Part of the Atlas of Small Regular Polytopes

Polytope of Type {60,6}

Atlas Canonical Name {60,6}*1440c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1440,5871)
Rank
3
Schläfli Type
{60,6}
Vertices, edges, …
120, 360, 12
Order of s0s1s2
30
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

5-fold

6-fold

10-fold

12-fold

15-fold

20-fold

30-fold

36-fold

40-fold

60-fold

72-fold

120-fold

180-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*s1*s2*(s1*s0)^2*(s2*s1)^2*s0*s1*s2> of order 2

8 facets

60 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^2*(s1*s2)^2*s1> of order 2

6 facets

60 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s1*s0*(s2*s1)^2> of order 2

6 facets

60 vertex figures

Representations

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

References

None.

to this polytope.

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