Part of the Atlas of Small Regular Polytopes

Polytope of Type {15,6}

Atlas Canonical Name {15,6}*720e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(720,794)
Rank
3
Schläfli Type
{15,6}
Vertices, edges, …
60, 180, 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

12-fold

15-fold

20-fold

30-fold

36-fold

60-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

12 facets

30 vertex figures

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

8 facets

30 vertex figures

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

6 facets

15 vertex figures

Representations

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

References

None.

to this polytope.

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