Polytope of Type {6,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,20}*1920a
if this polytope has a name.
Group : SmallGroup(1920,238598)
Rank : 3
Schlafli Type : {6,20}
Number of vertices, edges, etc : 48, 480, 160
Order of s0s1s2 : 30
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {6,20}*480c
   5-fold quotients : {6,4}*384a
   8-fold quotients : {6,20}*240b
   10-fold quotients : {6,4}*192a
   16-fold quotients : {6,10}*120
   20-fold quotients : {6,4}*96
   40-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {2,10}*40
   80-fold quotients : {3,4}*24, {6,2}*24
   96-fold quotients : {2,5}*20
   160-fold quotients : {3,2}*12
   240-fold quotients : {2,2}*8
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*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 2.
      80 facets:
         80 of {6}*12
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 2.
      80 facets:
         80 of {6}*12
      28 vertex figures:
         20 of {20}*40
         8 of {10}*20
   P/N, where N=<s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      80 facets:
         80 of {6}*12
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      80 facets:
         80 of {6}*12
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 2.
      80 facets:
         80 of {6}*12
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      80 facets:
         80 of {6}*12
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      80 facets:
         80 of {6}*12
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 2.
      80 facets:
         80 of {6}*12
      24 vertex figures:
         24 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      18 vertex figures:
         12 of {10}*20
         6 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2> of order 4.
      40 facets:
         40 of {6}*12
      16 vertex figures:
         8 of {20}*40
         8 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      16 vertex figures:
         8 of {20}*40
         8 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      14 vertex figures:
         10 of {20}*40
         4 of {10}*20
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 4.
      40 facets:
         40 of {6}*12
      14 vertex figures:
         10 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {6}*12
      14 vertex figures:
         10 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      14 vertex figures:
         10 of {20}*40
         4 of {10}*20
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      14 vertex figures:
         10 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      14 vertex figures:
         10 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 8.
      20 facets:
         20 of {6}*12
      8 vertex figures:
         4 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 8.
      20 facets:
         20 of {6}*12
      8 vertex figures:
         4 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0, s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 8.
      20 facets:
         20 of {6}*12
      10 vertex figures:
         2 of {20}*40
         8 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 8.
      20 facets:
         20 of {6}*12
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 8.
      20 facets:
         20 of {6}*12
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 8.
      20 facets:
         20 of {6}*12
      9 vertex figures:
         6 of {10}*20
         3 of {20}*40
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1> of order 8.
      20 facets:
         20 of {6}*12
      8 vertex figures:
         4 of {20}*40
         4 of {10}*20
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*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 {6}*12
      8 vertex figures:
         4 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 8.
      20 facets:
         20 of {6}*12
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 8.
      20 facets:
         20 of {6}*12
      7 vertex figures:
         5 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 8.
      20 facets:
         20 of {6}*12
      6 vertex figures:
         6 of {20}*40

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

Twisty Puzzle