Polytope of Type {2,6,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,24}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30374)
Rank : 4
Schlafli Type : {2,6,24}
Number of vertices, edges, etc : 2, 18, 216, 72
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,6,12}*864f
3-fold quotients : {2,6,8}*576
4-fold quotients : {2,6,12}*432b
6-fold quotients : {2,6,4}*288
12-fold quotients : {2,6,4}*144
27-fold quotients : {2,2,8}*64
54-fold quotients : {2,2,4}*32
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,165)( 4,166)( 5,167)( 6,173)( 7,171)( 8,172)( 9,169)( 10,170)
( 11,168)( 12,183)( 13,184)( 14,185)( 15,191)( 16,189)( 17,190)( 18,187)
( 19,188)( 20,186)( 21,174)( 22,175)( 23,176)( 24,182)( 25,180)( 26,181)
( 27,178)( 28,179)( 29,177)( 30,192)( 31,193)( 32,194)( 33,200)( 34,198)
( 35,199)( 36,196)( 37,197)( 38,195)( 39,210)( 40,211)( 41,212)( 42,218)
( 43,216)( 44,217)( 45,214)( 46,215)( 47,213)( 48,201)( 49,202)( 50,203)
( 51,209)( 52,207)( 53,208)( 54,205)( 55,206)( 56,204)( 57,138)( 58,139)
( 59,140)( 60,146)( 61,144)( 62,145)( 63,142)( 64,143)( 65,141)( 66,156)
( 67,157)( 68,158)( 69,164)( 70,162)( 71,163)( 72,160)( 73,161)( 74,159)
( 75,147)( 76,148)( 77,149)( 78,155)( 79,153)( 80,154)( 81,151)( 82,152)
( 83,150)( 84,111)( 85,112)( 86,113)( 87,119)( 88,117)( 89,118)( 90,115)
( 91,116)( 92,114)( 93,129)( 94,130)( 95,131)( 96,137)( 97,135)( 98,136)
( 99,133)(100,134)(101,132)(102,120)(103,121)(104,122)(105,128)(106,126)
(107,127)(108,124)(109,125)(110,123)(219,381)(220,382)(221,383)(222,389)
(223,387)(224,388)(225,385)(226,386)(227,384)(228,399)(229,400)(230,401)
(231,407)(232,405)(233,406)(234,403)(235,404)(236,402)(237,390)(238,391)
(239,392)(240,398)(241,396)(242,397)(243,394)(244,395)(245,393)(246,408)
(247,409)(248,410)(249,416)(250,414)(251,415)(252,412)(253,413)(254,411)
(255,426)(256,427)(257,428)(258,434)(259,432)(260,433)(261,430)(262,431)
(263,429)(264,417)(265,418)(266,419)(267,425)(268,423)(269,424)(270,421)
(271,422)(272,420)(273,354)(274,355)(275,356)(276,362)(277,360)(278,361)
(279,358)(280,359)(281,357)(282,372)(283,373)(284,374)(285,380)(286,378)
(287,379)(288,376)(289,377)(290,375)(291,363)(292,364)(293,365)(294,371)
(295,369)(296,370)(297,367)(298,368)(299,366)(300,327)(301,328)(302,329)
(303,335)(304,333)(305,334)(306,331)(307,332)(308,330)(309,345)(310,346)
(311,347)(312,353)(313,351)(314,352)(315,349)(316,350)(317,348)(318,336)
(319,337)(320,338)(321,344)(322,342)(323,343)(324,340)(325,341)(326,339);;
s2 := ( 3, 12)( 4, 14)( 5, 13)( 6, 17)( 7, 16)( 8, 15)( 9, 19)( 10, 18)
( 11, 20)( 22, 23)( 24, 26)( 27, 28)( 30, 39)( 31, 41)( 32, 40)( 33, 44)
( 34, 43)( 35, 42)( 36, 46)( 37, 45)( 38, 47)( 49, 50)( 51, 53)( 54, 55)
( 57, 93)( 58, 95)( 59, 94)( 60, 98)( 61, 97)( 62, 96)( 63,100)( 64, 99)
( 65,101)( 66, 84)( 67, 86)( 68, 85)( 69, 89)( 70, 88)( 71, 87)( 72, 91)
( 73, 90)( 74, 92)( 75,102)( 76,104)( 77,103)( 78,107)( 79,106)( 80,105)
( 81,109)( 82,108)( 83,110)(111,147)(112,149)(113,148)(114,152)(115,151)
(116,150)(117,154)(118,153)(119,155)(120,138)(121,140)(122,139)(123,143)
