Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,8}

Atlas Canonical Name {6,8}*1152b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1152,157849)
Rank
3
Schläfli Type
{6,8}
Vertices, edges, …
72, 288, 96
Order of s0s1s2
8
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-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*s2)^3*s1)^2> of order 2

48 facets

36 vertex figures

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

48 facets

40 vertex figures

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

48 facets

36 vertex figures

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

32 facets

24 vertex figures

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

24 facets

18 vertex figures

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

24 facets

24 vertex figures

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

24 facets

22 vertex figures

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

24 facets

18 vertex figures

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

24 facets

18 vertex figures

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

24 facets

20 vertex figures

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

16 facets

12 vertex figures

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

12 facets

14 vertex figures

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

12 facets

12 vertex figures

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

12 facets

11 vertex figures

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

12 facets

10 vertex figures

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

8 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle