Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,12,6}

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

Overview

Group
SmallGroup(1728,46671)
Rank
4
Schläfli Type
{4,12,6}
Vertices, edges, …
12, 72, 108, 6
Order of s0s1s2s3
12
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

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=<s0*(s1*s2)^3*s1*s0*s2> of order 3

6 facets

8 vertex figures

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

6 facets

4 vertex figures

Representations

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

References

None.

to this polytope.