Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,36,6}

Atlas Canonical Name {2,36,6}*864b

Overview

Group
SmallGroup(864,2437)
Rank
4
Schläfli Type
{2,36,6}
Vertices, edges, …
2, 36, 108, 6
Order of s0s1s2s3
36
Order of s0s1s2s3s2s1
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

27-fold

36-fold

54-fold

Covers minimal covers in bold

2-fold

Representations

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