Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,12}

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

Overview

Group
SmallGroup(1728,46674)
Rank
4
Schläfli Type
{4,6,12}
Vertices, edges, …
4, 36, 108, 36
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

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=<(s1*s2)^2*s1*s3*s2*s1*s2*s3> of order 3

12 facets

4 vertex figures

P/N, where N=<(s1*s2*s3*s2)^2> of order 3

12 facets

4 vertex figures

Representations

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

References

None.

to this polytope.