*960b
   3-fold covers : {4,6}*1440b, {12,6}*1440c, {12,6}*1440d
   4-fold covers : {4,12}*1920a, {4,24}*1920a, {8,6}*1920a, {4,24}*1920b, {8,12}*1920a, {8,12}*1920b, {4,24}*1920c, {4,24}*1920d, {8,12}*1920c, {8,6}*1920b, {8,12}*1920d, {4,12}*1920b, {4,6}*1920
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s2*s1*s2> of order 2.
      30 facets:
         30 of {4}*8
      22 vertex figures:
         4 of {3}*6
         18 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 2.
      30 facets:
         30 of {4}*8
      20 vertex figures:
         20 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      32 facets:
         28 of {4}*8
         4 of {2}*4
      20 vertex figures:
         20 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      30 facets:
         30 of {4}*8
      20 vertex figures:
         20 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      20 facets:
         20 of {4}*8
      16 vertex figures:
         12 of {6}*12
         4 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      18 facets:
         6 of {2}*4
         12 of {4}*8
      10 vertex figures:
         10 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 4.
      16 facets:
         14 of {4}*8
         2 of {2}*4
      10 vertex figures:
         10 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      16 facets:
         14 of {4}*8
         2 of {2}*4
      10 vertex figures:
         10 of {6}*12
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 4.
      16 facets:
         14 of {4}*8
         2 of {2}*4
      12 vertex figures:
         4 of {3}*6
         8 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
      10 facets:
         10 of {4}*8
      8 vertex figures:
         6 of {6}*12
         2 of {2}*4
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2> of order 6.
      10 facets:
         10 of {4}*8
      10 vertex figures:
         4 of {3}*6
         4 of {6}*12
         2 of {2}*4
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      10 facets:
         10 of {4}*8
      8 vertex figures:
         6 of {6}*12
         2 of {2}*4
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 6.
      12 facets:
         8 of {4}*8
         4 of {2}*4
      8 vertex figures:
         6 of {6}*12
         2 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s2> of order 8.
      9 facets:
         3 of {2}*4
         6 of {4}*8
      6 vertex figures:
         2 of {3}*6
         4 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
      6 facets:
         2 of {2}*4
         4 of {4}*8
      6 vertex figures:
         2 of {6}*12
         4 of {2}*4
   P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2> of order 24.
      3 facets:
         1 of {2}*4
         2 of {4}*8
      4 vertex figures:
         2 of {3}*6
         2 of {2}*4

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {4,6}

Twisty Puzzle