Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*648g

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Overview

Group
SmallGroup(648,555)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
54, 162, 54
Order of s0s1s2
6
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

Covers minimal covers in bold

2-fold

3-fold

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

30 facets

27 vertex figures

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

27 facets

27 vertex figures

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

27 facets

30 vertex figures

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

18 facets

18 vertex figures

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

18 facets

24 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

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

24 facets

18 vertex figures

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

9 facets

12 vertex figures

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

12 facets

9 vertex figures

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

10 facets

6 vertex figures

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

6 facets

10 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);;
s1 := ( 1, 4)( 2, 5)( 3, 6)(10,23)(11,24)(12,22)(13,20)(14,21)(15,19)(16,26)(17,27)(18,25);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(27)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26);
s1 := Sym(27)!( 1, 4)( 2, 5)( 3, 6)(10,23)(11,24)(12,22)(13,20)(14,21)(15,19)(16,26)(17,27)(18,25);
s2 := Sym(27)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26);
poly := sub<Sym(27)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

References

None.

to this polytope.

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