Polytope of Type {4,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20}*720
if this polytope has a name.
Group : SmallGroup(720,784)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 18, 180, 90
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,20,2} of size 1440
Vertex Figure Of :
   {2,4,20} of size 1440
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {4,4}*144
   10-fold quotients : {4,4}*72
   18-fold quotients : {2,10}*40
   36-fold quotients : {2,5}*20
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,20}*1440
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 2.
      45 facets:
         45 of {4}*8
      9 vertex figures:
         9 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      45 facets:
         45 of {4}*8
      9 vertex figures:
         9 of {20}*40
   P/N, where N=<s0*s1*s0*s1> of order 2.
      50 facets:
         10 of {2}*4
         40 of {4}*8
      10 vertex figures:
         8 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      30 facets:
         30 of {4}*8
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      25 facets:
         5 of {2}*4
         20 of {4}*8
      5 vertex figures:
         4 of {20}*40
         1 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      20 facets:
         10 of {2}*4
         10 of {4}*8
      4 vertex figures:
         2 of {20}*40
         2 of {10}*20

Permutation Representation (GAP) :

s0 := ( 6,36)( 7,37)( 8,38)( 9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35);;
s1 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(16,41)(17,45)(18,44)(19,43)(20,42)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37);;
s2 := ( 1,22)( 2,21)( 3,25)( 4,24)( 5,23)( 6,32)( 7,31)( 8,35)( 9,34)(10,33)(11,12)(13,15)(16,37)(17,36)(18,40)(19,39)(20,38)(26,27)(28,30)(41,42)(43,45);;
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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(45)!( 6,36)( 7,37)( 8,38)( 9,39)(10,40)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35);
s1 := Sym(45)!( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(16,41)(17,45)(18,44)(19,43)(20,42)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37);
s2 := Sym(45)!( 1,22)( 2,21)( 3,25)( 4,24)( 5,23)( 6,32)( 7,31)( 8,35)( 9,34)(10,33)(11,12)(13,15)(16,37)(17,36)(18,40)(19,39)(20,38)(26,27)(28,30)(41,42)(43,45);
poly := sub<Sym(45)|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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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*s0 >;

References : None.
to this polytope

Polytope of Type {4,20}

Twisty Puzzle