Polytope of Type {6,30}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*1440g
if this polytope has a name.
Group : SmallGroup(1440,5871)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 24, 360, 120
Order of s₀s₁s₂ : 60
Order of s₀s₁s₂s₁ : 6
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) :
   3-fold quotients : {6,30}*480
   4-fold quotients : {6,30}*360a
   5-fold quotients : {6,6}*288b
   6-fold quotients : {6,15}*240
   10-fold quotients : {3,6}*144
   12-fold quotients : {6,10}*120
   15-fold quotients : {6,6}*96
   20-fold quotients : {6,6}*72c
   30-fold quotients : {3,6}*48, {6,3}*48
   36-fold quotients : {2,10}*40
   40-fold quotients : {3,6}*36
   60-fold quotients : {3,3}*24, {6,2}*24
   72-fold quotients : {2,5}*20
   120-fold quotients : {3,2}*12
   180-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*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      12 vertex figures:
         12 of {30}*60
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      12 vertex figures:
         12 of {30}*60
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      60 facets:
         60 of {6}*12
      16 vertex figures:
         8 of {30}*60
         8 of {15}*30
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      40 facets:
         40 of {6}*12
      12 vertex figures:
         6 of {30}*60
         6 of {10}*20
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      30 facets:
         30 of {6}*12
      6 vertex figures:
         6 of {30}*60
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 4.
      30 facets:
         30 of {6}*12
      6 vertex figures:
         6 of {30}*60
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
      30 facets:
         30 of {6}*12
      6 vertex figures:
         6 of {30}*60

Permutation Representation (GAP) :

s0 := ( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(21,41)(22,42)(23,44)(24,43)(25,45)(26,46)(27,48)(28,47)(29,49)(30,50)(31,52)(32,51)(33,53)(34,54)(35,56)(36,55)(37,57)(38,58)(39,60)(40,59);;
s1 := ( 1,21)( 2,24)( 3,23)( 4,22)( 5,37)( 6,40)( 7,39)( 8,38)( 9,33)(10,36)(11,35)(12,34)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(42,44)(45,57)(46,60)(47,59)(48,58)(49,53)(50,56)(51,55)(52,54);;
s2 := ( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,18)(10,17)(11,19)(12,20)(13,14)(21,26)(22,25)(23,27)(24,28)(29,38)(30,37)(31,39)(32,40)(33,34)(41,46)(42,45)(43,47)(44,48)(49,58)(50,57)(51,59)(52,60)(53,54);;
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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*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*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(60)!( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(21,41)(22,42)(23,44)(24,43)(25,45)(26,46)(27,48)(28,47)(29,49)(30,50)(31,52)(32,51)(33,53)(34,54)(35,56)(36,55)(37,57)(38,58)(39,60)(40,59);
s1 := Sym(60)!( 1,21)( 2,24)( 3,23)( 4,22)( 5,37)( 6,40)( 7,39)( 8,38)( 9,33)(10,36)(11,35)(12,34)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(42,44)(45,57)(46,60)(47,59)(48,58)(49,53)(50,56)(51,55)(52,54);
s2 := Sym(60)!( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,18)(10,17)(11,19)(12,20)(13,14)(21,26)(22,25)(23,27)(24,28)(29,38)(30,37)(31,39)(32,40)(33,34)(41,46)(42,45)(43,47)(44,48)(49,58)(50,57)(51,59)(52,60)(53,54);
poly := sub<Sym(60)|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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*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*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 >;

References : None.
to this polytope

Polytope of Type {6,30}

Twisty Puzzle