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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}*1440a
if this polytope has a name.
Group : SmallGroup(1440,1584)
Rank : 4
Schlafli Type : {2,18,20}
Number of vertices, edges, etc : 2, 18, 180, 20
Order of s0s1s2s3 : 180
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   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) :
   2-fold quotients : {2,18,10}*720
   3-fold quotients : {2,6,20}*480a
   5-fold quotients : {2,18,4}*288a
   6-fold quotients : {2,6,10}*240
   9-fold quotients : {2,2,20}*160
   10-fold quotients : {2,18,2}*144
   15-fold quotients : {2,6,4}*96a
   18-fold quotients : {2,2,10}*80
   20-fold quotients : {2,9,2}*72
   30-fold quotients : {2,6,2}*48
   36-fold quotients : {2,2,5}*40
   45-fold quotients : {2,2,4}*32
   60-fold quotients : {2,3,2}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 18, 35)( 19, 34)( 20, 33)
( 21, 38)( 22, 37)( 23, 36)( 24, 41)( 25, 40)( 26, 39)( 27, 44)( 28, 43)
( 29, 42)( 30, 47)( 31, 46)( 32, 45)( 49, 50)( 52, 53)( 55, 56)( 58, 59)
( 61, 62)( 63, 80)( 64, 79)( 65, 78)( 66, 83)( 67, 82)( 68, 81)( 69, 86)
( 70, 85)( 71, 84)( 72, 89)( 73, 88)( 74, 87)( 75, 92)( 76, 91)( 77, 90)
( 94, 95)( 97, 98)(100,101)(103,104)(106,107)(108,125)(109,124)(110,123)
(111,128)(112,127)(113,126)(114,131)(115,130)(116,129)(117,134)(118,133)
(119,132)(120,137)(121,136)(122,135)(139,140)(142,143)(145,146)(148,149)
(151,152)(153,170)(154,169)(155,168)(156,173)(157,172)(158,171)(159,176)
(160,175)(161,174)(162,179)(163,178)(164,177)(165,182)(166,181)(167,180);;
s2 := (  3, 18)(  4, 20)(  5, 19)(  6, 30)(  7, 32)(  8, 31)(  9, 27)( 10, 29)
( 11, 28)( 12, 24)( 13, 26)( 14, 25)( 15, 21)( 16, 23)( 17, 22)( 33, 35)
( 36, 47)( 37, 46)( 38, 45)( 39, 44)( 40, 43)( 41, 42)( 48, 63)( 49, 65)
( 50, 64)( 51, 75)( 52, 77)( 53, 76)( 54, 72)( 55, 74)( 56, 73)( 57, 69)
( 58, 71)( 59, 70)( 60, 66)( 61, 68)( 62, 67)( 78, 80)( 81, 92)( 82, 91)
( 83, 90)( 84, 89)( 85, 88)( 86, 87)( 93,153)( 94,155)( 95,154)( 96,165)
( 97,167)( 98,166)( 99,162)(100,164)(101,163)(102,159)(103,161)(104,160)
(105,156)(106,158)(107,157)(108,138)(109,140)(110,139)(111,150)(112,152)
(113,151)(114,147)(115,149)(116,148)(117,144)(118,146)(119,145)(120,141)
(121,143)(122,142)(123,170)(124,169)(125,168)(126,182)(127,181)(128,180)
(129,179)(130,178)(131,177)(132,176)(133,175)(134,174)(135,173)(136,172)
(137,171);;
s3 := (  3, 96)(  4, 97)(  5, 98)(  6, 93)(  7, 94)(  8, 95)(  9,105)( 10,106)
( 11,107)( 12,102)( 13,103)( 14,104)( 15, 99)( 16,100)( 17,101)( 18,111)
( 19,112)( 20,113)( 21,108)( 22,109)( 23,110)( 24,120)( 25,121)( 26,122)
( 27,117)( 28,118)( 29,119)( 30,114)( 31,115)( 32,116)( 33,126)( 34,127)
( 35,128)( 36,123)( 37,124)( 38,125)( 39,135)( 40,136)( 41,137)( 42,132)
( 43,133)( 44,134)( 45,129)( 46,130)( 47,131)( 48,141)( 49,142)( 50,143)
( 51,138)( 52,139)( 53,140)( 54,150)( 55,151)( 56,152)( 57,147)( 58,148)
( 59,149)( 60,144)( 61,145)( 62,146)( 63,156)( 64,157)( 65,158)( 66,153)
