Polytope of Type {3,6,3} with group SmallGroup(108,17)

Overview

Group: SmallGroup(108,17)
Rank: 4
Schläfli Type: {3,6,3}
Vertices, edges, …: 3, 9, 9, 3
Order of s₀s₁s₂s₃: 3
Order of s₀s₁s₂s₃s₂s₁: 6
Also known as: ₇T⁴_(1,1)(1,1). if this polytope has another name.

Special Properties

Universal
Locally Toroidal
Orientable
Flat
Self-Dual

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

3-fold

{3,2,3}*36

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

9-fold

10-fold

11-fold

12-fold

13-fold

14-fold

15-fold

16-fold

17-fold

18-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (4,5)(6,7)(8,9);;
s1 := (2,4)(3,6)(8,9);;
s2 := (1,2)(4,9)(5,8)(6,7);;
s3 := (2,3)(4,6)(5,7)(8,9);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(9)!(4,5)(6,7)(8,9);
s1 := Sym(9)!(2,4)(3,6)(8,9);
s2 := Sym(9)!(1,2)(4,9)(5,8)(6,7);
s3 := Sym(9)!(2,3)(4,6)(5,7)(8,9);
poly := sub<Sym(9)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 >;

References

Notes following Theorem 11E7, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.