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Polytope of Type {9,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,18}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2339)
Rank : 4
Schlafli Type : {9,6,18}
Number of vertices, edges, etc : 9, 27, 54, 18
Order of s0s1s2s3 : 18
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) :
   3-fold quotients : {9,2,18}*648, {3,6,18}*648b, {9,6,6}*648b
   6-fold quotients : {9,2,9}*324
   9-fold quotients : {3,2,18}*216, {9,2,6}*216, {9,6,2}*216, {3,6,6}*216b
   18-fold quotients : {3,2,9}*108, {9,2,3}*108
   27-fold quotients : {9,2,2}*72, {3,2,6}*72, {3,6,2}*72
   54-fold quotients : {3,2,3}*36
   81-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  4,  7)(  5,  8)(  6,  9)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)
( 32, 62)( 33, 63)( 34, 58)( 35, 59)( 36, 60)( 37, 76)( 38, 77)( 39, 78)
( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)( 47, 68)
( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)( 85, 88)
( 86, 89)( 87, 90)( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)
( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,142)(113,143)
(114,144)(115,139)(116,140)(117,141)(118,157)(119,158)(120,159)(121,154)
(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)(129,150)
(130,145)(131,146)(132,147)(133,151)(134,152)(135,153)(166,169)(167,170)
(168,171)(172,184)(173,185)(174,186)(175,181)(176,182)(177,183)(178,187)
(179,188)(180,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)
(196,220)(197,221)(198,222)(199,238)(200,239)(201,240)(202,235)(203,236)
(204,237)(205,241)(206,242)(207,243)(208,229)(209,230)(210,231)(211,226)
(212,227)(213,228)(214,232)(215,233)(216,234);;
s1 := (  1, 37)(  2, 38)(  3, 39)(  4, 43)(  5, 44)(  6, 45)(  7, 40)(  8, 41)
(  9, 42)( 10, 28)( 11, 29)( 12, 30)( 13, 34)( 14, 35)( 15, 36)( 16, 31)
( 17, 32)( 18, 33)( 19, 49)( 20, 50)( 21, 51)( 22, 46)( 23, 47)( 24, 48)
( 25, 52)( 26, 53)( 27, 54)( 55, 64)( 56, 65)( 57, 66)( 58, 70)( 59, 71)
( 60, 72)( 61, 67)( 62, 68)( 63, 69)( 73, 76)( 74, 77)( 75, 78)( 82,118)
( 83,119)( 84,120)( 85,124)( 86,125)( 87,126)( 88,121)( 89,122)( 90,123)
( 91,109)( 92,110)( 93,111)( 94,115)( 95,116)( 96,117)( 97,112)( 98,113)
( 99,114)(100,130)(101,131)(102,132)(103,127)(104,128)(105,129)(106,133)
(107,134)(108,135)(136,145)(137,146)(138,147)(139,151)(140,152)(141,153)
(142,148)(143,149)(144,150)(154,157)(155,158)(156,159)(163,199)(164,200)
(165,201)(166,205)(167,206)(168,207)(169,202)(170,203)(171,204)(172,190)
(173,191)(174,192)(175,196)(176,197)(177,198)(178,193)(179,194)(180,195)
(181,211)(182,212)(183,213)(184,208)(185,209)(186,210)(187,214)(188,215)
(189,216)(217,226)(218,227)(219,228)(220,232)(221,233)(222,234)(223,229)
(224,230)(225,231)(235,238)(236,239)(237,240);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)( 32, 60)( 33, 59)( 34, 61)
( 35, 63)( 36, 62)( 37, 64)( 38, 66)( 39, 65)( 40, 67)( 41, 69)( 42, 68)
( 43, 70)( 44, 72)( 45, 71)( 46, 73)( 47, 75)( 48, 74)( 49, 76)( 50, 78)
( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 82,164)( 83,163)( 84,165)( 85,167)
( 86,166)( 87,168)( 88,170)( 89,169)( 90,171)( 91,173)( 92,172)( 93,174)
( 94,176)( 95,175)( 96,177)( 97,179)( 98,178)( 99,180)(100,182)(101,181)
(102,183)(103,185)(104,184)(105,186)(106,188)(107,187)(108,189)(109,218)
(110,217)(111,219)(112,221)(113,220)(114,222)(115,224)(116,223)(117,225)
(118,227)(119,226)(120,228)(121,230)(122,229)(123,231)(124,233)(125,232)
(126,234)(127,236)(128,235)(129,237)(130,239)(131,238)(132,240)(133,242)
(134,241)(135,243)(136,191)(137,190)(138,192)(139,194)(140,193)(141,195)
(142,197)(143,196)(144,198)(145,200)(146,199)(147,201)(148,203)(149,202)
(150,204)(151,206)(152,205)(153,207)(154,209)(155,208)(156,210)(157,212)
(158,211)(159,213)(160,215)(161,214)(162,216);;
s3 := (  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)(  8, 90)
(  9, 89)( 10, 91)( 11, 93)( 12, 92)( 13, 94)( 14, 96)( 15, 95)( 16, 97)
( 17, 99)( 18, 98)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)( 24,104)
( 25,106)( 26,108)( 27,107)( 28,109)( 29,111)( 30,110)( 31,112)( 32,114)
( 33,113)( 34,115)( 35,117)( 36,116)( 37,118)( 38,120)( 39,119)( 40,121)
( 41,123)( 42,122)( 43,124)( 44,126)( 45,125)( 46,127)( 47,129)( 48,128)
( 49,130)( 50,132)( 51,131)( 52,133)( 53,135)( 54,134)( 55,136)( 56,138)
( 57,137)( 58,139)( 59,141)( 60,140)( 61,142)( 62,144)( 63,143)( 64,145)
( 65,147)( 66,146)( 67,148)( 68,150)( 69,149)( 70,151)( 71,153)( 72,152)
( 73,154)( 74,156)( 75,155)( 76,157)( 77,159)( 78,158)( 79,160)( 80,162)
( 81,161)(163,164)(166,167)(169,170)(172,173)(175,176)(178,179)(181,182)
(184,185)(187,188)(190,191)(193,194)(196,197)(199,200)(202,203)(205,206)
(208,209)(211,212)(214,215)(217,218)(220,221)(223,224)(226,227)(229,230)
(232,233)(235,236)(238,239)(241,242);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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(243)!(  4,  7)(  5,  8)(  6,  9)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 55)( 29, 56)( 30, 57)
( 31, 61)( 32, 62)( 33, 63)( 34, 58)( 35, 59)( 36, 60)( 37, 76)( 38, 77)
( 39, 78)( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)
( 47, 68)( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)
( 85, 88)( 86, 89)( 87, 90)( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)
( 96,102)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,157)(119,158)(120,159)
(121,154)(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)
(129,150)(130,145)(131,146)(132,147)(133,151)(134,152)(135,153)(166,169)
(167,170)(168,171)(172,184)(173,185)(174,186)(175,181)(176,182)(177,183)
(178,187)(179,188)(180,189)(190,217)(191,218)(192,219)(193,223)(194,224)
(195,225)(196,220)(197,221)(198,222)(199,238)(200,239)(201,240)(202,235)
(203,236)(204,237)(205,241)(206,242)(207,243)(208,229)(209,230)(210,231)
(211,226)(212,227)(213,228)(214,232)(215,233)(216,234);
s1 := Sym(243)!(  1, 37)(  2, 38)(  3, 39)(  4, 43)(  5, 44)(  6, 45)(  7, 40)
(  8, 41)(  9, 42)( 10, 28)( 11, 29)( 12, 30)( 13, 34)( 14, 35)( 15, 36)
( 16, 31)( 17, 32)( 18, 33)( 19, 49)( 20, 50)( 21, 51)( 22, 46)( 23, 47)
( 24, 48)( 25, 52)( 26, 53)( 27, 54)( 55, 64)( 56, 65)( 57, 66)( 58, 70)
