Polytope of Type {6,6,15} with group SmallGroup(1080,539)

Overview

Group: SmallGroup(1080,539)
Rank: 4
Schläfli Type: {6,6,15}
Vertices, edges, …: 6, 18, 45, 15
Order of s₀s₁s₂s₃: 30
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

5-fold

{6,6,3}*216b

6-fold

{3,2,15}*180

9-fold

15-fold

18-fold

{3,2,5}*60

27-fold

{2,2,5}*40

30-fold

{3,2,3}*36

45-fold

{2,2,3}*24

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := ( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135);;
s1 := (  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 51,111)( 52,112)( 53,113)( 54,114)( 55,115)( 56,116)( 57,117)( 58,118)( 59,119)( 60,120)( 61, 91)( 62, 92)( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)( 69, 99)( 70,100)( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,121)( 77,122)( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);;
s2 := (  1, 46)(  2, 50)(  3, 49)(  4, 48)(  5, 47)(  6, 56)(  7, 60)(  8, 59)(  9, 58)( 10, 57)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 61)( 17, 65)( 18, 64)( 19, 63)( 20, 62)( 21, 71)( 22, 75)( 23, 74)( 24, 73)( 25, 72)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 76)( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36, 86)( 37, 90)( 38, 89)( 39, 88)( 40, 87)( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)( 99,103)(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)(115,117)(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132);;
s3 := (  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 22)( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 27)( 28, 30)( 31, 37)( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 97)( 47, 96)( 48,100)( 49, 99)( 50, 98)( 51, 92)( 52, 91)( 53, 95)( 54, 94)( 55, 93)( 56,102)( 57,101)( 58,105)( 59,104)( 60,103)( 61,112)( 62,111)( 63,115)( 64,114)( 65,113)( 66,107)( 67,106)( 68,110)( 69,109)( 70,108)( 71,117)( 72,116)( 73,120)( 74,119)( 75,118)( 76,127)( 77,126)( 78,130)( 79,129)( 80,128)( 81,122)( 82,121)( 83,125)( 84,124)( 85,123)( 86,132)( 87,131)( 88,135)( 89,134)( 90,133);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(135)!( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135);
s1 := Sym(135)!(  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 51,111)( 52,112)( 53,113)( 54,114)( 55,115)( 56,116)( 57,117)( 58,118)( 59,119)( 60,120)( 61, 91)( 62, 92)( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)( 69, 99)( 70,100)( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,121)( 77,122)( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);
s2 := Sym(135)!(  1, 46)(  2, 50)(  3, 49)(  4, 48)(  5, 47)(  6, 56)(  7, 60)(  8, 59)(  9, 58)( 10, 57)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 61)( 17, 65)( 18, 64)( 19, 63)( 20, 62)( 21, 71)( 22, 75)( 23, 74)( 24, 73)( 25, 72)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 76)( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36, 86)( 37, 90)( 38, 89)( 39, 88)( 40, 87)( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)( 99,103)(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)(115,117)(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132);
s3 := Sym(135)!(  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 22)( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 27)( 28, 30)( 31, 37)( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 97)( 47, 96)( 48,100)( 49, 99)( 50, 98)( 51, 92)( 52, 91)( 53, 95)( 54, 94)( 55, 93)( 56,102)( 57,101)( 58,105)( 59,104)( 60,103)( 61,112)( 62,111)( 63,115)( 64,114)( 65,113)( 66,107)( 67,106)( 68,110)( 69,109)( 70,108)( 71,117)( 72,116)( 73,120)( 74,119)( 75,118)( 76,127)( 77,126)( 78,130)( 79,129)( 80,128)( 81,122)( 82,121)( 83,125)( 84,124)( 85,123)( 86,132)( 87,131)( 88,135)( 89,134)( 90,133);
poly := sub<Sym(135)|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, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References