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Polytope of Type {2,18,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,20}*1440b
if this polytope has a name.
Group : SmallGroup(1440,4569)
Rank : 4
Schlafli Type : {2,18,20}
Number of vertices, edges, etc : 2, 18, 180, 20
Order of s0s1s2s3 : 90
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
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 : {2,6,20}*480b
   5-fold quotients : {2,18,4}*288c
   10-fold quotients : {2,9,4}*144
   15-fold quotients : {2,6,4}*96b
   30-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 40, 41)
( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 52, 53)( 55, 59)( 56, 61)( 57, 60)
( 58, 62)( 63,127)( 64,129)( 65,128)( 66,130)( 67,123)( 68,125)( 69,124)
( 70,126)( 71,131)( 72,133)( 73,132)( 74,134)( 75,139)( 76,141)( 77,140)
( 78,142)( 79,135)( 80,137)( 81,136)( 82,138)( 83,143)( 84,145)( 85,144)
( 86,146)( 87,151)( 88,153)( 89,152)( 90,154)( 91,147)( 92,149)( 93,148)
( 94,150)( 95,155)( 96,157)( 97,156)( 98,158)( 99,163)(100,165)(101,164)
(102,166)(103,159)(104,161)(105,160)(106,162)(107,167)(108,169)(109,168)
(110,170)(111,175)(112,177)(113,176)(114,178)(115,171)(116,173)(117,172)
(118,174)(119,179)(120,181)(121,180)(122,182);;
s2 := (  3, 63)(  4, 64)(  5, 66)(  6, 65)(  7, 71)(  8, 72)(  9, 74)( 10, 73)
( 11, 67)( 12, 68)( 13, 70)( 14, 69)( 15,111)( 16,112)( 17,114)( 18,113)
( 19,119)( 20,120)( 21,122)( 22,121)( 23,115)( 24,116)( 25,118)( 26,117)
( 27, 99)( 28,100)( 29,102)( 30,101)( 31,107)( 32,108)( 33,110)( 34,109)
( 35,103)( 36,104)( 37,106)( 38,105)( 39, 87)( 40, 88)( 41, 90)( 42, 89)
( 43, 95)( 44, 96)( 45, 98)( 46, 97)( 47, 91)( 48, 92)( 49, 94)( 50, 93)
( 51, 75)( 52, 76)( 53, 78)( 54, 77)( 55, 83)( 56, 84)( 57, 86)( 58, 85)
( 59, 79)( 60, 80)( 61, 82)( 62, 81)(123,127)(124,128)(125,130)(126,129)
(133,134)(135,175)(136,176)(137,178)(138,177)(139,171)(140,172)(141,174)
(142,173)(143,179)(144,180)(145,182)(146,181)(147,163)(148,164)(149,166)
(150,165)(151,159)(152,160)(153,162)(154,161)(155,167)(156,168)(157,170)
(158,169);;
s3 := (  3, 18)(  4, 17)(  5, 16)(  6, 15)(  7, 22)(  8, 21)(  9, 20)( 10, 19)
( 11, 26)( 12, 25)( 13, 24)( 14, 23)( 27, 54)( 28, 53)( 29, 52)( 30, 51)
( 31, 58)( 32, 57)( 33, 56)( 34, 55)( 35, 62)( 36, 61)( 37, 60)( 38, 59)
( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)( 63, 78)( 64, 77)
( 65, 76)( 66, 75)( 67, 82)( 68, 81)( 69, 80)( 70, 79)( 71, 86)( 72, 85)
( 73, 84)( 74, 83)( 87,114)( 88,113)( 89,112)( 90,111)( 91,118)( 92,117)
( 93,116)( 94,115)( 95,122)( 96,121)( 97,120)( 98,119)( 99,102)(100,101)
(103,106)(104,105)(107,110)(108,109)(123,138)(124,137)(125,136)(126,135)
(127,142)(128,141)(129,140)(130,139)(131,146)(132,145)(133,144)(134,143)
(147,174)(148,173)(149,172)(150,171)(151,178)(152,177)(153,176)(154,175)
