Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*1728d

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Overview

Group
SmallGroup(1728,46101)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
144, 432, 144
Order of s0s1s2
6
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

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

Quotients maximal quotients in bold

3-fold

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*s2*s1)^3> of order 2

72 facets

72 vertex figures

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

72 facets

72 vertex figures

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

36 facets

36 vertex figures

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

36 facets

36 vertex figures

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

36 facets

36 vertex figures

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

36 facets

36 vertex figures

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

36 facets

36 vertex figures

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

36 facets

36 vertex figures

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

36 facets

36 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

P/N, where N=<s0*(s1*s0*s2)^2*s1*s2, (s0*s2*s1)^3, (s0*s1)^2*s0*s2*s1*s0*(s2*s1)^2*s2, s0*s2*(s1*s0)^2*s2*s1*s0*(s2*s1)^2> of order 16

9 facets

9 vertex figures

P/N, where N=<(s0*s2*s1)^3, s0*s1*s2*(s1*s0)^2*s2*s1*s2> of order 16

9 facets

9 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,21)(14,23)(15,22)(16,24)(18,19)(25,33)(26,36)(27,35)(28,34)(30,32);;
s1 := ( 1,13)( 2,14)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,21)(10,22)(11,24)(12,23)(27,28)(31,32)(35,36);;
s2 := ( 5, 9)( 6,10)( 7,11)( 8,12)(13,35)(14,34)(15,36)(16,33)(17,31)(18,30)(19,32)(20,29)(21,27)(22,26)(23,28)(24,25);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0, 
s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,21)(14,23)(15,22)(16,24)(18,19)(25,33)(26,36)(27,35)(28,34)(30,32);
s1 := Sym(36)!( 1,13)( 2,14)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,21)(10,22)(11,24)(12,23)(27,28)(31,32)(35,36);
s2 := Sym(36)!( 5, 9)( 6,10)( 7,11)( 8,12)(13,35)(14,34)(15,36)(16,33)(17,31)(18,30)(19,32)(20,29)(21,27)(22,26)(23,28)(24,25);
poly := sub<Sym(36)|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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0, 
s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1 >;

References

None.

to this polytope.

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