Polytope of Type {6,40}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,40}*1920c
if this polytope has a name.
Group : SmallGroup(1920,238599)
Rank : 3
Schlafli Type : {6,40}
Number of vertices, edges, etc : 24, 480, 160
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 8
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 : {6,8}*384e
   8-fold quotients : {6,20}*240b
   10-fold quotients : {3,8}*192
   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*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      80 facets:
         80 of {6}*12
      12 vertex figures:
         12 of {40}*80
   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
      16 vertex figures:
         8 of {40}*80
         8 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1> of order 2.
      80 facets:
         80 of {6}*12
      12 vertex figures:
         12 of {40}*80
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      8 vertex figures:
         4 of {40}*80
         4 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1, s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      8 vertex figures:
         4 of {40}*80
         4 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         4 of {40}*80
         8 of {10}*20
   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
      8 vertex figures:
         4 of {40}*80
         4 of {20}*40
   P/N, where N=<s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      40 facets:
         40 of {6}*12
      8 vertex figures:
         4 of {40}*80
         4 of {20}*40

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {6,40}

Twisty Puzzle