Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,8}

Atlas Canonical Name {4,6,8}*1920

Overview

Group
SmallGroup(1920,240560)
Rank
4
Schläfli Type
{4,6,8}
Vertices, edges, …
20, 60, 120, 8
Order of s0s1s2s3
40
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

Covers minimal covers in bold

None in this atlas.

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*(s1*s2)^2> of order 3

8 facets

8 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 5)( 6,10)( 7,11)( 8,12)( 9,13)(14,18)(15,19)(16,20)(17,21)(22,26)(23,27)(24,28)(25,29)(30,34)(31,35)(32,36)(33,37);;
s1 := (2,3)(4,5);;
s2 := ( 1, 2)( 6,14)( 7,15)( 8,17)( 9,16)(10,18)(11,19)(12,21)(13,20)(22,30)(23,31)(24,33)(25,32)(26,34)(27,35)(28,37)(29,36);;
s3 := ( 6,26)( 7,27)( 8,29)( 9,28)(10,22)(11,23)(12,25)(13,24)(14,36)(15,37)(16,34)(17,35)(18,32)(19,33)(20,30)(21,31);;
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*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(37)!( 3, 5)( 6,10)( 7,11)( 8,12)( 9,13)(14,18)(15,19)(16,20)(17,21)(22,26)(23,27)(24,28)(25,29)(30,34)(31,35)(32,36)(33,37);
s1 := Sym(37)!(2,3)(4,5);
s2 := Sym(37)!( 1, 2)( 6,14)( 7,15)( 8,17)( 9,16)(10,18)(11,19)(12,21)(13,20)(22,30)(23,31)(24,33)(25,32)(26,34)(27,35)(28,37)(29,36);
s3 := Sym(37)!( 6,26)( 7,27)( 8,29)( 9,28)(10,22)(11,23)(12,25)(13,24)(14,36)(15,37)(16,34)(17,35)(18,32)(19,33)(20,30)(21,31);
poly := sub<Sym(37)|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*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 >;

References

None.

to this polytope.