(124,142)(125,141)(126,145)(127,144)(128,146)(129,156)(130,158)(131,157)
(132,161)(133,160)(134,159)(135,163)(136,162)(137,164)(165,174)(166,176)
(167,175)(168,179)(169,178)(170,177)(171,181)(172,180)(173,182)(184,185)
(186,188)(189,190)(192,201)(193,203)(194,202)(195,206)(196,205)(197,204)
(198,208)(199,207)(200,209)(211,212)(213,215)(216,217)(219,282)(220,284)
(221,283)(222,287)(223,286)(224,285)(225,289)(226,288)(227,290)(228,273)
(229,275)(230,274)(231,278)(232,277)(233,276)(234,280)(235,279)(236,281)
(237,291)(238,293)(239,292)(240,296)(241,295)(242,294)(243,298)(244,297)
(245,299)(246,309)(247,311)(248,310)(249,314)(250,313)(251,312)(252,316)
(253,315)(254,317)(255,300)(256,302)(257,301)(258,305)(259,304)(260,303)
(261,307)(262,306)(263,308)(264,318)(265,320)(266,319)(267,323)(268,322)
(269,321)(270,325)(271,324)(272,326)(327,417)(328,419)(329,418)(330,422)
(331,421)(332,420)(333,424)(334,423)(335,425)(336,408)(337,410)(338,409)
(339,413)(340,412)(341,411)(342,415)(343,414)(344,416)(345,426)(346,428)
(347,427)(348,431)(349,430)(350,429)(351,433)(352,432)(353,434)(354,390)
(355,392)(356,391)(357,395)(358,394)(359,393)(360,397)(361,396)(362,398)
(363,381)(364,383)(365,382)(366,386)(367,385)(368,384)(369,388)(370,387)
(371,389)(372,399)(373,401)(374,400)(375,404)(376,403)(377,402)(378,406)
(379,405)(380,407);;
s3 := ( 3,274)( 4,273)( 5,275)( 6,291)( 7,293)( 8,292)( 9,284)( 10,283)
( 11,282)( 12,281)( 13,280)( 14,279)( 15,299)( 16,298)( 17,297)( 18,290)
( 19,289)( 20,288)( 21,276)( 22,278)( 23,277)( 24,295)( 25,294)( 26,296)
( 27,287)( 28,286)( 29,285)( 30,301)( 31,300)( 32,302)( 33,318)( 34,320)
( 35,319)( 36,311)( 37,310)( 38,309)( 39,308)( 40,307)( 41,306)( 42,326)
( 43,325)( 44,324)( 45,317)( 46,316)( 47,315)( 48,303)( 49,305)( 50,304)
( 51,322)( 52,321)( 53,323)( 54,314)( 55,313)( 56,312)( 57,220)( 58,219)
( 59,221)( 60,237)( 61,239)( 62,238)( 63,230)( 64,229)( 65,228)( 66,227)
( 67,226)( 68,225)( 69,245)( 70,244)( 71,243)( 72,236)( 73,235)( 74,234)
( 75,222)( 76,224)( 77,223)( 78,241)( 79,240)( 80,242)( 81,233)( 82,232)
( 83,231)( 84,247)( 85,246)( 86,248)( 87,264)( 88,266)( 89,265)( 90,257)
( 91,256)( 92,255)( 93,254)( 94,253)( 95,252)( 96,272)( 97,271)( 98,270)
( 99,263)(100,262)(101,261)(102,249)(103,251)(104,250)(105,268)(106,267)
(107,269)(108,260)(109,259)(110,258)(111,409)(112,408)(113,410)(114,426)
(115,428)(116,427)(117,419)(118,418)(119,417)(120,416)(121,415)(122,414)
(123,434)(124,433)(125,432)(126,425)(127,424)(128,423)(129,411)(130,413)
(131,412)(132,430)(133,429)(134,431)(135,422)(136,421)(137,420)(138,382)
(139,381)(140,383)(141,399)(142,401)(143,400)(144,392)(145,391)(146,390)
(147,389)(148,388)(149,387)(150,407)(151,406)(152,405)(153,398)(154,397)
(155,396)(156,384)(157,386)(158,385)(159,403)(160,402)(161,404)(162,395)
(163,394)(164,393)(165,355)(166,354)(167,356)(168,372)(169,374)(170,373)
(171,365)(172,364)(173,363)(174,362)(175,361)(176,360)(177,380)(178,379)
(179,378)(180,371)(181,370)(182,369)(183,357)(184,359)(185,358)(186,376)
(187,375)(188,377)(189,368)(190,367)(191,366)(192,328)(193,327)(194,329)
(195,345)(196,347)(197,346)(198,338)(199,337)(200,336)(201,335)(202,334)
