Polytope of Type {6,30} with group SmallGroup(1080,539)

Overview

Group: SmallGroup(1080,539)
Rank: 3
Schläfli Type: {6,30}
Vertices, edges, …: 18, 270, 90
Order of s₀s₁s₂: 30
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

5-fold

{6,6}*216d

6-fold

{6,15}*180

9-fold

15-fold

18-fold

{2,15}*60

27-fold

{2,10}*40

30-fold

45-fold

54-fold

{2,5}*20

90-fold

135-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s0*(s2*s1)^3*s0*(s2*s1)^11*s2> of order 2

45 facets

45 of {6}*12

12 vertex figures

6 of {30}*60
6 of {15}*30

P/N, where N=<(s0*s1)^2> of order 3

60 facets

45 of {2}*4
15 of {6}*12

6 vertex figures

6 of {30}*60

P/N, where N=<s0*(s1*s2)^3*s1*s0*(s2*s1)^6*s2> of order 3

30 facets

30 of {6}*12

12 vertex figures

3 of {30}*60
9 of {10}*20

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

References

None.

to this polytope.