(155,182)(156,181)(157,180)(158,179)(159,162)(160,161)(163,166)(164,165)
(167,170)(168,169);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(182)!(1,2);
s1 := Sym(182)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 16, 17)( 19, 23)
( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)
( 40, 41)( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 52, 53)( 55, 59)( 56, 61)
( 57, 60)( 58, 62)( 63,127)( 64,129)( 65,128)( 66,130)( 67,123)( 68,125)
( 69,124)( 70,126)( 71,131)( 72,133)( 73,132)( 74,134)( 75,139)( 76,141)
( 77,140)( 78,142)( 79,135)( 80,137)( 81,136)( 82,138)( 83,143)( 84,145)
( 85,144)( 86,146)( 87,151)( 88,153)( 89,152)( 90,154)( 91,147)( 92,149)
( 93,148)( 94,150)( 95,155)( 96,157)( 97,156)( 98,158)( 99,163)(100,165)
(101,164)(102,166)(103,159)(104,161)(105,160)(106,162)(107,167)(108,169)
(109,168)(110,170)(111,175)(112,177)(113,176)(114,178)(115,171)(116,173)
(117,172)(118,174)(119,179)(120,181)(121,180)(122,182);
s2 := Sym(182)!(  3, 63)(  4, 64)(  5, 66)(  6, 65)(  7, 71)(  8, 72)(  9, 74)
( 10, 73)( 11, 67)( 12, 68)( 13, 70)( 14, 69)( 15,111)( 16,112)( 17,114)
( 18,113)( 19,119)( 20,120)( 21,122)( 22,121)( 23,115)( 24,116)( 25,118)
( 26,117)( 27, 99)( 28,100)( 29,102)( 30,101)( 31,107)( 32,108)( 33,110)
( 34,109)( 35,103)( 36,104)( 37,106)( 38,105)( 39, 87)( 40, 88)( 41, 90)
( 42, 89)( 43, 95)( 44, 96)( 45, 98)( 46, 97)( 47, 91)( 48, 92)( 49, 94)
( 50, 93)( 51, 75)( 52, 76)( 53, 78)( 54, 77)( 55, 83)( 56, 84)( 57, 86)
( 58, 85)( 59, 79)( 60, 80)( 61, 82)( 62, 81)(123,127)(124,128)(125,130)
(126,129)(133,134)(135,175)(136,176)(137,178)(138,177)(139,171)(140,172)
(141,174)(142,173)(143,179)(144,180)(145,182)(146,181)(147,163)(148,164)
(149,166)(150,165)(151,159)(152,160)(153,162)(154,161)(155,167)(156,168)
(157,170)(158,169);
s3 := Sym(182)!(  3, 18)(  4, 17)(  5, 16)(  6, 15)(  7, 22)(  8, 21)(  9, 20)
( 10, 19)( 11, 26)( 12, 25)( 13, 24)( 14, 23)( 27, 54)( 28, 53)( 29, 52)
( 30, 51)( 31, 58)( 32, 57)( 33, 56)( 34, 55)( 35, 62)( 36, 61)( 37, 60)
( 38, 59)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)( 63, 78)
( 64, 77)( 65, 76)( 66, 75)( 67, 82)( 68, 81)( 69, 80)( 70, 79)( 71, 86)
( 72, 85)( 73, 84)( 74, 83)( 87,114)( 88,113)( 89,112)( 90,111)( 91,118)
( 92,117)( 93,116)( 94,115)( 95,122)( 96,121)( 97,120)( 98,119)( 99,102)
(100,101)(103,106)(104,105)(107,110)(108,109)(123,138)(124,137)(125,136)
(126,135)(127,142)(128,141)(129,140)(130,139)(131,146)(132,145)(133,144)
(134,143)(147,174)(148,173)(149,172)(150,171)(151,178)(152,177)(153,176)
(154,175)(155,182)(156,181)(157,180)(158,179)(159,162)(160,161)(163,166)
(164,165)(167,170)(168,169);
poly := sub<Sym(182)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3 >; 
 

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