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Polytope of Type {18,15}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,15}*1620
if this polytope has a name.
Group : SmallGroup(1620,140)
Rank : 3
Schlafli Type : {18,15}
Number of vertices, edges, etc : 54, 405, 45
Order of s0s1s2 : 30
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {6,15}*540
   5-fold quotients : {18,3}*324
   9-fold quotients : {6,15}*180
   15-fold quotients : {6,3}*108
   27-fold quotients : {2,15}*60
   45-fold quotients : {6,3}*36
   81-fold quotients : {2,5}*20
   135-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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, 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);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 49)(  5, 51)(  6, 50)(  7, 52)(  8, 54)
(  9, 53)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)( 16, 88)
( 17, 90)( 18, 89)( 19, 73)( 20, 75)( 21, 74)( 22, 76)( 23, 78)( 24, 77)
( 25, 79)( 26, 81)( 27, 80)( 28, 64)( 29, 66)( 30, 65)( 31, 67)( 32, 69)
( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)( 40, 58)
( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 91, 92)( 94, 95)( 97, 98)
(100,128)(101,127)(102,129)(103,131)(104,130)(105,132)(106,134)(107,133)
(108,135)(109,119)(110,118)(111,120)(112,122)(113,121)(114,123)(115,125)
(116,124)(117,126);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)(  8, 17)
(  9, 16)( 19, 37)( 20, 39)( 21, 38)( 22, 41)( 23, 40)( 24, 42)( 25, 45)
( 26, 44)( 27, 43)( 29, 30)( 31, 32)( 34, 36)( 46,108)( 47,107)( 48,106)
( 49,100)( 50,102)( 51,101)( 52,104)( 53,103)( 54,105)( 55, 99)( 56, 98)
( 57, 97)( 58, 91)( 59, 93)( 60, 92)( 61, 95)( 62, 94)( 63, 96)( 64,135)
( 65,134)( 66,133)( 67,127)( 68,129)( 69,128)( 70,131)( 71,130)( 72,132)
( 73,126)( 74,125)( 75,124)( 76,118)( 77,120)( 78,119)( 79,122)( 80,121)
( 81,123)( 82,117)( 83,116)( 84,115)( 85,109)( 86,111)( 87,110)( 88,113)
( 89,112)( 90,114);;
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*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(  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);
s1 := Sym(135)!(  1, 46)(  2, 48)(  3, 47)(  4, 49)(  5, 51)(  6, 50)(  7, 52)
(  8, 54)(  9, 53)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)
( 16, 88)( 17, 90)( 18, 89)( 19, 73)( 20, 75)( 21, 74)( 22, 76)( 23, 78)
( 24, 77)( 25, 79)( 26, 81)( 27, 80)( 28, 64)( 29, 66)( 30, 65)( 31, 67)
( 32, 69)( 33, 68)( 34, 70)( 35, 72)( 36, 71)( 37, 55)( 38, 57)( 39, 56)
( 40, 58)( 41, 60)( 42, 59)( 43, 61)( 44, 63)( 45, 62)( 91, 92)( 94, 95)
( 97, 98)(100,128)(101,127)(102,129)(103,131)(104,130)(105,132)(106,134)
(107,133)(108,135)(109,119)(110,118)(111,120)(112,122)(113,121)(114,123)
(115,125)(116,124)(117,126);
s2 := Sym(135)!(  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)
(  8, 17)(  9, 16)( 19, 37)( 20, 39)( 21, 38)( 22, 41)( 23, 40)( 24, 42)
( 25, 45)( 26, 44)( 27, 43)( 29, 30)( 31, 32)( 34, 36)( 46,108)( 47,107)
( 48,106)( 49,100)( 50,102)( 51,101)( 52,104)( 53,103)( 54,105)( 55, 99)
( 56, 98)( 57, 97)( 58, 91)( 59, 93)( 60, 92)( 61, 95)( 62, 94)( 63, 96)
( 64,135)( 65,134)( 66,133)( 67,127)( 68,129)( 69,128)( 70,131)( 71,130)
( 72,132)( 73,126)( 74,125)( 75,124)( 76,118)( 77,120)( 78,119)( 79,122)
( 80,121)( 81,123)( 82,117)( 83,116)( 84,115)( 85,109)( 86,111)( 87,110)
( 88,113)( 89,112)( 90,114);
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*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope