Part of the Atlas of Small Regular Polytopes

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

Atlas Canonical Name {2,2,2,4,27}*1728

Overview

Group
SmallGroup(1728,20782)
Rank
6
Schläfli Type
{2,2,2,4,27}
Vertices, edges, …
2, 2, 2, 4, 54, 27
Order of s0s1s2s3s4s5
54
Order of s0s1s2s3s4s5s4s3s2s1
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

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (3,4);;
s2 := (5,6);;
s3 := (  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114);;
s4 := (  8,  9)( 11, 15)( 12, 17)( 13, 16)( 14, 18)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)( 25, 32)( 26, 34)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 99)( 48,101)( 49,100)( 50,102)( 51, 95)( 52, 97)( 53, 96)( 54, 98)( 55, 79)( 56, 81)( 57, 80)( 58, 82)( 59, 87)( 60, 89)( 61, 88)( 62, 90)( 63, 83)( 64, 85)( 65, 84)( 66, 86)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)( 72,105)( 73,104)( 74,106)( 75,111)( 76,113)( 77,112)( 78,114);;
s5 := (  7, 43)(  8, 44)(  9, 46)( 10, 45)( 11, 51)( 12, 52)( 13, 54)( 14, 53)( 15, 47)( 16, 48)( 17, 50)( 18, 49)( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)( 26, 69)( 27, 75)( 28, 76)( 29, 78)( 30, 77)( 31, 59)( 32, 60)( 33, 62)( 34, 61)( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 39, 63)( 40, 64)( 41, 66)( 42, 65)( 79, 91)( 80, 92)( 81, 94)( 82, 93)( 83, 99)( 84,100)( 85,102)( 86,101)( 87, 95)( 88, 96)( 89, 98)( 90, 97)(103,107)(104,108)(105,110)(106,109)(113,114);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5, 
s3*s5*s3*s5, s3*s4*s3*s4*s3*s4*s3*s4, 
s3*s4*s5*s4*s3*s4*s5*s3*s4, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(114)!(1,2);
s1 := Sym(114)!(3,4);
s2 := Sym(114)!(5,6);
s3 := Sym(114)!(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114);
s4 := Sym(114)!(  8,  9)( 11, 15)( 12, 17)( 13, 16)( 14, 18)( 19, 35)( 20, 37)( 21, 36)( 22, 38)( 23, 31)( 24, 33)( 25, 32)( 26, 34)( 27, 39)( 28, 41)( 29, 40)( 30, 42)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 99)( 48,101)( 49,100)( 50,102)( 51, 95)( 52, 97)( 53, 96)( 54, 98)( 55, 79)( 56, 81)( 57, 80)( 58, 82)( 59, 87)( 60, 89)( 61, 88)( 62, 90)( 63, 83)( 64, 85)( 65, 84)( 66, 86)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)( 72,105)( 73,104)( 74,106)( 75,111)( 76,113)( 77,112)( 78,114);
s5 := Sym(114)!(  7, 43)(  8, 44)(  9, 46)( 10, 45)( 11, 51)( 12, 52)( 13, 54)( 14, 53)( 15, 47)( 16, 48)( 17, 50)( 18, 49)( 19, 71)( 20, 72)( 21, 74)( 22, 73)( 23, 67)( 24, 68)( 25, 70)( 26, 69)( 27, 75)( 28, 76)( 29, 78)( 30, 77)( 31, 59)( 32, 60)( 33, 62)( 34, 61)( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 39, 63)( 40, 64)( 41, 66)( 42, 65)( 79, 91)( 80, 92)( 81, 94)( 82, 93)( 83, 99)( 84,100)( 85,102)( 86,101)( 87, 95)( 88, 96)( 89, 98)( 90, 97)(103,107)(104,108)(105,110)(106,109)(113,114);
poly := sub<Sym(114)|s0,s1,s2,s3,s4,s5>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s5*s0*s5, s1*s5*s1*s5, 
s2*s5*s2*s5, s3*s5*s3*s5, s3*s4*s3*s4*s3*s4*s3*s4, 
s3*s4*s5*s4*s3*s4*s5*s3*s4, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >;