Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,12}

Atlas Canonical Name {4,12}*432b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(432,741)
Rank
3
Schläfli Type
{4,12}
Vertices, edges, …
18, 108, 54
Order of s0s1s2
6
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

6-fold

18-fold

36-fold

54-fold

Covers minimal covers in bold

2-fold

3-fold

4-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*(s2*s1)^2)^2*s0*s1*s2> of order 2

27 facets

9 vertex figures

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

27 facets

9 vertex figures

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

30 facets

10 vertex figures

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

18 facets

6 vertex figures

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

18 facets

6 vertex figures

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

15 facets

5 vertex figures

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

12 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (4,7)(5,8)(6,9);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,17)(11,16)(12,18)(13,14);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
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(18)!(4,7)(5,8)(6,9);
s1 := Sym(18)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17);
s2 := Sym(18)!( 1, 2)( 4, 5)( 7, 8)(10,17)(11,16)(12,18)(13,14);
poly := sub<Sym(18)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

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