Polytope of Type {18,30} with group SmallGroup(1620,132)

Overview

Group: SmallGroup(1620,132)
Rank: 3
Schläfli Type: {18,30}
Vertices, edges, …: 27, 405, 45
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

{6,30}*540

5-fold

{18,6}*324a

15-fold

{6,6}*108

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

Permutation Representation (Magma)

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

References