Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,12}

Atlas Canonical Name {12,12}*864k

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(864,4321)
Rank
3
Schläfli Type
{12,12}
Vertices, edges, …
36, 216, 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
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

27-fold

36-fold

54-fold

72-fold

108-fold

Covers minimal covers in bold

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

18 facets

18 vertex figures

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

12 facets

24 vertex figures

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

12 facets

12 vertex figures

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

12 facets

12 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  2,  3)(  4, 22)(  5, 24)(  6, 23)(  7, 16)(  8, 18)(  9, 17)( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 26, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)( 34, 43)( 35, 45)( 36, 44)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 53, 54)( 56, 57)( 58, 76)( 59, 78)( 60, 77)( 61, 70)( 62, 72)( 63, 71)( 64, 73)( 65, 75)( 66, 74)( 68, 69)( 80, 81)( 83, 84)( 85,103)( 86,105)( 87,104)( 88, 97)( 89, 99)( 90, 98)( 91,100)( 92,102)( 93,101)( 95, 96)(107,108);;
s1 := (  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 14)( 11, 13)( 12, 15)( 16, 17)( 19, 26)( 20, 25)( 21, 27)( 22, 23)( 28, 29)( 31, 35)( 32, 34)( 33, 36)( 37, 41)( 38, 40)( 39, 42)( 43, 44)( 46, 53)( 47, 52)( 48, 54)( 49, 50)( 55, 83)( 56, 82)( 57, 84)( 58, 89)( 59, 88)( 60, 90)( 61, 86)( 62, 85)( 63, 87)( 64, 95)( 65, 94)( 66, 96)( 67, 92)( 68, 91)( 69, 93)( 70, 98)( 71, 97)( 72, 99)( 73,107)( 74,106)( 75,108)( 76,104)( 77,103)( 78,105)( 79,101)( 80,100)( 81,102);;
s2 := (  1, 67)(  2, 68)(  3, 69)(  4, 64)(  5, 65)(  6, 66)(  7, 70)(  8, 71)(  9, 72)( 10, 58)( 11, 59)( 12, 60)( 13, 55)( 14, 56)( 15, 57)( 16, 61)( 17, 62)( 18, 63)( 19, 76)( 20, 77)( 21, 78)( 22, 73)( 23, 74)( 24, 75)( 25, 79)( 26, 80)( 27, 81)( 28, 94)( 29, 95)( 30, 96)( 31, 91)( 32, 92)( 33, 93)( 34, 97)( 35, 98)( 36, 99)( 37, 85)( 38, 86)( 39, 87)( 40, 82)( 41, 83)( 42, 84)( 43, 88)( 44, 89)( 45, 90)( 46,103)( 47,104)( 48,105)( 49,100)( 50,101)( 51,102)( 52,106)( 53,107)( 54,108);;
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*s2*s1*s2*s0*s1*s2*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*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(108)!(  2,  3)(  4, 22)(  5, 24)(  6, 23)(  7, 16)(  8, 18)(  9, 17)( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 26, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)( 34, 43)( 35, 45)( 36, 44)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 53, 54)( 56, 57)( 58, 76)( 59, 78)( 60, 77)( 61, 70)( 62, 72)( 63, 71)( 64, 73)( 65, 75)( 66, 74)( 68, 69)( 80, 81)( 83, 84)( 85,103)( 86,105)( 87,104)( 88, 97)( 89, 99)( 90, 98)( 91,100)( 92,102)( 93,101)( 95, 96)(107,108);
s1 := Sym(108)!(  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 14)( 11, 13)( 12, 15)( 16, 17)( 19, 26)( 20, 25)( 21, 27)( 22, 23)( 28, 29)( 31, 35)( 32, 34)( 33, 36)( 37, 41)( 38, 40)( 39, 42)( 43, 44)( 46, 53)( 47, 52)( 48, 54)( 49, 50)( 55, 83)( 56, 82)( 57, 84)( 58, 89)( 59, 88)( 60, 90)( 61, 86)( 62, 85)( 63, 87)( 64, 95)( 65, 94)( 66, 96)( 67, 92)( 68, 91)( 69, 93)( 70, 98)( 71, 97)( 72, 99)( 73,107)( 74,106)( 75,108)( 76,104)( 77,103)( 78,105)( 79,101)( 80,100)( 81,102);
s2 := Sym(108)!(  1, 67)(  2, 68)(  3, 69)(  4, 64)(  5, 65)(  6, 66)(  7, 70)(  8, 71)(  9, 72)( 10, 58)( 11, 59)( 12, 60)( 13, 55)( 14, 56)( 15, 57)( 16, 61)( 17, 62)( 18, 63)( 19, 76)( 20, 77)( 21, 78)( 22, 73)( 23, 74)( 24, 75)( 25, 79)( 26, 80)( 27, 81)( 28, 94)( 29, 95)( 30, 96)( 31, 91)( 32, 92)( 33, 93)( 34, 97)( 35, 98)( 36, 99)( 37, 85)( 38, 86)( 39, 87)( 40, 82)( 41, 83)( 42, 84)( 43, 88)( 44, 89)( 45, 90)( 46,103)( 47,104)( 48,105)( 49,100)( 50,101)( 51,102)( 52,106)( 53,107)( 54,108);
poly := sub<Sym(108)|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*s2*s1*s2*s0*s1*s2*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*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

Twisty Puzzle