Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,1788)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
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

3-fold

4-fold

9-fold

12-fold

27-fold

54-fold

Covers minimal covers in bold

None in this atlas.

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*s0*s2*(s1*s0)^5*s1*s2*s1> of order 2

27 facets

60 vertex figures

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

36 facets

54 vertex figures

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

22 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

11 facets

24 vertex figures

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

9 facets

24 vertex figures

Representations

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

References

None.

to this polytope.

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