Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,54,2}

Atlas Canonical Name {4,54,2}*864a

Overview

Group
SmallGroup(864,635)
Rank
4
Schläfli Type
{4,54,2}
Vertices, edges, …
4, 108, 54, 2
Order of s0s1s2s3
108
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 := ( 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, 62)(  5, 61)(  6, 63)(  7, 59)(  8, 58)(  9, 60)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)( 16, 80)( 17, 79)( 18, 81)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)( 24, 65)( 25, 71)( 26, 70)( 27, 72)( 28, 82)( 29, 84)( 30, 83)( 31, 89)( 32, 88)( 33, 90)( 34, 86)( 35, 85)( 36, 87)( 37,103)( 38,105)( 39,104)( 40,100)( 41,102)( 42,101)( 43,107)( 44,106)( 45,108)( 46, 94)( 47, 96)( 48, 95)( 49, 91)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 99);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 17)(  5, 16)(  6, 18)(  7, 14)(  8, 13)(  9, 15)( 19, 22)( 20, 24)( 21, 23)( 25, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 49)( 47, 51)( 48, 50)( 52, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 76)( 74, 78)( 75, 77)( 79, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)(100,103)(101,105)(102,104)(106,107);;
s3 := (109,110);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*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*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*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)!( 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(110)!(  1, 55)(  2, 57)(  3, 56)(  4, 62)(  5, 61)(  6, 63)(  7, 59)(  8, 58)(  9, 60)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)( 16, 80)( 17, 79)( 18, 81)( 19, 67)( 20, 69)( 21, 68)( 22, 64)( 23, 66)( 24, 65)( 25, 71)( 26, 70)( 27, 72)( 28, 82)( 29, 84)( 30, 83)( 31, 89)( 32, 88)( 33, 90)( 34, 86)( 35, 85)( 36, 87)( 37,103)( 38,105)( 39,104)( 40,100)( 41,102)( 42,101)( 43,107)( 44,106)( 45,108)( 46, 94)( 47, 96)( 48, 95)( 49, 91)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 99);
s2 := Sym(110)!(  1, 10)(  2, 12)(  3, 11)(  4, 17)(  5, 16)(  6, 18)(  7, 14)(  8, 13)(  9, 15)( 19, 22)( 20, 24)( 21, 23)( 25, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 49)( 47, 51)( 48, 50)( 52, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 76)( 74, 78)( 75, 77)( 79, 80)( 82, 91)( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)(100,103)(101,105)(102,104)(106,107);
s3 := Sym(110)!(109,110);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*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*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*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 >;