( 67,154)( 68,155)( 69,165)( 70,166)( 71,167)( 72,162)( 73,163)( 74,164)
( 75,159)( 76,160)( 77,161)( 78,171)( 79,172)( 80,173)( 81,168)( 82,169)
( 83,170)( 84,180)( 85,181)( 86,182)( 87,177)( 88,178)( 89,179)( 90,174)
( 91,175)( 92,176);;
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*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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(182)!(1,2);
s1 := Sym(182)!(  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 18, 35)( 19, 34)
( 20, 33)( 21, 38)( 22, 37)( 23, 36)( 24, 41)( 25, 40)( 26, 39)( 27, 44)
( 28, 43)( 29, 42)( 30, 47)( 31, 46)( 32, 45)( 49, 50)( 52, 53)( 55, 56)
( 58, 59)( 61, 62)( 63, 80)( 64, 79)( 65, 78)( 66, 83)( 67, 82)( 68, 81)
( 69, 86)( 70, 85)( 71, 84)( 72, 89)( 73, 88)( 74, 87)( 75, 92)( 76, 91)
( 77, 90)( 94, 95)( 97, 98)(100,101)(103,104)(106,107)(108,125)(109,124)
(110,123)(111,128)(112,127)(113,126)(114,131)(115,130)(116,129)(117,134)
(118,133)(119,132)(120,137)(121,136)(122,135)(139,140)(142,143)(145,146)
(148,149)(151,152)(153,170)(154,169)(155,168)(156,173)(157,172)(158,171)
(159,176)(160,175)(161,174)(162,179)(163,178)(164,177)(165,182)(166,181)
(167,180);
s2 := Sym(182)!(  3, 18)(  4, 20)(  5, 19)(  6, 30)(  7, 32)(  8, 31)(  9, 27)
( 10, 29)( 11, 28)( 12, 24)( 13, 26)( 14, 25)( 15, 21)( 16, 23)( 17, 22)
( 33, 35)( 36, 47)( 37, 46)( 38, 45)( 39, 44)( 40, 43)( 41, 42)( 48, 63)
( 49, 65)( 50, 64)( 51, 75)( 52, 77)( 53, 76)( 54, 72)( 55, 74)( 56, 73)
( 57, 69)( 58, 71)( 59, 70)( 60, 66)( 61, 68)( 62, 67)( 78, 80)( 81, 92)
( 82, 91)( 83, 90)( 84, 89)( 85, 88)( 86, 87)( 93,153)( 94,155)( 95,154)
( 96,165)( 97,167)( 98,166)( 99,162)(100,164)(101,163)(102,159)(103,161)
(104,160)(105,156)(106,158)(107,157)(108,138)(109,140)(110,139)(111,150)
(112,152)(113,151)(114,147)(115,149)(116,148)(117,144)(118,146)(119,145)
(120,141)(121,143)(122,142)(123,170)(124,169)(125,168)(126,182)(127,181)
(128,180)(129,179)(130,178)(131,177)(132,176)(133,175)(134,174)(135,173)
(136,172)(137,171);
s3 := Sym(182)!(  3, 96)(  4, 97)(  5, 98)(  6, 93)(  7, 94)(  8, 95)(  9,105)
( 10,106)( 11,107)( 12,102)( 13,103)( 14,104)( 15, 99)( 16,100)( 17,101)
( 18,111)( 19,112)( 20,113)( 21,108)( 22,109)( 23,110)( 24,120)( 25,121)
( 26,122)( 27,117)( 28,118)( 29,119)( 30,114)( 31,115)( 32,116)( 33,126)
( 34,127)( 35,128)( 36,123)( 37,124)( 38,125)( 39,135)( 40,136)( 41,137)
( 42,132)( 43,133)( 44,134)( 45,129)( 46,130)( 47,131)( 48,141)( 49,142)
( 50,143)( 51,138)( 52,139)( 53,140)( 54,150)( 55,151)( 56,152)( 57,147)
( 58,148)( 59,149)( 60,144)( 61,145)( 62,146)( 63,156)( 64,157)( 65,158)
( 66,153)( 67,154)( 68,155)( 69,165)( 70,166)( 71,167)( 72,162)( 73,163)
( 74,164)( 75,159)( 76,160)( 77,161)( 78,171)( 79,172)( 80,173)( 81,168)
( 82,169)( 83,170)( 84,180)( 85,181)( 86,182)( 87,177)( 88,178)( 89,179)
( 90,174)( 91,175)( 92,176);
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*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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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