Polytope of Type {20,6}

Play with this polytope as a twisty puzzle

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

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