(203,333)(204,353)(205,352)(206,351)(207,344)(208,343)(209,342)(210,330)
(211,332)(212,331)(213,349)(214,348)(215,350)(216,341)(217,340)(218,339);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(434)!(1,2);
s1 := Sym(434)!( 3,165)( 4,166)( 5,167)( 6,173)( 7,171)( 8,172)( 9,169)
( 10,170)( 11,168)( 12,183)( 13,184)( 14,185)( 15,191)( 16,189)( 17,190)
( 18,187)( 19,188)( 20,186)( 21,174)( 22,175)( 23,176)( 24,182)( 25,180)
( 26,181)( 27,178)( 28,179)( 29,177)( 30,192)( 31,193)( 32,194)( 33,200)
( 34,198)( 35,199)( 36,196)( 37,197)( 38,195)( 39,210)( 40,211)( 41,212)
( 42,218)( 43,216)( 44,217)( 45,214)( 46,215)( 47,213)( 48,201)( 49,202)
( 50,203)( 51,209)( 52,207)( 53,208)( 54,205)( 55,206)( 56,204)( 57,138)
( 58,139)( 59,140)( 60,146)( 61,144)( 62,145)( 63,142)( 64,143)( 65,141)
( 66,156)( 67,157)( 68,158)( 69,164)( 70,162)( 71,163)( 72,160)( 73,161)
( 74,159)( 75,147)( 76,148)( 77,149)( 78,155)( 79,153)( 80,154)( 81,151)
( 82,152)( 83,150)( 84,111)( 85,112)( 86,113)( 87,119)( 88,117)( 89,118)
( 90,115)( 91,116)( 92,114)( 93,129)( 94,130)( 95,131)( 96,137)( 97,135)
( 98,136)( 99,133)(100,134)(101,132)(102,120)(103,121)(104,122)(105,128)
(106,126)(107,127)(108,124)(109,125)(110,123)(219,381)(220,382)(221,383)
(222,389)(223,387)(224,388)(225,385)(226,386)(227,384)(228,399)(229,400)
(230,401)(231,407)(232,405)(233,406)(234,403)(235,404)(236,402)(237,390)
(238,391)(239,392)(240,398)(241,396)(242,397)(243,394)(244,395)(245,393)
(246,408)(247,409)(248,410)(249,416)(250,414)(251,415)(252,412)(253,413)
(254,411)(255,426)(256,427)(257,428)(258,434)(259,432)(260,433)(261,430)
(262,431)(263,429)(264,417)(265,418)(266,419)(267,425)(268,423)(269,424)
(270,421)(271,422)(272,420)(273,354)(274,355)(275,356)(276,362)(277,360)
(278,361)(279,358)(280,359)(281,357)(282,372)(283,373)(284,374)(285,380)
(286,378)(287,379)(288,376)(289,377)(290,375)(291,363)(292,364)(293,365)
(294,371)(295,369)(296,370)(297,367)(298,368)(299,366)(300,327)(301,328)
(302,329)(303,335)(304,333)(305,334)(306,331)(307,332)(308,330)(309,345)
(310,346)(311,347)(312,353)(313,351)(314,352)(315,349)(316,350)(317,348)
(318,336)(319,337)(320,338)(321,344)(322,342)(323,343)(324,340)(325,341)
(326,339);
s2 := Sym(434)!( 3, 12)( 4, 14)( 5, 13)( 6, 17)( 7, 16)( 8, 15)( 9, 19)
( 10, 18)( 11, 20)( 22, 23)( 24, 26)( 27, 28)( 30, 39)( 31, 41)( 32, 40)
( 33, 44)( 34, 43)( 35, 42)( 36, 46)( 37, 45)( 38, 47)( 49, 50)( 51, 53)
( 54, 55)( 57, 93)( 58, 95)( 59, 94)( 60, 98)( 61, 97)( 62, 96)( 63,100)
( 64, 99)( 65,101)( 66, 84)( 67, 86)( 68, 85)( 69, 89)( 70, 88)( 71, 87)
( 72, 91)( 73, 90)( 74, 92)( 75,102)( 76,104)( 77,103)( 78,107)( 79,106)
( 80,105)( 81,109)( 82,108)( 83,110)(111,147)(112,149)(113,148)(114,152)
(115,151)(116,150)(117,154)(118,153)(119,155)(120,138)(121,140)(122,139)
(123,143)(124,142)(125,141)(126,145)(127,144)(128,146)(129,156)(130,158)
(131,157)(132,161)(133,160)(134,159)(135,163)(136,162)(137,164)(165,174)
(166,176)(167,175)(168,179)(169,178)(170,177)(171,181)(172,180)(173,182)
(184,185)(186,188)(189,190)(192,201)(193,203)(194,202)(195,206)(196,205)
(197,204)(198,208)(199,207)(200,209)(211,212)(213,215)(216,217)(219,282)
(220,284)(221,283)(222,287)(223,286)(224,285)(225,289)(226,288)(227,290)
(228,273)(229,275)(230,274)(231,278)(232,277)(233,276)(234,280)(235,279)
(236,281)(237,291)(238,293)(239,292)(240,296)(241,295)(242,294)(243,298)
(244,297)(245,299)(246,309)(247,311)(248,310)(249,314)(250,313)(251,312)
(252,316)(253,315)(254,317)(255,300)(256,302)(257,301)(258,305)(259,304)
(260,303)(261,307)(262,306)(263,308)(264,318)(265,320)(266,319)(267,323)
(268,322)(269,321)(270,325)(271,324)(272,326)(327,417)(328,419)(329,418)
(330,422)(331,421)(332,420)(333,424)(334,423)(335,425)(336,408)(337,410)
(338,409)(339,413)(340,412)(341,411)(342,415)(343,414)(344,416)(345,426)
(346,428)(347,427)(348,431)(349,430)(350,429)(351,433)(352,432)(353,434)
(354,390)(355,392)(356,391)(357,395)(358,394)(359,393)(360,397)(361,396)
(362,398)(363,381)(364,383)(365,382)(366,386)(367,385)(368,384)(369,388)
(370,387)(371,389)(372,399)(373,401)(374,400)(375,404)(376,403)(377,402)
(378,406)(379,405)(380,407);
s3 := Sym(434)!( 3,274)( 4,273)( 5,275)( 6,291)( 7,293)( 8,292)( 9,284)
( 10,283)( 11,282)( 12,281)( 13,280)( 14,279)( 15,299)( 16,298)( 17,297)
( 18,290)( 19,289)( 20,288)( 21,276)( 22,278)( 23,277)( 24,295)( 25,294)
( 26,296)( 27,287)( 28,286)( 29,285)( 30,301)( 31,300)( 32,302)( 33,318)
( 34,320)( 35,319)( 36,311)( 37,310)( 38,309)( 39,308)( 40,307)( 41,306)
( 42,326)( 43,325)( 44,324)( 45,317)( 46,316)( 47,315)( 48,303)( 49,305)
( 50,304)( 51,322)( 52,321)( 53,323)( 54,314)( 55,313)( 56,312)( 57,220)
( 58,219)( 59,221)( 60,237)( 61,239)( 62,238)( 63,230)( 64,229)( 65,228)
( 66,227)( 67,226)( 68,225)( 69,245)( 70,244)( 71,243)( 72,236)( 73,235)
( 74,234)( 75,222)( 76,224)( 77,223)( 78,241)( 79,240)( 80,242)( 81,233)
( 82,232)( 83,231)( 84,247)( 85,246)( 86,248)( 87,264)( 88,266)( 89,265)
( 90,257)( 91,256)( 92,255)( 93,254)( 94,253)( 95,252)( 96,272)( 97,271)
( 98,270)( 99,263)(100,262)(101,261)(102,249)(103,251)(104,250)(105,268)
(106,267)(107,269)(108,260)(109,259)(110,258)(111,409)(112,408)(113,410)
(114,426)(115,428)(116,427)(117,419)(118,418)(119,417)(120,416)(121,415)
(122,414)(123,434)(124,433)(125,432)(126,425)(127,424)(128,423)(129,411)
(130,413)(131,412)(132,430)(133,429)(134,431)(135,422)(136,421)(137,420)
(138,382)(139,381)(140,383)(141,399)(142,401)(143,400)(144,392)(145,391)
(146,390)(147,389)(148,388)(149,387)(150,407)(151,406)(152,405)(153,398)
(154,397)(155,396)(156,384)(157,386)(158,385)(159,403)(160,402)(161,404)
(162,395)(163,394)(164,393)(165,355)(166,354)(167,356)(168,372)(169,374)
(170,373)(171,365)(172,364)(173,363)(174,362)(175,361)(176,360)(177,380)
(178,379)(179,378)(180,371)(181,370)(182,369)(183,357)(184,359)(185,358)
(186,376)(187,375)(188,377)(189,368)(190,367)(191,366)(192,328)(193,327)
(194,329)(195,345)(196,347)(197,346)(198,338)(199,337)(200,336)(201,335)
(202,334)(203,333)(204,353)(205,352)(206,351)(207,344)(208,343)(209,342)
(210,330)(211,332)(212,331)(213,349)(214,348)(215,350)(216,341)(217,340)
(218,339);
poly := sub<Sym(434)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3 >;
to this polytope