Polytope of Type {20,4}

Play with this polytope as a twisty puzzle

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

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {20,4}

Twisty Puzzle