Polytope of Type {5,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,20}*1600
if this polytope has a name.
Group : SmallGroup(1600,10261)
Rank : 3
Schlafli Type : {5,20}
Number of vertices, edges, etc : 40, 400, 160
Order of s0s1s2 : 10
Order of s0s1s2s1 : 20
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) :
   5-fold quotients : {5,4}*320
   10-fold quotients : {5,4}*160
   16-fold quotients : {5,10}*100
   80-fold quotients : {5,2}*20
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*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
      80 facets:
         80 of {5}*10
      24 vertex figures:
         16 of {20}*40
         8 of {10}*20
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
      80 facets:
         80 of {5}*10
      20 vertex figures:
         20 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 2.
      80 facets:
         80 of {5}*10
      20 vertex figures:
         20 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 2.
      80 facets:
         80 of {5}*10
      20 vertex figures:
         20 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 2.
      80 facets:
         80 of {5}*10
      20 vertex figures:
         20 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      80 facets:
         80 of {5}*10
      20 vertex figures:
         20 of {20}*40
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 4.
      40 facets:
         40 of {5}*10
      12 vertex figures:
         4 of {10}*20
         8 of {20}*40
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      12 vertex figures:
         8 of {20}*40
         4 of {10}*20
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      12 vertex figures:
         4 of {10}*20
         8 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      14 vertex figures:
         6 of {20}*40
         8 of {10}*20
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      40 facets:
         40 of {5}*10
      14 vertex figures:
         6 of {20}*40
         8 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 4.
      40 facets:
         40 of {5}*10
      12 vertex figures:
         8 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      12 vertex figures:
         8 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      12 vertex figures:
         8 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1> of order 4.
      40 facets:
         40 of {5}*10
      12 vertex figures:
         8 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {5}*10
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 8.
      20 facets:
         20 of {5}*10
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0> of order 8.
      20 facets:
         20 of {5}*10
      8 vertex figures:
         2 of {20}*40
         6 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*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2> of order 8.
      20 facets:
         20 of {5}*10
      8 vertex figures:
         6 of {10}*20
         2 of {20}*40
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 8.
      20 facets:
         20 of {5}*10
      6 vertex figures:
         2 of {10}*20
         4 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 8.
      20 facets:
         20 of {5}*10
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 8.
      20 facets:
         20 of {5}*10
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 8.
      20 facets:
         20 of {5}*10
      7 vertex figures:
         3 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2> of order 8.
      20 facets:
         20 of {5}*10
      7 vertex figures:
         3 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0> of order 8.
      20 facets:
         20 of {5}*10
      5 vertex figures:
         5 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 8.
      20 facets:
         20 of {5}*10
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20

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

Twisty Puzzle