Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6,12}

Atlas Canonical Name {6,6,12}*864g

Overview

Group
SmallGroup(864,4406)
Rank
4
Schläfli Type
{6,6,12}
Vertices, edges, …
6, 18, 36, 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

4-fold

6-fold

9-fold

12-fold

18-fold

27-fold

36-fold

54-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.