Polytope of Type {6,6} with group SmallGroup(216,102)

Overview

Group: SmallGroup(216,102)
Rank: 3
Schläfli Type: {6,6}
Vertices, edges, …: 18, 54, 18
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

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

2-fold

{3,6}*108

3-fold

{6,6}*72c

6-fold

{3,6}*36

9-fold

{6,2}*24

18-fold

{3,2}*12

27-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

9-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

6 facets

6 of {6}*12

10 vertex figures

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

Representations

Permutation Representation (GAP)

s0 := ( 1, 7)( 2, 8)( 3, 9)(10,16)(11,17)(12,18);;
s1 := ( 1,10)( 2,11)( 3,12)( 4,17)( 5,18)( 6,16)( 7,15)( 8,13)( 9,14);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);;
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*s0*s1*s2*s1*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(18)!( 1, 7)( 2, 8)( 3, 9)(10,16)(11,17)(12,18);
s1 := Sym(18)!( 1,10)( 2,11)( 3,12)( 4,17)( 5,18)( 6,16)( 7,15)( 8,13)( 9,14);
s2 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);
poly := sub<Sym(18)|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*s0*s1*s2*s1*s0*s1*s0*s1*s2*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 >;

References

None.

to this polytope.