Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,4}

Atlas Canonical Name {8,4}*1344f

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1344,11684)
Rank
3
Schläfli Type
{8,4}
Vertices, edges, …
168, 336, 84
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*s2*(s1*s0)^2*s2*s1*s0*s1> of order 2

42 facets

84 vertex figures

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

44 facets

88 vertex figures

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

42 facets

84 vertex figures

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

42 facets

84 vertex figures

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

24 facets

48 vertex figures

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

24 facets

44 vertex figures

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

22 facets

44 vertex figures

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

22 facets

44 vertex figures

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

22 facets

44 vertex figures

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

22 facets

44 vertex figures

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

16 facets

32 vertex figures

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

12 facets

22 vertex figures

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

12 facets

22 vertex figures

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

12 facets

22 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 8

14 facets

26 vertex figures

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

12 facets

24 vertex figures

P/N, where N=<(s0*s1)^2, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*(s1*s0)^3*s1*s2*s1> of order 24

6 facets

10 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle