Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,9}

Atlas Canonical Name {6,9}*1296b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,1785)
Rank
3
Schläfli Type
{6,9}
Vertices, edges, …
72, 324, 108
Order of s0s1s2
36
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*(s1*s0)^2*(s2*s1*s0)^2*s1*s2> of order 2

54 facets

36 vertex figures

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

42 facets

24 vertex figures

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

36 facets

24 vertex figures

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

27 facets

18 vertex figures

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

27 facets

18 vertex figures

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

18 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

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