Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,2,4,6,9}

Atlas Canonical Name {2,2,4,6,9}*1728

Overview

Group
SmallGroup(1728,30872)
Rank
6
Schläfli Type
{2,2,4,6,9}
Vertices, edges, …
2, 2, 4, 12, 27, 9
Order of s0s1s2s3s4s5
36
Order of s0s1s2s3s4s5s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

18-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (3,4);;
s2 := (  5, 59)(  6, 60)(  7, 61)(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)( 57,111)( 58,112);;
s3 := (  8, 11)(  9, 12)( 10, 13)( 17, 20)( 18, 21)( 19, 22)( 26, 29)( 27, 30)( 28, 31)( 35, 38)( 36, 39)( 37, 40)( 44, 47)( 45, 48)( 46, 49)( 53, 56)( 54, 57)( 55, 58)( 59, 86)( 60, 87)( 61, 88)( 62, 92)( 63, 93)( 64, 94)( 65, 89)( 66, 90)( 67, 91)( 68, 95)( 69, 96)( 70, 97)( 71,101)( 72,102)( 73,103)( 74, 98)( 75, 99)( 76,100)( 77,104)( 78,105)( 79,106)( 80,110)( 81,111)( 82,112)( 83,107)( 84,108)( 85,109);;
s4 := (  5,  8)(  6, 10)(  7,  9)( 12, 13)( 14, 27)( 15, 26)( 16, 28)( 17, 24)( 18, 23)( 19, 25)( 20, 30)( 21, 29)( 22, 31)( 32, 35)( 33, 37)( 34, 36)( 39, 40)( 41, 54)( 42, 53)( 43, 55)( 44, 51)( 45, 50)( 46, 52)( 47, 57)( 48, 56)( 49, 58)( 59, 62)( 60, 64)( 61, 63)( 66, 67)( 68, 81)( 69, 80)( 70, 82)( 71, 78)( 72, 77)( 73, 79)( 74, 84)( 75, 83)( 76, 85)( 86, 89)( 87, 91)( 88, 90)( 93, 94)( 95,108)( 96,107)( 97,109)( 98,105)( 99,104)(100,106)(101,111)(102,110)(103,112);;
s5 := (  5, 14)(  6, 16)(  7, 15)(  8, 20)(  9, 22)( 10, 21)( 11, 17)( 12, 19)( 13, 18)( 23, 24)( 26, 30)( 27, 29)( 28, 31)( 32, 41)( 33, 43)( 34, 42)( 35, 47)( 36, 49)( 37, 48)( 38, 44)( 39, 46)( 40, 45)( 50, 51)( 53, 57)( 54, 56)( 55, 58)( 59, 68)( 60, 70)( 61, 69)( 62, 74)( 63, 76)( 64, 75)( 65, 71)( 66, 73)( 67, 72)( 77, 78)( 80, 84)( 81, 83)( 82, 85)( 86, 95)( 87, 97)( 88, 96)( 89,101)( 90,103)( 91,102)( 92, 98)( 93,100)( 94, 99)(104,105)(107,111)(108,110)(109,112);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s5*s3*s4*s3*s4*s5*s3*s4, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(112)!(1,2);
s1 := Sym(112)!(3,4);
s2 := Sym(112)!(  5, 59)(  6, 60)(  7, 61)(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)( 57,111)( 58,112);
s3 := Sym(112)!(  8, 11)(  9, 12)( 10, 13)( 17, 20)( 18, 21)( 19, 22)( 26, 29)( 27, 30)( 28, 31)( 35, 38)( 36, 39)( 37, 40)( 44, 47)( 45, 48)( 46, 49)( 53, 56)( 54, 57)( 55, 58)( 59, 86)( 60, 87)( 61, 88)( 62, 92)( 63, 93)( 64, 94)( 65, 89)( 66, 90)( 67, 91)( 68, 95)( 69, 96)( 70, 97)( 71,101)( 72,102)( 73,103)( 74, 98)( 75, 99)( 76,100)( 77,104)( 78,105)( 79,106)( 80,110)( 81,111)( 82,112)( 83,107)( 84,108)( 85,109);
s4 := Sym(112)!(  5,  8)(  6, 10)(  7,  9)( 12, 13)( 14, 27)( 15, 26)( 16, 28)( 17, 24)( 18, 23)( 19, 25)( 20, 30)( 21, 29)( 22, 31)( 32, 35)( 33, 37)( 34, 36)( 39, 40)( 41, 54)( 42, 53)( 43, 55)( 44, 51)( 45, 50)( 46, 52)( 47, 57)( 48, 56)( 49, 58)( 59, 62)( 60, 64)( 61, 63)( 66, 67)( 68, 81)( 69, 80)( 70, 82)( 71, 78)( 72, 77)( 73, 79)( 74, 84)( 75, 83)( 76, 85)( 86, 89)( 87, 91)( 88, 90)( 93, 94)( 95,108)( 96,107)( 97,109)( 98,105)( 99,104)(100,106)(101,111)(102,110)(103,112);
s5 := Sym(112)!(  5, 14)(  6, 16)(  7, 15)(  8, 20)(  9, 22)( 10, 21)( 11, 17)( 12, 19)( 13, 18)( 23, 24)( 26, 30)( 27, 29)( 28, 31)( 32, 41)( 33, 43)( 34, 42)( 35, 47)( 36, 49)( 37, 48)( 38, 44)( 39, 46)( 40, 45)( 50, 51)( 53, 57)( 54, 56)( 55, 58)( 59, 68)( 60, 70)( 61, 69)( 62, 74)( 63, 76)( 64, 75)( 65, 71)( 66, 73)( 67, 72)( 77, 78)( 80, 84)( 81, 83)( 82, 85)( 86, 95)( 87, 97)( 88, 96)( 89,101)( 90,103)( 91,102)( 92, 98)( 93,100)( 94, 99)(104,105)(107,111)(108,110)(109,112);
poly := sub<Sym(112)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5, 
s3*s5*s3*s5, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s5*s3*s4*s3*s4*s5*s3*s4, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;