Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,12}

Atlas Canonical Name {4,4,12}*1728a

Overview

Group
SmallGroup(1728,30394)
Rank
4
Schläfli Type
{4,4,12}
Vertices, edges, …
4, 36, 108, 54
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

4-fold

6-fold

12-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> of order 2

30 facets

4 vertex figures

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

18 facets

4 vertex figures

P/N, where N=<(s1*s2)^2, (s1*s2*s3*s2)^2> of order 6

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

References

None.

to this polytope.