Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296p

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,3490)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
Order of s0s1s2
6
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

27-fold

54-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=<s1*s0*s1*s2*(s1*s0)^2*s1*s2*(s1*s0)^3*s2*s1> of order 2

27 facets

54 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

30 facets

36 vertex figures

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

15 facets

18 vertex figures

P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1, (s1*s0)^2*s2*s1*s0*s1*s2> of order 9

6 facets

12 vertex figures

P/N, where N=<(s0*s1)^4, (s0*s2*s1)^2*s0*(s1*s2)^2> of order 9

10 facets

12 vertex figures

P/N, where N=<s0*s2*(s1*s0)^3*s1*s2, (s0*s2*s1)^2*s0*(s1*s2)^2> of order 9

14 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle