Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12,4}

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

Overview

Group
SmallGroup(1728,46671)
Rank
4
Schläfli Type
{6,12,4}
Vertices, edges, …
6, 108, 72, 12
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=<(s1*s2)^4> of order 3

8 facets

6 vertex figures

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

4 facets

6 vertex figures

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

References

None.

to this polytope.