Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,27,4,2}

Atlas Canonical Name {2,27,4,2}*864

Overview

Group
SmallGroup(864,1916)
Rank
5
Schläfli Type
{2,27,4,2}
Vertices, edges, …
2, 27, 54, 4, 2
Order of s0s1s2s3s4
54
Order of s0s1s2s3s4s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

Covers minimal covers in bold

2-fold

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 95)( 44, 97)( 45, 96)( 46, 98)( 47, 91)( 48, 93)( 49, 92)( 50, 94)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 59, 79)( 60, 81)( 61, 80)( 62, 82)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)( 70,102)( 71,107)( 72,109)( 73,108)( 74,110);;
s2 := (  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 47)(  8, 48)(  9, 50)( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)( 18, 69)( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)( 26, 73)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)( 34, 53)( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 75, 87)( 76, 88)( 77, 90)( 78, 89)( 79, 95)( 80, 96)( 81, 98)( 82, 97)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 99,103)(100,104)(101,106)(102,105)(109,110);;
s3 := (  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109);;
s4 := (111,112);;
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, 
s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*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 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(112)!(1,2);
s1 := Sym(112)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 95)( 44, 97)( 45, 96)( 46, 98)( 47, 91)( 48, 93)( 49, 92)( 50, 94)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 59, 79)( 60, 81)( 61, 80)( 62, 82)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)( 70,102)( 71,107)( 72,109)( 73,108)( 74,110);
s2 := Sym(112)!(  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 47)(  8, 48)(  9, 50)( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)( 18, 69)( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)( 26, 73)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)( 34, 53)( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 75, 87)( 76, 88)( 77, 90)( 78, 89)( 79, 95)( 80, 96)( 81, 98)( 82, 97)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 99,103)(100,104)(101,106)(102,105)(109,110);
s3 := Sym(112)!(  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109);
s4 := Sym(112)!(111,112);
poly := sub<Sym(112)|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, s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*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 >;