Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,9}

Atlas Canonical Name {6,9}*1728

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,12249)
Rank
3
Schläfli Type
{6,9}
Vertices, edges, …
96, 432, 144
Order of s0s1s2
72
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

16-fold

36-fold

48-fold

72-fold

144-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)^2)^2*s2*(s1*s0)^2*s2*s1> of order 2

72 facets

48 vertex figures

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

54 facets

32 vertex figures

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

36 facets

24 vertex figures

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

36 facets

24 vertex figures

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

36 facets

24 vertex figures

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

36 facets

24 vertex figures

P/N, where N=<(s1*s0)^2*s2*s1*s0*s1*s2> of order 8

18 facets

12 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s0*s1)^3*s2*(s1*s0)^2*(s2*s1*s0)^2*s1*s2> of order 8

18 facets

12 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2*s1, s0*s1*(s2*(s1*s0)^2)^2*s2*s1*s0*s1*s2> of order 8

18 facets

12 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  3,  4)(  5,  6)(  9, 14)( 10, 13)( 11, 15)( 12, 16)( 19, 20)( 21, 22)( 25, 30)( 26, 29)( 27, 31)( 28, 32)( 35, 36)( 37, 38)( 41, 46)( 42, 45)( 43, 47)( 44, 48)( 51, 52)( 53, 54)( 57, 62)( 58, 61)( 59, 63)( 60, 64)( 67, 68)( 69, 70)( 73, 78)( 74, 77)( 75, 79)( 76, 80)( 83, 84)( 85, 86)( 89, 94)( 90, 93)( 91, 95)( 92, 96)( 99,100)(101,102)(105,110)(106,109)(107,111)(108,112)(115,116)(117,118)(121,126)(122,125)(123,127)(124,128)(131,132)(133,134)(137,142)(138,141)(139,143)(140,144);;
s1 := (  2,  4)(  5, 13)(  6, 16)(  7, 15)(  8, 14)(  9, 11)( 17, 33)( 18, 36)( 19, 35)( 20, 34)( 21, 45)( 22, 48)( 23, 47)( 24, 46)( 25, 43)( 26, 42)( 27, 41)( 28, 44)( 29, 37)( 30, 40)( 31, 39)( 32, 38)( 49,129)( 50,132)( 51,131)( 52,130)( 53,141)( 54,144)( 55,143)( 56,142)( 57,139)( 58,138)( 59,137)( 60,140)( 61,133)( 62,136)( 63,135)( 64,134)( 65,113)( 66,116)( 67,115)( 68,114)( 69,125)( 70,128)( 71,127)( 72,126)( 73,123)( 74,122)( 75,121)( 76,124)( 77,117)( 78,120)( 79,119)( 80,118)( 81, 97)( 82,100)( 83, 99)( 84, 98)( 85,109)( 86,112)( 87,111)( 88,110)( 89,107)( 90,106)( 91,105)( 92,108)( 93,101)( 94,104)( 95,103)( 96,102);;
s2 := (  1, 55)(  2, 56)(  3, 54)(  4, 53)(  5, 52)(  6, 51)(  7, 49)(  8, 50)(  9, 58)( 10, 57)( 11, 59)( 12, 60)( 13, 62)( 14, 61)( 15, 63)( 16, 64)( 17, 87)( 18, 88)( 19, 86)( 20, 85)( 21, 84)( 22, 83)( 23, 81)( 24, 82)( 25, 90)( 26, 89)( 27, 91)( 28, 92)( 29, 94)( 30, 93)( 31, 95)( 32, 96)( 33, 71)( 34, 72)( 35, 70)( 36, 69)( 37, 68)( 38, 67)( 39, 65)( 40, 66)( 41, 74)( 42, 73)( 43, 75)( 44, 76)( 45, 78)( 46, 77)( 47, 79)( 48, 80)( 97,135)( 98,136)( 99,134)(100,133)(101,132)(102,131)(103,129)(104,130)(105,138)(106,137)(107,139)(108,140)(109,142)(110,141)(111,143)(112,144)(113,119)(114,120)(115,118)(116,117)(121,122)(125,126);;
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*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(144)!(  3,  4)(  5,  6)(  9, 14)( 10, 13)( 11, 15)( 12, 16)( 19, 20)( 21, 22)( 25, 30)( 26, 29)( 27, 31)( 28, 32)( 35, 36)( 37, 38)( 41, 46)( 42, 45)( 43, 47)( 44, 48)( 51, 52)( 53, 54)( 57, 62)( 58, 61)( 59, 63)( 60, 64)( 67, 68)( 69, 70)( 73, 78)( 74, 77)( 75, 79)( 76, 80)( 83, 84)( 85, 86)( 89, 94)( 90, 93)( 91, 95)( 92, 96)( 99,100)(101,102)(105,110)(106,109)(107,111)(108,112)(115,116)(117,118)(121,126)(122,125)(123,127)(124,128)(131,132)(133,134)(137,142)(138,141)(139,143)(140,144);
s1 := Sym(144)!(  2,  4)(  5, 13)(  6, 16)(  7, 15)(  8, 14)(  9, 11)( 17, 33)( 18, 36)( 19, 35)( 20, 34)( 21, 45)( 22, 48)( 23, 47)( 24, 46)( 25, 43)( 26, 42)( 27, 41)( 28, 44)( 29, 37)( 30, 40)( 31, 39)( 32, 38)( 49,129)( 50,132)( 51,131)( 52,130)( 53,141)( 54,144)( 55,143)( 56,142)( 57,139)( 58,138)( 59,137)( 60,140)( 61,133)( 62,136)( 63,135)( 64,134)( 65,113)( 66,116)( 67,115)( 68,114)( 69,125)( 70,128)( 71,127)( 72,126)( 73,123)( 74,122)( 75,121)( 76,124)( 77,117)( 78,120)( 79,119)( 80,118)( 81, 97)( 82,100)( 83, 99)( 84, 98)( 85,109)( 86,112)( 87,111)( 88,110)( 89,107)( 90,106)( 91,105)( 92,108)( 93,101)( 94,104)( 95,103)( 96,102);
s2 := Sym(144)!(  1, 55)(  2, 56)(  3, 54)(  4, 53)(  5, 52)(  6, 51)(  7, 49)(  8, 50)(  9, 58)( 10, 57)( 11, 59)( 12, 60)( 13, 62)( 14, 61)( 15, 63)( 16, 64)( 17, 87)( 18, 88)( 19, 86)( 20, 85)( 21, 84)( 22, 83)( 23, 81)( 24, 82)( 25, 90)( 26, 89)( 27, 91)( 28, 92)( 29, 94)( 30, 93)( 31, 95)( 32, 96)( 33, 71)( 34, 72)( 35, 70)( 36, 69)( 37, 68)( 38, 67)( 39, 65)( 40, 66)( 41, 74)( 42, 73)( 43, 75)( 44, 76)( 45, 78)( 46, 77)( 47, 79)( 48, 80)( 97,135)( 98,136)( 99,134)(100,133)(101,132)(102,131)(103,129)(104,130)(105,138)(106,137)(107,139)(108,140)(109,142)(110,141)(111,143)(112,144)(113,119)(114,120)(115,118)(116,117)(121,122)(125,126);
poly := sub<Sym(144)|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*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

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