Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*864c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(864,4673)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
72, 216, 36
Order of s0s1s2
6
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

24-fold

36-fold

72-fold

108-fold

Covers minimal covers in bold

2-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)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 2

24 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

48 vertex figures

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

24 facets

24 vertex figures

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

12 facets

24 vertex figures

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

9 facets

24 vertex figures

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

12 facets

24 vertex figures

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

12 facets

18 vertex figures

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

18 facets

18 vertex figures

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

6 facets

12 vertex figures

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

12 facets

12 vertex figures

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

16 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

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