Polytope of Type {10,20}

Play with this polytope as a twisty puzzle

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

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