Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,3}

Atlas Canonical Name {12,3}*648

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(648,703)
Rank
3
Schläfli Type
{12,3}
Vertices, edges, …
108, 162, 27
Order of s0s1s2
9
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

27-fold

Covers minimal covers in bold

2-fold

3-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)^4> of order 3

15 facets

36 vertex figures

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

9 facets

36 vertex figures

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

9 facets

36 vertex figures

P/N, where N=<s0*s1*s2*(s1*s0)^2*s2*s1, (s0*s1)^3*s2*(s1*s0)^3*s1*s2*s1> of order 9

3 facets

12 vertex figures

P/N, where N=<(s0*s1)^4, s0*s1*s2*(s1*s0)^2*s2*s1> of order 9

5 facets

12 vertex figures

P/N, where N=<(s0*s1)^4, s1*s0*s2*(s1*s0)^3*s1*s2*s1> of order 9

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

References

None.

to this polytope.

Twisty Puzzle