Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,4}

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

Overview

Group
SmallGroup(576,8399)
Rank
4
Schläfli Type
{4,4,4}
Vertices, edges, …
4, 36, 36, 18
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
2T4(2,0)(3,3), {{4,4|2},{4,4}6}. 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=<(s1*s2)^2> of order 2

10 facets

4 vertex figures

  • 4 of 2-fold non-regular quotient of {4,4}*144
P/N, where N=<(s2*s1*s2*s3)^2> of order 3

6 facets

4 vertex figures

  • 4 of 3-fold non-regular quotient of {4,4}*144
P/N, where N=<(s1*s2)^2, s1*s3*s2*s1*s2*s3> of order 6

4 facets

4 vertex figures

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

Representations

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

References

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

to this polytope.