Polytope of Type {10,6,9} with group SmallGroup(1080,286)

Overview

Group: SmallGroup(1080,286)
Rank: 4
Schläfli Type: {10,6,9}
Vertices, edges, …: 10, 30, 27, 9
Order of s₀s₁s₂s₃: 90
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

{2,6,9}*216

6-fold

{5,2,9}*180

9-fold

{10,2,3}*120

15-fold

18-fold

{5,2,3}*60

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

Permutation Representation (Magma)

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

References