Part of the Atlas of Small Regular Polytopes

Polytope of Type {30,6}

Atlas Canonical Name {30,6}*1440g

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1440,5871)
Rank
3
Schläfli Type
{30,6}
Vertices, edges, …
120, 360, 24
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

12 facets

60 vertex figures

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

12 facets

60 vertex figures

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

16 facets

60 vertex figures

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

12 facets

40 vertex figures

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

6 facets

30 vertex figures

P/N, where N=<(s0*s1)^3*(s0*s2*s1)^2*s2> of order 4

6 facets

30 vertex figures

P/N, where N=<(s1*s0)^3*s2*s1*s0*s2*s1*s2, (s1*s0*s2)^2*(s1*s0)^2*s1*s2> of order 4

6 facets

30 vertex figures

Representations

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

References

None.

to this polytope.

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