Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,15}

Atlas Canonical Name {3,15}*1440

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Overview

Group
SmallGroup(1440,5848)
Rank
3
Schläfli Type
{3,15}
Vertices, edges, …
48, 360, 240
Order of s0s1s2
20
Order of s0s1s2s1
15
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

4-fold

12-fold

24-fold

60-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)^4*s1*s0*(s2*s1)^5*s2> of order 2

120 facets

24 vertex figures

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

120 facets

24 vertex figures

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

120 facets

24 vertex figures

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

80 facets

24 vertex figures

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

80 facets

16 vertex figures

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

60 facets

12 vertex figures

P/N, where N=<s1*s2*s1*(s0*(s2*s1)^3)^2*s0*s2*s1*s2, s0*(s1*s2)^4*s1*s0*(s2*s1)^5*s2> of order 4

60 facets

12 vertex figures

P/N, where N=<(s1*s2)^2*s1*(s0*(s2*s1)^3)^2*s2, s0*s1*s0*(s2*s1)^5*s0*(s2*s1)^4> of order 4

60 facets

12 vertex figures

P/N, where N=<s0*(s1*s0*s2)^3*s1*s2> of order 5

48 facets

16 vertex figures

P/N, where N=<s0*s2*s1*s0*(s2*s1)^4*s0*s2*s1*s2, s0*(s1*s2)^2*s1*(s0*(s2*s1)^3)^2*s2> of order 6

40 facets

12 vertex figures

P/N, where N=<s1*s2*s1*s0*(s2*s1)^3*s0*s2*s1*s2, s0*(s2*s1)^6*s0*(s2*s1)^3> of order 6

40 facets

8 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^2*(s2*s1)^2, s0*(s1*s2)^5*s1*s0*(s2*s1)^3*s2> of order 10

24 facets

8 vertex figures

P/N, where N=<s1*s0*(s2*s1)^3*s0*(s2*s1)^2*s2, s1*s2*s1*s0*(s2*s1)^3*s0*s2*s1*s2> of order 12

20 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (2,3)(4,5)(7,9);;
s1 := (1,2)(4,5)(8,9);;
s2 := (2,4)(3,5)(6,8);;
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, 
s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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(9)!(2,3)(4,5)(7,9);
s1 := Sym(9)!(1,2)(4,5)(8,9);
s2 := Sym(9)!(2,4)(3,5)(6,8);
poly := sub<Sym(9)|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, s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*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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