Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,8}

Atlas Canonical Name {4,8}*1344e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1344,11684)
Rank
3
Schläfli Type
{4,8}
Vertices, edges, …
84, 336, 168
Order of s0s1s2
14
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

2-fold

4-fold

168-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*s1)^2> of order 2

88 facets

44 vertex figures

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

84 facets

42 vertex figures

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

84 facets

42 vertex figures

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

84 facets

42 vertex figures

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

48 facets

24 vertex figures

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

44 facets

24 vertex figures

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

44 facets

22 vertex figures

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

44 facets

22 vertex figures

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

44 facets

22 vertex figures

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

44 facets

22 vertex figures

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

32 facets

16 vertex figures

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

22 facets

12 vertex figures

P/N, where N=<(s0*s1)^2, (s1*s2*s1*s0)^2*s1*s2, s0*s1*(s2*s1*s0)^5*s2*s1*s2> of order 8

22 facets

12 vertex figures

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

26 facets

14 vertex figures

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

22 facets

12 vertex figures

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

24 facets

12 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2, s0*(s1*s2)^2*s1*s0*s2*s1*s2> of order 24

10 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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