Polytope of Type {10,3}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,3}*720b
if this polytope has a name.
Group : SmallGroup(720,771)
Rank : 3
Schlafli Type : {10,3}
Number of vertices, edges, etc : 120, 180, 36
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {10,3,2} of size 1440
Vertex Figure Of :
   {2,10,3} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,3}*360
   3-fold quotients : {10,3}*240
   6-fold quotients : {5,3}*120, {10,3}*120a, {10,3}*120b
   12-fold quotients : {5,3}*60
   60-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,3}*1440a, {10,6}*1440f
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      24 facets:
         12 of {5}*10
         12 of {10}*20
      60 vertex figures:
         60 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 2.
      18 facets:
         18 of {10}*20
      60 vertex figures:
         60 of {3}*6
   P/N, where N=<s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {10}*20
      60 vertex figures:
         60 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 3.
      12 facets:
         12 of {10}*20
      40 vertex figures:
         40 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1> of order 4.
      9 facets:
         9 of {10}*20
      30 vertex figures:
         30 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 5.
      12 facets:
         6 of {10}*20
         6 of {2}*4
      24 vertex figures:
         24 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      6 facets:
         6 of {10}*20
      20 vertex figures:
         20 of {3}*6
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2> of order 10.
      6 facets:
         3 of {2}*4
         3 of {10}*20
      12 vertex figures:
         12 of {3}*6
   P/N, where N=<s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 12.
      3 facets:
         3 of {10}*20
      10 vertex figures:
         10 of {3}*6

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {10,3}

Twisty Puzzle