Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,15}

Atlas Canonical Name {6,15}*720e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(720,794)
Rank
3
Schläfli Type
{6,15}
Vertices, edges, …
24, 180, 60
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*s0)^2*s2*s1*s0*s1> of order 2

30 facets

12 vertex figures

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

30 facets

8 vertex figures

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

15 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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