Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,4}

Atlas Canonical Name {4,4,4}*1152f

Overview

Group
SmallGroup(1152,157849)
Rank
4
Schläfli Type
{4,4,4}
Vertices, edges, …
16, 72, 72, 9
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Locally Toroidal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

No regular quotients.

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)^3*s1*s2> of order 2

9 facets

  • 6 of 2-fold non-regular quotient of {4,4}*128
  • 3 of 2-fold non-regular quotient of {4,4}*128

8 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s2*s1*s0*s1*s2> of order 4

9 facets

  • 9 of 4-fold non-regular quotient of {4,4}*128

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 1,11)( 2,12)( 3, 9)( 4,10)( 5,15)( 6,16)( 7,13)( 8,14);;
s1 := ( 2, 5)( 3,13)( 4, 9)( 7,14)( 8,10)(11,16);;
s2 := ( 5,13)( 6,14)( 7,15)( 8,16);;
s3 := ( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15);;
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s1*s0*s1*s2*s0*s3*s2*s1*s0*s1*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(16)!( 1,11)( 2,12)( 3, 9)( 4,10)( 5,15)( 6,16)( 7,13)( 8,14);
s1 := Sym(16)!( 2, 5)( 3,13)( 4, 9)( 7,14)( 8,10)(11,16);
s2 := Sym(16)!( 5,13)( 6,14)( 7,15)( 8,16);
s3 := Sym(16)!( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15);
poly := sub<Sym(16)|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, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s1*s0*s1*s2*s0*s3*s2*s1*s0*s1*s2*s3 >;

References

None.

to this polytope.