Polytope of Type {2,4,8} with group SmallGroup(128,1728)

Overview

Group: SmallGroup(128,1728)
Rank: 4
Schläfli Type: {2,4,8}
Vertices, edges, …: 2, 4, 16, 8
Order of s₀s₁s₂s₃: 8
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Degenerate
Universal
Orientable
Flat

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

{2,2,2}*16

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

9-fold

10-fold

11-fold

13-fold

14-fold

15-fold

Representations

Permutation Representation (GAP)

s0 := (1,2);;
s1 := ( 4, 6)( 5, 8)(12,15)(14,17);;
s2 := ( 3, 4)( 5, 7)( 6, 9)( 8,11)(10,12)(13,15)(14,16)(17,18);;
s3 := ( 4, 5)( 6, 8)( 7,10)(11,13)(12,14)(15,17);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(18)!(1,2);
s1 := Sym(18)!( 4, 6)( 5, 8)(12,15)(14,17);
s2 := Sym(18)!( 3, 4)( 5, 7)( 6, 9)( 8,11)(10,12)(13,15)(14,16)(17,18);
s3 := Sym(18)!( 4, 5)( 6, 8)( 7,10)(11,13)(12,14)(15,17);
poly := sub<Sym(18)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;