Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,12}

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

Overview

Group
SmallGroup(1728,30394)
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

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

12 facets

4 vertex figures

Representations

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

References

None.

to this polytope.