Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,9}

Atlas Canonical Name {6,9}*1296d

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,1790)
Rank
3
Schläfli Type
{6,9}
Vertices, edges, …
72, 324, 108
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

3-fold

4-fold

9-fold

12-fold

27-fold

36-fold

54-fold

108-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=<((s1*s0)^2*s1*s2)^3> of order 2

54 facets

36 vertex figures

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

36 facets

24 vertex figures

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

36 facets

24 vertex figures

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

36 facets

24 vertex figures

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

36 facets

24 vertex figures

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

27 facets

18 vertex figures

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

27 facets

18 vertex figures

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

18 facets

12 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1, s0*s1*s2*(s1*s0)^2*s1*(s2*s1*s0)^3*s1> of order 6

18 facets

12 vertex figures

P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, s0*s1*(s2*(s1*s0)^2)^2*s2*s1*s0*s1> of order 6

18 facets

12 vertex figures

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

18 facets

12 vertex figures

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

18 facets

12 vertex figures

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

12 facets

8 vertex figures

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

12 facets

8 vertex figures

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

12 facets

8 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s1*s0)^2*s2*s1*s0*(s2*s1)^2> of order 12

9 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle