Polytope of Type {30,6} with group SmallGroup(1620,136)

Overview

Group: SmallGroup(1620,136)
Rank: 3
Schläfli Type: {30,6}
Vertices, edges, …: 135, 405, 27
Order of s₀s₁s₂: 45
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

{30,6}*540

5-fold

{6,6}*324b

15-fold

{6,6}*108

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=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 3

9 facets

9 of {30}*60

45 vertex figures

45 of {6}*12

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3

9 facets

9 of {30}*60

75 vertex figures

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

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.