Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*1296a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,3490)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
108, 324, 108
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
  • Self-Dual

Quotients maximal quotients in bold

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

56 facets

56 vertex figures

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

54 facets

54 vertex figures

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

36 facets

36 vertex figures

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

36 facets

36 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

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

20 facets

20 vertex figures

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

20 facets

20 vertex figures

Representations

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

References

None.

to this polytope.

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