Polytope of Type {10,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6}*1620
if this polytope has a name.
Group : SmallGroup(1620,422)
Rank : 3
Schlafli Type : {10,6}
Number of vertices, edges, etc : 135, 405, 81
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*s2*s1*s0*s2> of order 3.
      27 facets:
         27 of {10}*20
      63 vertex figures:
         36 of {6}*12
         27 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 3.
      27 facets:
         27 of {10}*20
      45 vertex figures:
         45 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      27 facets:
         27 of {10}*20
      45 vertex figures:
         45 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 3.
      27 facets:
         27 of {10}*20
      45 vertex figures:
         45 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
      27 facets:
         27 of {10}*20
      45 vertex figures:
         45 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      27 facets:
         27 of {10}*20
      45 vertex figures:
         45 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 9.
      9 facets:
         9 of {10}*20
      27 vertex figures:
         9 of {6}*12
         18 of {2}*4
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 9.
      9 facets:
         9 of {10}*20
      21 vertex figures:
         9 of {2}*4
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2> of order 9.
      9 facets:
         9 of {10}*20
      21 vertex figures:
         12 of {6}*12
         9 of {2}*4
   P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 9.
      9 facets:
         9 of {10}*20
      21 vertex figures:
         9 of {2}*4
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2> of order 9.
      9 facets:
         9 of {10}*20
      15 vertex figures:
         15 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 9.
      9 facets:
         9 of {10}*20
      15 vertex figures:
         15 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 9.
      9 facets:
         9 of {10}*20
      15 vertex figures:
         15 of {6}*12

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

Twisty Puzzle