Polytope of Type {9,8,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,8,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,30284)
Rank : 4
Schlafli Type : {9,8,6}
Number of vertices, edges, etc : 18, 72, 48, 6
Order of s0s1s2s3 : 36
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 : {9,4,6}*864
3-fold quotients : {9,8,2}*576, {3,8,6}*576
6-fold quotients : {9,4,2}*288, {3,4,6}*288
8-fold quotients : {9,2,6}*216
9-fold quotients : {3,8,2}*192
12-fold quotients : {9,4,2}*144
16-fold quotients : {9,2,3}*108
18-fold quotients : {3,4,2}*96
24-fold quotients : {9,2,2}*72, {3,2,6}*72
36-fold quotients : {3,4,2}*48
48-fold quotients : {3,2,3}*36
72-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)( 53, 55)
( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)( 63, 69)
( 64, 70)( 73,161)( 74,162)( 75,164)( 76,163)( 77,167)( 78,168)( 79,165)
( 80,166)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)( 86,160)( 87,157)
( 88,158)( 89,145)( 90,146)( 91,148)( 92,147)( 93,151)( 94,152)( 95,149)
( 96,150)( 97,185)( 98,186)( 99,188)(100,187)(101,191)(102,192)(103,189)
(104,190)(105,177)(106,178)(107,180)(108,179)(109,183)(110,184)(111,181)
(112,182)(113,169)(114,170)(115,172)(116,171)(117,175)(118,176)(119,173)
(120,174)(121,209)(122,210)(123,212)(124,211)(125,215)(126,216)(127,213)
(128,214)(129,201)(130,202)(131,204)(132,203)(133,207)(134,208)(135,205)
(136,206)(137,193)(138,194)(139,196)(140,195)(141,199)(142,200)(143,197)
(144,198)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)(228,236)
(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)(249,258)
(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)(265,266)
(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)(278,287)
(279,286)(280,285)(289,378)(290,377)(291,379)(292,380)(293,384)(294,383)
(295,382)(296,381)(297,370)(298,369)(299,371)(300,372)(301,376)(302,375)
(303,374)(304,373)(305,362)(306,361)(307,363)(308,364)(309,368)(310,367)
(311,366)(312,365)(313,402)(314,401)(315,403)(316,404)(317,408)(318,407)
(319,406)(320,405)(321,394)(322,393)(323,395)(324,396)(325,400)(326,399)
(327,398)(328,397)(329,386)(330,385)(331,387)(332,388)(333,392)(334,391)
(335,390)(336,389)(337,426)(338,425)(339,427)(340,428)(341,432)(342,431)
(343,430)(344,429)(345,418)(346,417)(347,419)(348,420)(349,424)(350,423)
(351,422)(352,421)(353,410)(354,409)(355,411)(356,412)(357,416)(358,415)
(359,414)(360,413);;
s1 := ( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)( 8, 76)
( 9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)( 16, 92)
( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)( 24, 84)
( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)( 32,100)
( 33,113)( 34,114)( 35,119)( 36,120)( 37,118)( 38,117)( 39,115)( 40,116)
( 41,105)( 42,106)( 43,111)( 44,112)( 45,110)( 46,109)( 47,107)( 48,108)
( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)( 56,124)
( 57,137)( 58,138)( 59,143)( 60,144)( 61,142)( 62,141)( 63,139)( 64,140)
( 65,129)( 66,130)( 67,135)( 68,136)( 69,134)( 70,133)( 71,131)( 72,132)
(145,161)(146,162)(147,167)(148,168)(149,166)(150,165)(151,163)(152,164)
(155,159)(156,160)(157,158)(169,185)(170,186)(171,191)(172,192)(173,190)
(174,189)(175,187)(176,188)(179,183)(180,184)(181,182)(193,209)(194,210)
(195,215)(196,216)(197,214)(198,213)(199,211)(200,212)(203,207)(204,208)
(205,206)(217,290)(218,289)(219,296)(220,295)(221,293)(222,294)(223,292)
(224,291)(225,306)(226,305)(227,312)(228,311)(229,309)(230,310)(231,308)
(232,307)(233,298)(234,297)(235,304)(236,303)(237,301)(238,302)(239,300)
(240,299)(241,314)(242,313)(243,320)(244,319)(245,317)(246,318)(247,316)
(248,315)(249,330)(250,329)(251,336)(252,335)(253,333)(254,334)(255,332)
