Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,4,4}

Atlas Canonical Name {6,4,4}*1728b

Overview

Group
SmallGroup(1728,46671)
Rank
4
Schläfli Type
{6,4,4}
Vertices, edges, …
6, 108, 72, 36
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
{{6,4|2},{4,4|6}}. if this polytope has another name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

12-fold

18-fold

27-fold

36-fold

54-fold

72-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*s2)^2*(s3*s2*s1*s2)^2*s3> of order 2

18 facets

6 vertex figures

  • 6 of 2-fold non-regular quotient of {4,4}*288
P/N, where N=<s2*(s3*s2*s1)^4*s3*s2*s3> of order 2

18 facets

6 vertex figures

  • 6 of 2-fold non-regular quotient of {4,4}*288
P/N, where N=<(s1*s2)^2> of order 2

20 facets

6 vertex figures

  • 6 of 2-fold non-regular quotient of {4,4}*288
P/N, where N=<s1*(s2*s1*s3)^3*s2*s3> of order 3

12 facets

6 vertex figures

  • 6 of 3-fold non-regular quotient of {4,4}*288
P/N, where N=<(s1*s2*s3*s2)^2> of order 3

12 facets

6 vertex figures

  • 6 of 3-fold non-regular quotient of {4,4}*288
P/N, where N=<(s1*s2)^2*(s3*s2*s1*s2)^2*s3, s1*s2*s1*(s3*s2*s1*s2)^2*s3*s2> of order 4

9 facets

6 vertex figures

  • 6 of 4-fold non-regular quotient of {4,4}*288
P/N, where N=<(s1*s2*s3*s2)^2, (s1*s2)^2*(s3*s2*s1*s2)^2*s3> of order 6

6 facets

6 vertex figures

  • 6 of 6-fold non-regular quotient of {4,4}*288
P/N, where N=<(s1*s2)^2, (s1*s2*s3*s2)^2> of order 6

8 facets

6 vertex figures

  • 6 of 6-fold non-regular quotient of {4,4}*288
P/N, where N=<(s2*s3)^2, (s1*s2)^2*(s3*s2*s1)^2> of order 6

6 facets

6 vertex figures

  • 6 of 6-fold non-regular quotient of {4,4}*288
P/N, where N=<(s1*s2*s3*s2)^2, s3*s2*s1*s2*s3*s2*s1*s3*s2*s3> of order 6

6 facets

6 vertex figures

  • 6 of 6-fold non-regular quotient of {4,4}*288
P/N, where N=<(s1*s2)^2, s2*(s3*s2*s1)^2*s3*s2*s3> of order 6

8 facets

6 vertex figures

  • 6 of 6-fold non-regular quotient of {4,4}*288

Representations

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

References

None.

to this polytope.