Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*432i

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(432,756)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
18, 108, 36
Order of s0s1s2
12
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

2-fold

3-fold

6-fold

9-fold

18-fold

27-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*s1)^2> of order 3

24 facets

6 vertex figures

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

12 facets

6 vertex figures

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

12 facets

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,31)(29,33)(30,32)(35,36)(37,40)(38,42)(39,41)(44,45)(46,49)(47,51)(48,50)(53,54);;
s2 := ( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);;
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*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(54)!( 1,28)( 2,29)( 3,30)( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,46)(11,47)(12,48)(13,52)(14,53)(15,54)(16,49)(17,50)(18,51)(19,37)(20,38)(21,39)(22,43)(23,44)(24,45)(25,40)(26,41)(27,42);
s1 := Sym(54)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,31)(29,33)(30,32)(35,36)(37,40)(38,42)(39,41)(44,45)(46,49)(47,51)(48,50)(53,54);
s2 := Sym(54)!( 1, 2)( 4,11)( 5,10)( 6,12)( 7,20)( 8,19)( 9,21)(13,14)(16,23)(17,22)(18,24)(25,26)(28,29)(31,38)(32,37)(33,39)(34,47)(35,46)(36,48)(40,41)(43,50)(44,49)(45,51)(52,53);
poly := sub<Sym(54)|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*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.

Twisty Puzzle