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Polytope of Type {18,10,4}

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