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Polytope of Type {15,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,18}*1620
if this polytope has a name.
Group : SmallGroup(1620,140)
Rank : 3
Schlafli Type : {15,18}
Number of vertices, edges, etc : 45, 405, 54
Order of s0s1s2 : 30
Order of s0s1s2s1 : 18
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 : {3,18}*324
   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, 92)( 47, 91)( 48, 93)
( 49, 95)( 50, 94)( 51, 96)( 52, 98)( 53, 97)( 54, 99)( 55,128)( 56,127)
( 57,129)( 58,131)( 59,130)( 60,132)( 61,134)( 62,133)( 63,135)( 64,119)
( 65,118)( 66,120)( 67,122)( 68,121)( 69,123)( 70,125)( 71,124)( 72,126)
( 73,110)( 74,109)( 75,111)( 76,113)( 77,112)( 78,114)( 79,116)( 80,115)
( 81,117)( 82,101)( 83,100)( 84,102)( 85,104)( 86,103)( 87,105)( 88,107)
( 89,106)( 90,108);;
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 := (  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, 92)( 47, 91)( 48, 93)( 49, 98)
( 50, 97)( 51, 99)( 52, 95)( 53, 94)( 54, 96)( 55,101)( 56,100)( 57,102)
( 58,107)( 59,106)( 60,108)( 61,104)( 62,103)( 63,105)( 64,110)( 65,109)
( 66,111)( 67,116)( 68,115)( 69,117)( 70,113)( 71,112)( 72,114)( 73,119)
( 74,118)( 75,120)( 76,125)( 77,124)( 78,126)( 79,122)( 80,121)( 81,123)
( 82,128)( 83,127)( 84,129)( 85,134)( 86,133)( 87,135)( 88,131)( 89,130)
( 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, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
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, 92)( 47, 91)
( 48, 93)( 49, 95)( 50, 94)( 51, 96)( 52, 98)( 53, 97)( 54, 99)( 55,128)
( 56,127)( 57,129)( 58,131)( 59,130)( 60,132)( 61,134)( 62,133)( 63,135)
( 64,119)( 65,118)( 66,120)( 67,122)( 68,121)( 69,123)( 70,125)( 71,124)
( 72,126)( 73,110)( 74,109)( 75,111)( 76,113)( 77,112)( 78,114)( 79,116)
( 80,115)( 81,117)( 82,101)( 83,100)( 84,102)( 85,104)( 86,103)( 87,105)
( 88,107)( 89,106)( 90,108);
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)!(  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, 92)( 47, 91)( 48, 93)
( 49, 98)( 50, 97)( 51, 99)( 52, 95)( 53, 94)( 54, 96)( 55,101)( 56,100)
( 57,102)( 58,107)( 59,106)( 60,108)( 61,104)( 62,103)( 63,105)( 64,110)
( 65,109)( 66,111)( 67,116)( 68,115)( 69,117)( 70,113)( 71,112)( 72,114)
( 73,119)( 74,118)( 75,120)( 76,125)( 77,124)( 78,126)( 79,122)( 80,121)
( 81,123)( 82,128)( 83,127)( 84,129)( 85,134)( 86,133)( 87,135)( 88,131)
( 89,130)( 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, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 >; 
 
References : None.
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