( 59, 71)( 60, 72)( 61, 67)( 62, 68)( 63, 69)( 73, 76)( 74, 77)( 75, 78)
( 82,118)( 83,119)( 84,120)( 85,124)( 86,125)( 87,126)( 88,121)( 89,122)
( 90,123)( 91,109)( 92,110)( 93,111)( 94,115)( 95,116)( 96,117)( 97,112)
( 98,113)( 99,114)(100,130)(101,131)(102,132)(103,127)(104,128)(105,129)
(106,133)(107,134)(108,135)(136,145)(137,146)(138,147)(139,151)(140,152)
(141,153)(142,148)(143,149)(144,150)(154,157)(155,158)(156,159)(163,199)
(164,200)(165,201)(166,205)(167,206)(168,207)(169,202)(170,203)(171,204)
(172,190)(173,191)(174,192)(175,196)(176,197)(177,198)(178,193)(179,194)
(180,195)(181,211)(182,212)(183,213)(184,208)(185,209)(186,210)(187,214)
(188,215)(189,216)(217,226)(218,227)(219,228)(220,232)(221,233)(222,234)
(223,229)(224,230)(225,231)(235,238)(236,239)(237,240);
s2 := Sym(243)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)( 32, 60)( 33, 59)
( 34, 61)( 35, 63)( 36, 62)( 37, 64)( 38, 66)( 39, 65)( 40, 67)( 41, 69)
( 42, 68)( 43, 70)( 44, 72)( 45, 71)( 46, 73)( 47, 75)( 48, 74)( 49, 76)
( 50, 78)( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 82,164)( 83,163)( 84,165)
( 85,167)( 86,166)( 87,168)( 88,170)( 89,169)( 90,171)( 91,173)( 92,172)
( 93,174)( 94,176)( 95,175)( 96,177)( 97,179)( 98,178)( 99,180)(100,182)
(101,181)(102,183)(103,185)(104,184)(105,186)(106,188)(107,187)(108,189)
(109,218)(110,217)(111,219)(112,221)(113,220)(114,222)(115,224)(116,223)
(117,225)(118,227)(119,226)(120,228)(121,230)(122,229)(123,231)(124,233)
(125,232)(126,234)(127,236)(128,235)(129,237)(130,239)(131,238)(132,240)
(133,242)(134,241)(135,243)(136,191)(137,190)(138,192)(139,194)(140,193)
(141,195)(142,197)(143,196)(144,198)(145,200)(146,199)(147,201)(148,203)
(149,202)(150,204)(151,206)(152,205)(153,207)(154,209)(155,208)(156,210)
(157,212)(158,211)(159,213)(160,215)(161,214)(162,216);
s3 := Sym(243)!(  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)
(  8, 90)(  9, 89)( 10, 91)( 11, 93)( 12, 92)( 13, 94)( 14, 96)( 15, 95)
( 16, 97)( 17, 99)( 18, 98)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)
( 24,104)( 25,106)( 26,108)( 27,107)( 28,109)( 29,111)( 30,110)( 31,112)
( 32,114)( 33,113)( 34,115)( 35,117)( 36,116)( 37,118)( 38,120)( 39,119)
( 40,121)( 41,123)( 42,122)( 43,124)( 44,126)( 45,125)( 46,127)( 47,129)
( 48,128)( 49,130)( 50,132)( 51,131)( 52,133)( 53,135)( 54,134)( 55,136)
( 56,138)( 57,137)( 58,139)( 59,141)( 60,140)( 61,142)( 62,144)( 63,143)
( 64,145)( 65,147)( 66,146)( 67,148)( 68,150)( 69,149)( 70,151)( 71,153)
( 72,152)( 73,154)( 74,156)( 75,155)( 76,157)( 77,159)( 78,158)( 79,160)
( 80,162)( 81,161)(163,164)(166,167)(169,170)(172,173)(175,176)(178,179)
(181,182)(184,185)(187,188)(190,191)(193,194)(196,197)(199,200)(202,203)
(205,206)(208,209)(211,212)(214,215)(217,218)(220,221)(223,224)(226,227)
(229,230)(232,233)(235,236)(238,239)(241,242);
poly := sub<Sym(243)|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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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.
to this polytope