Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,12,6}

Atlas Canonical Name {2,6,12,6}*1728f

Overview

Group
SmallGroup(1728,47409)
Rank
5
Schläfli Type
{2,6,12,6}
Vertices, edges, …
2, 6, 36, 36, 6
Order of s0s1s2s3s4
12
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

9-fold

12-fold

18-fold

24-fold

27-fold

36-fold

54-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)( 70, 82)( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)( 93,102)( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)(101,107);;
s2 := (  3, 69)(  4, 70)(  5, 71)(  6, 66)(  7, 67)(  8, 68)(  9, 72)( 10, 73)( 11, 74)( 12, 60)( 13, 61)( 14, 62)( 15, 57)( 16, 58)( 17, 59)( 18, 63)( 19, 64)( 20, 65)( 21, 78)( 22, 79)( 23, 80)( 24, 75)( 25, 76)( 26, 77)( 27, 81)( 28, 82)( 29, 83)( 30, 96)( 31, 97)( 32, 98)( 33, 93)( 34, 94)( 35, 95)( 36, 99)( 37,100)( 38,101)( 39, 87)( 40, 88)( 41, 89)( 42, 84)( 43, 85)( 44, 86)( 45, 90)( 46, 91)( 47, 92)( 48,105)( 49,106)( 50,107)( 51,102)( 52,103)( 53,104)( 54,108)( 55,109)( 56,110);;
s3 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)( 81,105)( 82,107)( 83,106);;
s4 := (  3,  4)(  6,  7)(  9, 10)( 12, 13)( 15, 16)( 18, 19)( 21, 22)( 24, 25)( 27, 28)( 30, 31)( 33, 34)( 36, 37)( 39, 40)( 42, 43)( 45, 46)( 48, 49)( 51, 52)( 54, 55)( 57, 58)( 60, 61)( 63, 64)( 66, 67)( 69, 70)( 72, 73)( 75, 76)( 78, 79)( 81, 82)( 84, 85)( 87, 88)( 90, 91)( 93, 94)( 96, 97)( 99,100)(102,103)(105,106)(108,109);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)( 70, 82)( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)( 93,102)( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)(101,107);
s2 := Sym(110)!(  3, 69)(  4, 70)(  5, 71)(  6, 66)(  7, 67)(  8, 68)(  9, 72)( 10, 73)( 11, 74)( 12, 60)( 13, 61)( 14, 62)( 15, 57)( 16, 58)( 17, 59)( 18, 63)( 19, 64)( 20, 65)( 21, 78)( 22, 79)( 23, 80)( 24, 75)( 25, 76)( 26, 77)( 27, 81)( 28, 82)( 29, 83)( 30, 96)( 31, 97)( 32, 98)( 33, 93)( 34, 94)( 35, 95)( 36, 99)( 37,100)( 38,101)( 39, 87)( 40, 88)( 41, 89)( 42, 84)( 43, 85)( 44, 86)( 45, 90)( 46, 91)( 47, 92)( 48,105)( 49,106)( 50,107)( 51,102)( 52,103)( 53,104)( 54,108)( 55,109)( 56,110);
s3 := Sym(110)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)( 81,105)( 82,107)( 83,106);
s4 := Sym(110)!(  3,  4)(  6,  7)(  9, 10)( 12, 13)( 15, 16)( 18, 19)( 21, 22)( 24, 25)( 27, 28)( 30, 31)( 33, 34)( 36, 37)( 39, 40)( 42, 43)( 45, 46)( 48, 49)( 51, 52)( 54, 55)( 57, 58)( 60, 61)( 63, 64)( 66, 67)( 69, 70)( 72, 73)( 75, 76)( 78, 79)( 81, 82)( 84, 85)( 87, 88)( 90, 91)( 93, 94)( 96, 97)( 99,100)(102,103)(105,106)(108,109);
poly := sub<Sym(110)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;