Polytope of Type {6,10}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10}*1620
if this polytope has a name.
Group : SmallGroup(1620,422)
Rank : 3
Schlafli Type : {6,10}
Number of vertices, edges, etc : 81, 405, 135
Order of s0s1s2 : 5
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      45 facets:
         45 of {6}*12
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s1*s0*s1> of order 3.
      63 facets:
         27 of {2}*4
         36 of {6}*12
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      45 facets:
         45 of {6}*12
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      45 facets:
         45 of {6}*12
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 3.
      45 facets:
         45 of {6}*12
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 3.
      45 facets:
         45 of {6}*12
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      15 facets:
         15 of {6}*12
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 9.
      27 facets:
         18 of {2}*4
         9 of {6}*12
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
      21 facets:
         9 of {2}*4
         12 of {6}*12
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      21 facets:
         9 of {2}*4
         12 of {6}*12
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
      15 facets:
         15 of {6}*12
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 9.
      21 facets:
         9 of {2}*4
         12 of {6}*12
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 9.
      15 facets:
         15 of {6}*12
      9 vertex figures:
         9 of {10}*20

Permutation Representation (GAP) : 
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68);;
s1 := ( 1, 2)( 4,71)( 5,70)( 6,72)( 7,50)( 8,49)( 9,51)(10,35)(11,34)(12,36)(13,14)(16,74)(17,73)(18,75)(19,59)(20,58)(21,60)(22,38)(23,37)(24,39)(25,26)(28,80)(29,79)(30,81)(31,32)(40,56)(41,55)(42,57)(43,44)(46,47)(52,68)(53,67)(54,69)(61,62)(64,65)(76,77);;
s2 := ( 2,36)( 3,59)( 4,31)( 5,57)( 6, 8)( 7,61)( 9,29)(10,15)(11,38)(12,70)(13,45)(14,68)(16,66)(18,40)(19,26)(20,49)(21,75)(22,47)(23,79)(25,77)(27,54)(28,55)(30,32)(33,62)(35,60)(37,69)(39,43)(42,64)(44,71)(46,80)(50,52)(51,78)(53,73)(56,63)(67,72)(74,76);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
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(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68);
s1 := Sym(81)!( 1, 2)( 4,71)( 5,70)( 6,72)( 7,50)( 8,49)( 9,51)(10,35)(11,34)(12,36)(13,14)(16,74)(17,73)(18,75)(19,59)(20,58)(21,60)(22,38)(23,37)(24,39)(25,26)(28,80)(29,79)(30,81)(31,32)(40,56)(41,55)(42,57)(43,44)(46,47)(52,68)(53,67)(54,69)(61,62)(64,65)(76,77);
s2 := Sym(81)!( 2,36)( 3,59)( 4,31)( 5,57)( 6, 8)( 7,61)( 9,29)(10,15)(11,38)(12,70)(13,45)(14,68)(16,66)(18,40)(19,26)(20,49)(21,75)(22,47)(23,79)(25,77)(27,54)(28,55)(30,32)(33,62)(35,60)(37,69)(39,43)(42,64)(44,71)(46,80)(50,52)(51,78)(53,73)(56,63)(67,72)(74,76);
poly := sub<Sym(81)|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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
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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