Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,30}

Atlas Canonical Name {6,30}*1440g

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1440,5871)
Rank
3
Schläfli Type
{6,30}
Vertices, edges, …
24, 360, 120
Order of s0s1s2
60
Order of s0s1s2s1
6
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)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 2

60 facets

12 vertex figures

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

60 facets

12 vertex figures

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

60 facets

16 vertex figures

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 3

40 facets

12 vertex figures

P/N, where N=<(s0*(s2*s1)^2)^2*s2> of order 4

30 facets

6 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*s0*(s2*s1)^3*s2> of order 4

30 facets

6 vertex figures

P/N, where N=<s0*(s1*s2)^2*(s1*s0)^2*s2*s1*s2, (s0*s2*s1)^2*s0*(s1*s2)^2*s1> of order 4

30 facets

6 vertex figures

Representations

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.

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