Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,4}

Atlas Canonical Name {12,4}*432a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(432,530)
Rank
3
Schläfli Type
{12,4}
Vertices, edges, …
54, 108, 18
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

2-fold

3-fold

6-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=<s1*s0*(s2*s1*s0*s1)^2*s2> of order 2

9 facets

27 vertex figures

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

10 facets

30 vertex figures

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

6 facets

18 vertex figures

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

5 facets

15 vertex figures

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

4 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle