Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6,4}

Atlas Canonical Name {12,6,4}*1728f

Overview

Group
SmallGroup(1728,30394)
Rank
4
Schläfli Type
{12,6,4}
Vertices, edges, …
36, 108, 36, 4
Order of s0s1s2s3
4
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

12-fold

27-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*s0*s2*(s1*s0)^2*s2*s1> of order 3

4 facets

12 vertex figures

Representations

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

References

None.

to this polytope.