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Polytope of Type {45,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {45,6}*1620b
if this polytope has a name.
Group : SmallGroup(1620,135)
Rank : 3
Schlafli Type : {45,6}
Number of vertices, edges, etc : 135, 405, 18
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {15,6}*540
   5-fold quotients : {9,6}*324c
   9-fold quotients : {15,6}*180
   15-fold quotients : {3,6}*108
   27-fold quotients : {15,2}*60
   45-fold quotients : {3,6}*36
   81-fold quotients : {5,2}*20
   135-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  5,  6)(  8,  9)( 10, 37)( 11, 39)( 12, 38)( 13, 40)( 14, 42)
( 15, 41)( 16, 43)( 17, 45)( 18, 44)( 19, 28)( 20, 30)( 21, 29)( 22, 31)
( 23, 33)( 24, 32)( 25, 34)( 26, 36)( 27, 35)( 46, 91)( 47, 93)( 48, 92)
( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)( 55,127)( 56,129)
( 57,128)( 58,130)( 59,132)( 60,131)( 61,133)( 62,135)( 63,134)( 64,118)
( 65,120)( 66,119)( 67,121)( 68,123)( 69,122)( 70,124)( 71,126)( 72,125)
( 73,109)( 74,111)( 75,110)( 76,112)( 77,114)( 78,113)( 79,115)( 80,117)
( 81,116)( 82,100)( 83,102)( 84,101)( 85,103)( 86,105)( 87,104)( 88,106)
( 89,108)( 90,107);;
s1 := (  1, 58)(  2, 60)(  3, 59)(  4, 63)(  5, 62)(  6, 61)(  7, 56)(  8, 55)
(  9, 57)( 10, 49)( 11, 51)( 12, 50)( 13, 54)( 14, 53)( 15, 52)( 16, 47)
( 17, 46)( 18, 48)( 19, 85)( 20, 87)( 21, 86)( 22, 90)( 23, 89)( 24, 88)
( 25, 83)( 26, 82)( 27, 84)( 28, 76)( 29, 78)( 30, 77)( 31, 81)( 32, 80)
( 33, 79)( 34, 74)( 35, 73)( 36, 75)( 37, 67)( 38, 69)( 39, 68)( 40, 72)
( 41, 71)( 42, 70)( 43, 65)( 44, 64)( 45, 66)( 91,100)( 92,102)( 93,101)
( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,127)(110,129)
(111,128)(112,132)(113,131)(114,130)(115,134)(116,133)(117,135)(119,120)
(121,123)(124,125);;
s2 := (  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);;
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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
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*s0*s2*s1*s0 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(135)!(  2,  3)(  5,  6)(  8,  9)( 10, 37)( 11, 39)( 12, 38)( 13, 40)
( 14, 42)( 15, 41)( 16, 43)( 17, 45)( 18, 44)( 19, 28)( 20, 30)( 21, 29)
( 22, 31)( 23, 33)( 24, 32)( 25, 34)( 26, 36)( 27, 35)( 46, 91)( 47, 93)
( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)( 55,127)
( 56,129)( 57,128)( 58,130)( 59,132)( 60,131)( 61,133)( 62,135)( 63,134)
( 64,118)( 65,120)( 66,119)( 67,121)( 68,123)( 69,122)( 70,124)( 71,126)
( 72,125)( 73,109)( 74,111)( 75,110)( 76,112)( 77,114)( 78,113)( 79,115)
( 80,117)( 81,116)( 82,100)( 83,102)( 84,101)( 85,103)( 86,105)( 87,104)
( 88,106)( 89,108)( 90,107);
s1 := Sym(135)!(  1, 58)(  2, 60)(  3, 59)(  4, 63)(  5, 62)(  6, 61)(  7, 56)
(  8, 55)(  9, 57)( 10, 49)( 11, 51)( 12, 50)( 13, 54)( 14, 53)( 15, 52)
( 16, 47)( 17, 46)( 18, 48)( 19, 85)( 20, 87)( 21, 86)( 22, 90)( 23, 89)
( 24, 88)( 25, 83)( 26, 82)( 27, 84)( 28, 76)( 29, 78)( 30, 77)( 31, 81)
( 32, 80)( 33, 79)( 34, 74)( 35, 73)( 36, 75)( 37, 67)( 38, 69)( 39, 68)
( 40, 72)( 41, 71)( 42, 70)( 43, 65)( 44, 64)( 45, 66)( 91,100)( 92,102)
( 93,101)( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,127)
(110,129)(111,128)(112,132)(113,131)(114,130)(115,134)(116,133)(117,135)
(119,120)(121,123)(124,125);
s2 := 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);
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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
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*s0*s2*s1*s0 >; 
 
References : None.
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