Polytope of Type {3,6} with group SmallGroup(144,183)

Overview

Group: SmallGroup(144,183)
Rank: 3
Schläfli Type: {3,6}
Vertices, edges, …: 12, 36, 24
Order of s₀s₁s₂: 12
Order of s₀s₁s₂s₁: 6
Also known as: {3,6}_(2,2). if this polytope has another name.

Special Properties

Toroidal
Locally Spherical
Orientable

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

3-fold

{3,6}*48

4-fold

{3,6}*36

6-fold

{3,3}*24

12-fold

{3,2}*12

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

{15,6}*720e

6-fold

7-fold

{21,6}*1008b

8-fold

9-fold

10-fold

11-fold

{33,6}*1584

12-fold

13-fold

{39,6}*1872

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

12 facets

12 of {3}*6

6 vertex figures

6 of {6}*12

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

12 facets

12 of {3}*6

8 vertex figures

4 of {6}*12
4 of {3}*6

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

8 facets

8 of {3}*6

6 vertex figures

3 of {6}*12
3 of {2}*4

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

6 facets

6 of {3}*6

3 vertex figures

3 of {6}*12

Representations

Permutation Representation (GAP)

s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11);;
s1 := ( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11);;
s2 := ( 1, 2)( 5, 6)( 9,10);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(12)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11);
s1 := Sym(12)!( 1, 5)( 2, 7)( 3, 6)( 4, 8)(10,11);
s2 := Sym(12)!( 1, 2)( 5, 6)( 9,10);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

References

None.

to this polytope.