(256,331)(257,322)(258,321)(259,328)(260,327)(261,325)(262,326)(263,324)
(264,323)(265,338)(266,337)(267,344)(268,343)(269,341)(270,342)(271,340)
(272,339)(273,354)(274,353)(275,360)(276,359)(277,357)(278,358)(279,356)
(280,355)(281,346)(282,345)(283,352)(284,351)(285,349)(286,350)(287,348)
(288,347)(361,378)(362,377)(363,384)(364,383)(365,381)(366,382)(367,380)
(368,379)(369,370)(371,376)(372,375)(385,402)(386,401)(387,408)(388,407)
(389,405)(390,406)(391,404)(392,403)(393,394)(395,400)(396,399)(409,426)
(410,425)(411,432)(412,431)(413,429)(414,430)(415,428)(416,427)(417,418)
(419,424)(420,423);;
s2 := ( 1,219)( 2,220)( 3,218)( 4,217)( 5,223)( 6,224)( 7,222)( 8,221)
( 9,227)( 10,228)( 11,226)( 12,225)( 13,231)( 14,232)( 15,230)( 16,229)
( 17,235)( 18,236)( 19,234)( 20,233)( 21,239)( 22,240)( 23,238)( 24,237)
( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)( 32,269)
( 33,275)( 34,276)( 35,274)( 36,273)( 37,279)( 38,280)( 39,278)( 40,277)
( 41,283)( 42,284)( 43,282)( 44,281)( 45,287)( 46,288)( 47,286)( 48,285)
( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)( 56,245)
( 57,251)( 58,252)( 59,250)( 60,249)( 61,255)( 62,256)( 63,254)( 64,253)
( 65,259)( 66,260)( 67,258)( 68,257)( 69,263)( 70,264)( 71,262)( 72,261)
( 73,291)( 74,292)( 75,290)( 76,289)( 77,295)( 78,296)( 79,294)( 80,293)
( 81,299)( 82,300)( 83,298)( 84,297)( 85,303)( 86,304)( 87,302)( 88,301)
( 89,307)( 90,308)( 91,306)( 92,305)( 93,311)( 94,312)( 95,310)( 96,309)
( 97,339)( 98,340)( 99,338)(100,337)(101,343)(102,344)(103,342)(104,341)
(105,347)(106,348)(107,346)(108,345)(109,351)(110,352)(111,350)(112,349)
(113,355)(114,356)(115,354)(116,353)(117,359)(118,360)(119,358)(120,357)
(121,315)(122,316)(123,314)(124,313)(125,319)(126,320)(127,318)(128,317)
(129,323)(130,324)(131,322)(132,321)(133,327)(134,328)(135,326)(136,325)
(137,331)(138,332)(139,330)(140,329)(141,335)(142,336)(143,334)(144,333)
(145,363)(146,364)(147,362)(148,361)(149,367)(150,368)(151,366)(152,365)
(153,371)(154,372)(155,370)(156,369)(157,375)(158,376)(159,374)(160,373)
(161,379)(162,380)(163,378)(164,377)(165,383)(166,384)(167,382)(168,381)
(169,411)(170,412)(171,410)(172,409)(173,415)(174,416)(175,414)(176,413)
(177,419)(178,420)(179,418)(180,417)(181,423)(182,424)(183,422)(184,421)
(185,427)(186,428)(187,426)(188,425)(189,431)(190,432)(191,430)(192,429)
(193,387)(194,388)(195,386)(196,385)(197,391)(198,392)(199,390)(200,389)
(201,395)(202,396)(203,394)(204,393)(205,399)(206,400)(207,398)(208,397)
(209,403)(210,404)(211,402)(212,401)(213,407)(214,408)(215,406)(216,405);;
s3 := ( 1, 25)( 2, 26)( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)( 8, 32)
( 9, 33)( 10, 34)( 11, 35)( 12, 36)( 13, 37)( 14, 38)( 15, 39)( 16, 40)
( 17, 41)( 18, 42)( 19, 43)( 20, 44)( 21, 45)( 22, 46)( 23, 47)( 24, 48)
( 73, 97)( 74, 98)( 75, 99)( 76,100)( 77,101)( 78,102)( 79,103)( 80,104)
( 81,105)( 82,106)( 83,107)( 84,108)( 85,109)( 86,110)( 87,111)( 88,112)
( 89,113)( 90,114)( 91,115)( 92,116)( 93,117)( 94,118)( 95,119)( 96,120)
(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)
(153,177)(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)
(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192)
(217,241)(218,242)(219,243)(220,244)(221,245)(222,246)(223,247)(224,248)
(225,249)(226,250)(227,251)(228,252)(229,253)(230,254)(231,255)(232,256)
(233,257)(234,258)(235,259)(236,260)(237,261)(238,262)(239,263)(240,264)
(289,313)(290,314)(291,315)(292,316)(293,317)(294,318)(295,319)(296,320)
(297,321)(298,322)(299,323)(300,324)(301,325)(302,326)(303,327)(304,328)
(305,329)(306,330)(307,331)(308,332)(309,333)(310,334)(311,335)(312,336)
(361,385)(362,386)(363,387)(364,388)(365,389)(366,390)(367,391)(368,392)
(369,393)(370,394)(371,395)(372,396)(373,397)(374,398)(375,399)(376,400)
(377,401)(378,402)(379,403)(380,404)(381,405)(382,406)(383,407)(384,408);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*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(432)!( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)
( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)
( 63, 69)( 64, 70)( 73,161)( 74,162)( 75,164)( 76,163)( 77,167)( 78,168)
( 79,165)( 80,166)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)( 86,160)
( 87,157)( 88,158)( 89,145)( 90,146)( 91,148)( 92,147)( 93,151)( 94,152)
( 95,149)( 96,150)( 97,185)( 98,186)( 99,188)(100,187)(101,191)(102,192)
(103,189)(104,190)(105,177)(106,178)(107,180)(108,179)(109,183)(110,184)
(111,181)(112,182)(113,169)(114,170)(115,172)(116,171)(117,175)(118,176)
(119,173)(120,174)(121,209)(122,210)(123,212)(124,211)(125,215)(126,216)
(127,213)(128,214)(129,201)(130,202)(131,204)(132,203)(133,207)(134,208)
(135,205)(136,206)(137,193)(138,194)(139,196)(140,195)(141,199)(142,200)
(143,197)(144,198)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)
(228,236)(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)
(249,258)(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)
(265,266)(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)
(278,287)(279,286)(280,285)(289,378)(290,377)(291,379)(292,380)(293,384)
(294,383)(295,382)(296,381)(297,370)(298,369)(299,371)(300,372)(301,376)
(302,375)(303,374)(304,373)(305,362)(306,361)(307,363)(308,364)(309,368)
(310,367)(311,366)(312,365)(313,402)(314,401)(315,403)(316,404)(317,408)
(318,407)(319,406)(320,405)(321,394)(322,393)(323,395)(324,396)(325,400)
(326,399)(327,398)(328,397)(329,386)(330,385)(331,387)(332,388)(333,392)
(334,391)(335,390)(336,389)(337,426)(338,425)(339,427)(340,428)(341,432)
(342,431)(343,430)(344,429)(345,418)(346,417)(347,419)(348,420)(349,424)
(350,423)(351,422)(352,421)(353,410)(354,409)(355,411)(356,412)(357,416)
(358,415)(359,414)(360,413);
s1 := Sym(432)!( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)
( 8, 76)( 9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)
( 16, 92)( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)
( 24, 84)( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)
( 32,100)( 33,113)( 34,114)( 35,119)( 36,120)( 37,118)( 38,117)( 39,115)
( 40,116)( 41,105)( 42,106)( 43,111)( 44,112)( 45,110)( 46,109)( 47,107)
( 48,108)( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)
( 56,124)( 57,137)( 58,138)( 59,143)( 60,144)( 61,142)( 62,141)( 63,139)
( 64,140)( 65,129)( 66,130)( 67,135)( 68,136)( 69,134)( 70,133)( 71,131)
( 72,132)(145,161)(146,162)(147,167)(148,168)(149,166)(150,165)(151,163)
(152,164)(155,159)(156,160)(157,158)(169,185)(170,186)(171,191)(172,192)
(173,190)(174,189)(175,187)(176,188)(179,183)(180,184)(181,182)(193,209)
(194,210)(195,215)(196,216)(197,214)(198,213)(199,211)(200,212)(203,207)
(204,208)(205,206)(217,290)(218,289)(219,296)(220,295)(221,293)(222,294)
(223,292)(224,291)(225,306)(226,305)(227,312)(228,311)(229,309)(230,310)
(231,308)(232,307)(233,298)(234,297)(235,304)(236,303)(237,301)(238,302)
(239,300)(240,299)(241,314)(242,313)(243,320)(244,319)(245,317)(246,318)
(247,316)(248,315)(249,330)(250,329)(251,336)(252,335)(253,333)(254,334)
(255,332)(256,331)(257,322)(258,321)(259,328)(260,327)(261,325)(262,326)
(263,324)(264,323)(265,338)(266,337)(267,344)(268,343)(269,341)(270,342)
(271,340)(272,339)(273,354)(274,353)(275,360)(276,359)(277,357)(278,358)
(279,356)(280,355)(281,346)(282,345)(283,352)(284,351)(285,349)(286,350)
(287,348)(288,347)(361,378)(362,377)(363,384)(364,383)(365,381)(366,382)
