Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,4}

Atlas Canonical Name {4,4,4}*576a

Overview

Group
SmallGroup(576,8399)
Rank
4
Schläfli Type
{4,4,4}
Vertices, edges, …
18, 36, 36, 4
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
2T4(3,3)(2,0), {{4,4}6,{4,4|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

18-fold

36-fold

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

4 facets

  • 4 of 2-fold non-regular quotient of {4,4}*144

10 vertex figures

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

4 facets

  • 4 of 3-fold non-regular quotient of {4,4}*144

6 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2> of order 6

4 facets

  • 4 of 6-fold non-regular quotient of {4,4}*144

4 vertex figures

Representations

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

References

  1. Theorem 10C2, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.