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Polytope of Type {4,10,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,10,18}*1440
Also Known As : {{4,10|2},{10,18|2}}. if this polytope has another name.
Group : SmallGroup(1440,1593)
Rank : 4
Schlafli Type : {4,10,18}
Number of vertices, edges, etc : 4, 20, 90, 18
Order of s0s1s2s3 : 180
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {2,10,18}*720
   3-fold quotients : {4,10,6}*480
   5-fold quotients : {4,2,18}*288
   6-fold quotients : {2,10,6}*240
   9-fold quotients : {4,10,2}*160
   10-fold quotients : {4,2,9}*144, {2,2,18}*144
   15-fold quotients : {4,2,6}*96
   18-fold quotients : {2,10,2}*80
   20-fold quotients : {2,2,9}*72
   30-fold quotients : {4,2,3}*48, {2,2,6}*48
   36-fold quotients : {2,5,2}*40
   45-fold quotients : {4,2,2}*32
   60-fold quotients : {2,2,3}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)
( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,151)
(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)
(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)
(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)
(131,176)(132,177)(133,178)(134,179)(135,180);;
s1 := (  1, 91)(  2, 92)(  3, 93)(  4,103)(  5,104)(  6,105)(  7,100)(  8,101)
(  9,102)( 10, 97)( 11, 98)( 12, 99)( 13, 94)( 14, 95)( 15, 96)( 16,106)
( 17,107)( 18,108)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)( 24,117)
( 25,112)( 26,113)( 27,114)( 28,109)( 29,110)( 30,111)( 31,121)( 32,122)
( 33,123)( 34,133)( 35,134)( 36,135)( 37,130)( 38,131)( 39,132)( 40,127)
( 41,128)( 42,129)( 43,124)( 44,125)( 45,126)( 46,136)( 47,137)( 48,138)
( 49,148)( 50,149)( 51,150)( 52,145)( 53,146)( 54,147)( 55,142)( 56,143)
( 57,144)( 58,139)( 59,140)( 60,141)( 61,151)( 62,152)( 63,153)( 64,163)
( 65,164)( 66,165)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)( 72,159)
( 73,154)( 74,155)( 75,156)( 76,166)( 77,167)( 78,168)( 79,178)( 80,179)
( 81,180)( 82,175)( 83,176)( 84,177)( 85,172)( 86,173)( 87,174)( 88,169)
( 89,170)( 90,171);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  7, 13)(  8, 15)(  9, 14)( 11, 12)( 16, 36)
( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 46, 49)( 47, 51)
( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 81)( 62, 80)( 63, 79)
( 64, 78)( 65, 77)( 66, 76)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 86)
( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 91, 94)( 92, 96)( 93, 95)( 97,103)
( 98,105)( 99,104)(101,102)(106,126)(107,125)(108,124)(109,123)(110,122)
(111,121)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)
(119,128)(120,127)(136,139)(137,141)(138,140)(142,148)(143,150)(144,149)
(146,147)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)(157,180)
(158,179)(159,178)(160,177)(161,176)(162,175)(163,174)(164,173)(165,172);;
s3 := (  1, 16)(  2, 18)(  3, 17)(  4, 19)(  5, 21)(  6, 20)(  7, 22)(  8, 24)
(  9, 23)( 10, 25)( 11, 27)( 12, 26)( 13, 28)( 14, 30)( 15, 29)( 31, 33)
( 34, 36)( 37, 39)( 40, 42)( 43, 45)( 46, 61)( 47, 63)( 48, 62)( 49, 64)
( 50, 66)( 51, 65)( 52, 67)( 53, 69)( 54, 68)( 55, 70)( 56, 72)( 57, 71)
( 58, 73)( 59, 75)( 60, 74)( 76, 78)( 79, 81)( 82, 84)( 85, 87)( 88, 90)
( 91,106)( 92,108)( 93,107)( 94,109)( 95,111)( 96,110)( 97,112)( 98,114)
( 99,113)(100,115)(101,117)(102,116)(103,118)(104,120)(105,119)(121,123)