(367,380)(368,379)(369,370)(371,376)(372,375)(385,402)(386,401)(387,408)
(388,407)(389,405)(390,406)(391,404)(392,403)(393,394)(395,400)(396,399)
(409,426)(410,425)(411,432)(412,431)(413,429)(414,430)(415,428)(416,427)
(417,418)(419,424)(420,423);
s2 := Sym(432)!( 1,219)( 2,220)( 3,218)( 4,217)( 5,223)( 6,224)( 7,222)
( 8,221)( 9,227)( 10,228)( 11,226)( 12,225)( 13,231)( 14,232)( 15,230)
( 16,229)( 17,235)( 18,236)( 19,234)( 20,233)( 21,239)( 22,240)( 23,238)
( 24,237)( 25,267)( 26,268)( 27,266)( 28,265)( 29,271)( 30,272)( 31,270)
( 32,269)( 33,275)( 34,276)( 35,274)( 36,273)( 37,279)( 38,280)( 39,278)
( 40,277)( 41,283)( 42,284)( 43,282)( 44,281)( 45,287)( 46,288)( 47,286)
( 48,285)( 49,243)( 50,244)( 51,242)( 52,241)( 53,247)( 54,248)( 55,246)
( 56,245)( 57,251)( 58,252)( 59,250)( 60,249)( 61,255)( 62,256)( 63,254)
( 64,253)( 65,259)( 66,260)( 67,258)( 68,257)( 69,263)( 70,264)( 71,262)
( 72,261)( 73,291)( 74,292)( 75,290)( 76,289)( 77,295)( 78,296)( 79,294)
( 80,293)( 81,299)( 82,300)( 83,298)( 84,297)( 85,303)( 86,304)( 87,302)
( 88,301)( 89,307)( 90,308)( 91,306)( 92,305)( 93,311)( 94,312)( 95,310)
( 96,309)( 97,339)( 98,340)( 99,338)(100,337)(101,343)(102,344)(103,342)
(104,341)(105,347)(106,348)(107,346)(108,345)(109,351)(110,352)(111,350)
(112,349)(113,355)(114,356)(115,354)(116,353)(117,359)(118,360)(119,358)
(120,357)(121,315)(122,316)(123,314)(124,313)(125,319)(126,320)(127,318)
(128,317)(129,323)(130,324)(131,322)(132,321)(133,327)(134,328)(135,326)
(136,325)(137,331)(138,332)(139,330)(140,329)(141,335)(142,336)(143,334)
(144,333)(145,363)(146,364)(147,362)(148,361)(149,367)(150,368)(151,366)
(152,365)(153,371)(154,372)(155,370)(156,369)(157,375)(158,376)(159,374)
(160,373)(161,379)(162,380)(163,378)(164,377)(165,383)(166,384)(167,382)
(168,381)(169,411)(170,412)(171,410)(172,409)(173,415)(174,416)(175,414)
(176,413)(177,419)(178,420)(179,418)(180,417)(181,423)(182,424)(183,422)
(184,421)(185,427)(186,428)(187,426)(188,425)(189,431)(190,432)(191,430)
(192,429)(193,387)(194,388)(195,386)(196,385)(197,391)(198,392)(199,390)
(200,389)(201,395)(202,396)(203,394)(204,393)(205,399)(206,400)(207,398)
(208,397)(209,403)(210,404)(211,402)(212,401)(213,407)(214,408)(215,406)
(216,405);
s3 := Sym(432)!( 1, 25)( 2, 26)( 3, 27)( 4, 28)( 5, 29)( 6, 30)( 7, 31)
( 8, 32)( 9, 33)( 10, 34)( 11, 35)( 12, 36)( 13, 37)( 14, 38)( 15, 39)
( 16, 40)( 17, 41)( 18, 42)( 19, 43)( 20, 44)( 21, 45)( 22, 46)( 23, 47)
( 24, 48)( 73, 97)( 74, 98)( 75, 99)( 76,100)( 77,101)( 78,102)( 79,103)
( 80,104)( 81,105)( 82,106)( 83,107)( 84,108)( 85,109)( 86,110)( 87,111)
( 88,112)( 89,113)( 90,114)( 91,115)( 92,116)( 93,117)( 94,118)( 95,119)
( 96,120)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)
(152,176)(153,177)(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)
(160,184)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)
(168,192)(217,241)(218,242)(219,243)(220,244)(221,245)(222,246)(223,247)
(224,248)(225,249)(226,250)(227,251)(228,252)(229,253)(230,254)(231,255)
(232,256)(233,257)(234,258)(235,259)(236,260)(237,261)(238,262)(239,263)
(240,264)(289,313)(290,314)(291,315)(292,316)(293,317)(294,318)(295,319)
(296,320)(297,321)(298,322)(299,323)(300,324)(301,325)(302,326)(303,327)
(304,328)(305,329)(306,330)(307,331)(308,332)(309,333)(310,334)(311,335)
(312,336)(361,385)(362,386)(363,387)(364,388)(365,389)(366,390)(367,391)
(368,392)(369,393)(370,394)(371,395)(372,396)(373,397)(374,398)(375,399)
(376,400)(377,401)(378,402)(379,403)(380,404)(381,405)(382,406)(383,407)
(384,408);
poly := sub<Sym(432)|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, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope