Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,3}

Atlas Canonical Name {12,3}*432

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(432,523)
Rank
3
Schläfli Type
{12,3}
Vertices, edges, …
72, 108, 18
Order of s0s1s2
6
Order of s0s1s2s1
12
Also known as
{12,3}6. if this polytope has another name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

18-fold

36-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*s0)^5*s1*s2> of order 2

12 facets

36 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^5*s1*s2*s1> of order 2

9 facets

36 vertex figures

P/N, where N=<(s0*s1)^4> of order 3

10 facets

24 vertex figures

P/N, where N=<(s0*s1)^4, s0*s2*(s1*s0)^5*s1*s2> of order 6

6 facets

12 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2> of order 6

8 facets

12 vertex figures

P/N, where N=<(s0*s1)^4, s1*s2*(s1*s0)^3*s2*(s1*s0)^2*s1*s2> of order 6

5 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

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