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