(124,126)(127,129)(130,132)(133,135)(136,151)(137,153)(138,152)(139,154)
(140,156)(141,155)(142,157)(143,159)(144,158)(145,160)(146,162)(147,161)
(148,163)(149,165)(150,164)(166,168)(169,171)(172,174)(175,177)(178,180);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, 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, 
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(180)!( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)
( 98,143)( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)
(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)
(122,167)(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)
(130,175)(131,176)(132,177)(133,178)(134,179)(135,180);
s1 := Sym(180)!(  1, 91)(  2, 92)(  3, 93)(  4,103)(  5,104)(  6,105)(  7,100)
(  8,101)(  9,102)( 10, 97)( 11, 98)( 12, 99)( 13, 94)( 14, 95)( 15, 96)
( 16,106)( 17,107)( 18,108)( 19,118)( 20,119)( 21,120)( 22,115)( 23,116)
( 24,117)( 25,112)( 26,113)( 27,114)( 28,109)( 29,110)( 30,111)( 31,121)
( 32,122)( 33,123)( 34,133)( 35,134)( 36,135)( 37,130)( 38,131)( 39,132)
( 40,127)( 41,128)( 42,129)( 43,124)( 44,125)( 45,126)( 46,136)( 47,137)
( 48,138)( 49,148)( 50,149)( 51,150)( 52,145)( 53,146)( 54,147)( 55,142)
( 56,143)( 57,144)( 58,139)( 59,140)( 60,141)( 61,151)( 62,152)( 63,153)
( 64,163)( 65,164)( 66,165)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)
( 72,159)( 73,154)( 74,155)( 75,156)( 76,166)( 77,167)( 78,168)( 79,178)
( 80,179)( 81,180)( 82,175)( 83,176)( 84,177)( 85,172)( 86,173)( 87,174)
( 88,169)( 89,170)( 90,171);
s2 := Sym(180)!(  1,  4)(  2,  6)(  3,  5)(  7, 13)(  8, 15)(  9, 14)( 11, 12)
( 16, 36)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 46, 49)
( 47, 51)( 48, 50)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 61, 81)( 62, 80)
( 63, 79)( 64, 78)( 65, 77)( 66, 76)( 67, 90)( 68, 89)( 69, 88)( 70, 87)
( 71, 86)( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 91, 94)( 92, 96)( 93, 95)
( 97,103)( 98,105)( 99,104)(101,102)(106,126)(107,125)(108,124)(109,123)
(110,122)(111,121)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)
(118,129)(119,128)(120,127)(136,139)(137,141)(138,140)(142,148)(143,150)
(144,149)(146,147)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)
(157,180)(158,179)(159,178)(160,177)(161,176)(162,175)(163,174)(164,173)
(165,172);
s3 := Sym(180)!(  1, 16)(  2, 18)(  3, 17)(  4, 19)(  5, 21)(  6, 20)(  7, 22)
(  8, 24)(  9, 23)( 10, 25)( 11, 27)( 12, 26)( 13, 28)( 14, 30)( 15, 29)
( 31, 33)( 34, 36)( 37, 39)( 40, 42)( 43, 45)( 46, 61)( 47, 63)( 48, 62)
( 49, 64)( 50, 66)( 51, 65)( 52, 67)( 53, 69)( 54, 68)( 55, 70)( 56, 72)
( 57, 71)( 58, 73)( 59, 75)( 60, 74)( 76, 78)( 79, 81)( 82, 84)( 85, 87)
( 88, 90)( 91,106)( 92,108)( 93,107)( 94,109)( 95,111)( 96,110)( 97,112)
( 98,114)( 99,113)(100,115)(101,117)(102,116)(103,118)(104,120)(105,119)
(121,123)(124,126)(127,129)(130,132)(133,135)(136,151)(137,153)(138,152)
(139,154)(140,156)(141,155)(142,157)(143,159)(144,158)(145,160)(146,162)
(147,161)(148,163)(149,165)(150,164)(166,168)(169,171)(172,174)(175,177)
(178,180);
poly := sub<Sym(180)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
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, 
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 >; 
 
References : None.
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