Polytope of Type {6,6} with group SmallGroup(324,39) - Atlas of Small Regular Polytopes

Overview

Group: SmallGroup(324,39)
Rank: 3
Schläfli Type: {6,6}
Vertices, edges, …: 27, 81, 27
Order of s₀s₁s₂: 9
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

3-fold

{6,6}*108

Covers minimal covers in bold

2-fold

{6,6}*648a

3-fold

4-fold

5-fold

6-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

9 facets

9 of {6}*12

15 vertex figures

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

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

9 facets

9 of {6}*12

9 vertex figures

9 of {6}*12

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

s0 := Sym(9)!(1,7)(2,8)(3,9);
s1 := Sym(9)!(2,3)(4,7)(5,9)(6,8);
s2 := Sym(9)!(2,3)(4,5)(8,9);
poly := sub<Sym(9)|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*s2*s0*s1*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 >;

References

None.

to this polytope.