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

Overview

Group: SmallGroup(1080,528)
Rank: 3
Schläfli Type: {60,6}
Vertices, edges, …: 90, 270, 9
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
Self-Petrie

Quotients maximal quotients in bold

3-fold

{20,6}*360

5-fold

{12,6}*216c

15-fold

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

References