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