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Polytope of Type {6,45}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,45}*540
if this polytope has a name.
Group : SmallGroup(540,52)
Rank : 3
Schlafli Type : {6,45}
Number of vertices, edges, etc : 6, 135, 45
Order of s0s1s2 : 90
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,45,2} of size 1080
Vertex Figure Of :
   {2,6,45} of size 1080
   {3,6,45} of size 1620
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,45}*180, {6,15}*180
   5-fold quotients : {6,9}*108
   9-fold quotients : {2,15}*60
   15-fold quotients : {2,9}*36, {6,3}*36
   27-fold quotients : {2,5}*20
   45-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,90}*1080b
   3-fold covers : {18,45}*1620, {6,45}*1620a, {6,135}*1620
Permutation Representation (GAP) :
s0 := ( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)
( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,106)
( 62,107)( 63,108)( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)( 69,114)
( 70,115)( 71,116)( 72,117)( 73,118)( 74,119)( 75,120)( 76,121)( 77,122)
( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)
( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 58)(  5, 60)(  6, 59)(  7, 55)(  8, 57)
(  9, 56)( 10, 52)( 11, 54)( 12, 53)( 13, 49)( 14, 51)( 15, 50)( 16, 77)
( 17, 76)( 18, 78)( 19, 89)( 20, 88)( 21, 90)( 22, 86)( 23, 85)( 24, 87)
( 25, 83)( 26, 82)( 27, 84)( 28, 80)( 29, 79)( 30, 81)( 31, 62)( 32, 61)
( 33, 63)( 34, 74)( 35, 73)( 36, 75)( 37, 71)( 38, 70)( 39, 72)( 40, 68)
( 41, 67)( 42, 69)( 43, 65)( 44, 64)( 45, 66)( 92, 93)( 94,103)( 95,105)
( 96,104)( 97,100)( 98,102)( 99,101)(106,122)(107,121)(108,123)(109,134)
(110,133)(111,135)(112,131)(113,130)(114,132)(115,128)(116,127)(117,129)
(118,125)(119,124)(120,126);;
s2 := (  1, 19)(  2, 21)(  3, 20)(  4, 16)(  5, 18)(  6, 17)(  7, 28)(  8, 30)
(  9, 29)( 10, 25)( 11, 27)( 12, 26)( 13, 22)( 14, 24)( 15, 23)( 31, 35)
( 32, 34)( 33, 36)( 37, 44)( 38, 43)( 39, 45)( 40, 41)( 46,109)( 47,111)
( 48,110)( 49,106)( 50,108)( 51,107)( 52,118)( 53,120)( 54,119)( 55,115)
( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61, 94)( 62, 96)( 63, 95)
( 64, 91)( 65, 93)( 66, 92)( 67,103)( 68,105)( 69,104)( 70,100)( 71,102)
( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76,125)( 77,124)( 78,126)( 79,122)
( 80,121)( 81,123)( 82,134)( 83,133)( 84,135)( 85,131)( 86,130)( 87,132)
( 88,128)( 89,127)( 90,129);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
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*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*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)!( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)
( 53, 98)( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)
( 61,106)( 62,107)( 63,108)( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)
( 69,114)( 70,115)( 71,116)( 72,117)( 73,118)( 74,119)( 75,120)( 76,121)
( 77,122)( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)
( 85,130)( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);
s1 := Sym(135)!(  1, 46)(  2, 48)(  3, 47)(  4, 58)(  5, 60)(  6, 59)(  7, 55)
(  8, 57)(  9, 56)( 10, 52)( 11, 54)( 12, 53)( 13, 49)( 14, 51)( 15, 50)
( 16, 77)( 17, 76)( 18, 78)( 19, 89)( 20, 88)( 21, 90)( 22, 86)( 23, 85)
( 24, 87)( 25, 83)( 26, 82)( 27, 84)( 28, 80)( 29, 79)( 30, 81)( 31, 62)
( 32, 61)( 33, 63)( 34, 74)( 35, 73)( 36, 75)( 37, 71)( 38, 70)( 39, 72)
( 40, 68)( 41, 67)( 42, 69)( 43, 65)( 44, 64)( 45, 66)( 92, 93)( 94,103)
( 95,105)( 96,104)( 97,100)( 98,102)( 99,101)(106,122)(107,121)(108,123)
(109,134)(110,133)(111,135)(112,131)(113,130)(114,132)(115,128)(116,127)
(117,129)(118,125)(119,124)(120,126);
s2 := Sym(135)!(  1, 19)(  2, 21)(  3, 20)(  4, 16)(  5, 18)(  6, 17)(  7, 28)
(  8, 30)(  9, 29)( 10, 25)( 11, 27)( 12, 26)( 13, 22)( 14, 24)( 15, 23)
( 31, 35)( 32, 34)( 33, 36)( 37, 44)( 38, 43)( 39, 45)( 40, 41)( 46,109)
( 47,111)( 48,110)( 49,106)( 50,108)( 51,107)( 52,118)( 53,120)( 54,119)
( 55,115)( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61, 94)( 62, 96)
( 63, 95)( 64, 91)( 65, 93)( 66, 92)( 67,103)( 68,105)( 69,104)( 70,100)
( 71,102)( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76,125)( 77,124)( 78,126)
( 79,122)( 80,121)( 81,123)( 82,134)( 83,133)( 84,135)( 85,131)( 86,130)
( 87,132)( 88,128)( 89,127)( 90,129);
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
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*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*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.
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