Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,8}

Atlas Canonical Name {4,8}*1152d

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1152,157849)
Rank
3
Schläfli Type
{4,8}
Vertices, edges, …
72, 288, 144
Order of s0s1s2
12
Order of s0s1s2s1
3
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

16-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)^2*(s0*s2*s1)^2*s0*(s2*s1)^2*s2> of order 2

72 facets

36 vertex figures

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

72 facets

40 vertex figures

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

72 facets

36 vertex figures

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

48 facets

24 vertex figures

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

36 facets

18 vertex figures

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

36 facets

24 vertex figures

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

36 facets

22 vertex figures

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

36 facets

18 vertex figures

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

36 facets

18 vertex figures

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

36 facets

20 vertex figures

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

24 facets

12 vertex figures

P/N, where N=<s0*s1*s2*s1*(s0*(s2*s1)^2)^2, s1*s0*s1*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2, (s0*s1)^2*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 8

18 facets

14 vertex figures

P/N, where N=<(s1*s2)^4, (s0*s1)^2*(s2*s1)^3*s0*s2*s1*s0, (s0*s2*s1)^6> of order 8

18 facets

12 vertex figures

P/N, where N=<s0*(s1*s2)^3*s1*s0*s2, s1*s0*(s1*s2)^3*s1*s0*s2*s1, (s0*s1)^2*(s2*s1*s0)^4*s2*s1*s2> of order 8

18 facets

11 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 8

18 facets

10 vertex figures

P/N, where N=<(s0*s2*s1)^4, (s0*s1)^2*(s2*s1)^2*s0*(s2*s1)^2*s2> of order 12

12 facets

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15);;
s1 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,16)(12,15);;
s2 := ( 2,13)( 3, 9)( 4, 5)( 6,16)( 7,12)(10,15);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(16)!( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15);
s1 := Sym(16)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,16)(12,15);
s2 := Sym(16)!( 2,13)( 3, 9)( 4, 5)( 6,16)( 7,12)(10,15);
poly := sub<Sym(16)|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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

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