Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,4,4}

Atlas Canonical Name {3,4,4}*192b

Overview

Group
SmallGroup(192,1472)
Rank
4
Schläfli Type
{3,4,4}
Vertices, edges, …
6, 12, 16, 4
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
Dual of 1T4(2,0), {{3,4},{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

8-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

9-fold

10-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

4 facets

  • 4 of 2-fold non-regular quotient of {3,4}*48

4 vertex figures

Representations

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

References

  1. Theorem 10B3, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.