Part of the Atlas of Small Regular Polytopes

Polytope of Type {9,6}

Atlas Canonical Name {9,6}*1296b

▶ Play as a twisty puzzle

Overview

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

36 facets

54 vertex figures

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

24 facets

42 vertex figures

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

24 facets

36 vertex figures

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

18 facets

27 vertex figures

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

18 facets

27 vertex figures

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

12 facets

18 vertex figures

Representations

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

References

None.

to this polytope.

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