Polytope of Type {6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*1620a
if this polytope has a name.
Group : SmallGroup(1620,135)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 27, 405, 135
Order of s₀s₁s₂ : 45
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,30}*540
   5-fold quotients : {6,6}*324b
   15-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
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