Polytope of Type {6,60} with group SmallGroup(1080,528)

Overview

Group: SmallGroup(1080,528)
Rank: 3
Schläfli Type: {6,60}
Vertices, edges, …: 9, 270, 90
Order of s₀s₁s₂: 60
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable

Quotients maximal quotients in bold

3-fold

{6,20}*360

5-fold

{6,12}*216c

15-fold

{6,4}*72

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

Finitely Presented Group Representation (GAP)

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

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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 >;

References