Polytope of Type {5,10}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,10}*1620
if this polytope has a name.
Group : SmallGroup(1620,422)
Rank : 3
Schlafli Type : {5,10}
Number of vertices, edges, etc : 81, 405, 162
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 3.
      54 facets:
         54 of {5}*10
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 3.
      54 facets:
         54 of {5}*10
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      54 facets:
         54 of {5}*10
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 3.
      54 facets:
         54 of {5}*10
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 3.
      54 facets:
         54 of {5}*10
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 3.
      54 facets:
         54 of {5}*10
      27 vertex figures:
         27 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 9.
      18 facets:
         18 of {5}*10
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 9.
      18 facets:
         18 of {5}*10
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 9.
      18 facets:
         18 of {5}*10
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 9.
      18 facets:
         18 of {5}*10
      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*s2*s1*s2*s1*s2*s1*s2> of order 9.
      18 facets:
         18 of {5}*10
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1> of order 9.
      18 facets:
         18 of {5}*10
      9 vertex figures:
         9 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 9.
      18 facets:
         18 of {5}*10
      9 vertex figures:
         9 of {10}*20

Permutation Representation (GAP) :

s0 := ( 4,49)( 5,50)( 6,51)( 7,70)( 8,71)( 9,72)(10,58)(11,59)(12,60)(13,25)(14,26)(15,27)(16,37)(17,38)(18,39)(19,34)(20,35)(21,36)(22,73)(23,74)(24,75)(28,40)(29,41)(30,42)(31,61)(32,62)(33,63)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(55,79)(56,80)(57,81);;
s1 := ( 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);;
s2 := ( 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);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {5,10}

Twisty Puzzle