Polytope of Type {6,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,20}*720
if this polytope has a name.
Group : SmallGroup(720,810)
Rank : 3
Schlafli Type : {6,20}
Number of vertices, edges, etc : 18, 180, 60
Order of s0s1s2 : 20
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,20,2} of size 1440
Vertex Figure Of :
   {2,6,20} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,20}*360
   5-fold quotients : {6,4}*144
   9-fold quotients : {2,20}*80
   10-fold quotients : {6,4}*72
   18-fold quotients : {2,10}*40
   36-fold quotients : {2,5}*20
   45-fold quotients : {2,4}*16
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,40}*1440, {12,20}*1440
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1> of order 3.
      40 facets:
         30 of {2}*4
         10 of {6}*12
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      20 facets:
         20 of {6}*12
      6 vertex figures:
         6 of {20}*40

Permutation Representation (GAP) : 
s0 := ( 1,46)( 2,47)( 3,48)( 4,49)( 5,50)( 6,56)( 7,57)( 8,58)( 9,59)(10,60)(11,51)(12,52)(13,53)(14,54)(15,55)(16,76)(17,77)(18,78)(19,79)(20,80)(21,86)(22,87)(23,88)(24,89)(25,90)(26,81)(27,82)(28,83)(29,84)(30,85)(31,61)(32,62)(33,63)(34,64)(35,65)(36,71)(37,72)(38,73)(39,74)(40,75)(41,66)(42,67)(43,68)(44,69)(45,70);;
s1 := ( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)(16,21)(17,25)(18,24)(19,23)(20,22)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(46,51)(47,55)(48,54)(49,53)(50,52)(57,60)(58,59)(61,66)(62,70)(63,69)(64,68)(65,67)(72,75)(73,74)(76,81)(77,85)(78,84)(79,83)(80,82)(87,90)(88,89);;
s2 := ( 1, 2)( 3, 5)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)(11,32)(12,31)(13,35)(14,34)(15,33)(21,22)(23,25)(26,37)(27,36)(28,40)(29,39)(30,38)(41,42)(43,45)(46,47)(48,50)(51,62)(52,61)(53,65)(54,64)(55,63)(56,77)(57,76)(58,80)(59,79)(60,78)(66,67)(68,70)(71,82)(72,81)(73,85)(74,84)(75,83)(86,87)(88,90);;
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*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(90)!( 1,46)( 2,47)( 3,48)( 4,49)( 5,50)( 6,56)( 7,57)( 8,58)( 9,59)(10,60)(11,51)(12,52)(13,53)(14,54)(15,55)(16,76)(17,77)(18,78)(19,79)(20,80)(21,86)(22,87)(23,88)(24,89)(25,90)(26,81)(27,82)(28,83)(29,84)(30,85)(31,61)(32,62)(33,63)(34,64)(35,65)(36,71)(37,72)(38,73)(39,74)(40,75)(41,66)(42,67)(43,68)(44,69)(45,70);
s1 := Sym(90)!( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)(16,21)(17,25)(18,24)(19,23)(20,22)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(46,51)(47,55)(48,54)(49,53)(50,52)(57,60)(58,59)(61,66)(62,70)(63,69)(64,68)(65,67)(72,75)(73,74)(76,81)(77,85)(78,84)(79,83)(80,82)(87,90)(88,89);
s2 := Sym(90)!( 1, 2)( 3, 5)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)(11,32)(12,31)(13,35)(14,34)(15,33)(21,22)(23,25)(26,37)(27,36)(28,40)(29,39)(30,38)(41,42)(43,45)(46,47)(48,50)(51,62)(52,61)(53,65)(54,64)(55,63)(56,77)(57,76)(58,80)(59,79)(60,78)(66,67)(68,70)(71,82)(72,81)(73,85)(74,84)(75,83)(86,87)(88,90);
poly := sub<Sym(90)|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*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
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Twisty Puzzle