Polytope of Type {2,8,27}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,27}*1728
if this polytope has a name.
Group : SmallGroup(1728,11367)
Rank : 4
Schlafli Type : {2,8,27}
Number of vertices, edges, etc : 2, 16, 216, 54
Order of s0s1s2s3 : 108
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 : {2,4,27}*864
3-fold quotients : {2,8,9}*576
4-fold quotients : {2,4,27}*432
6-fold quotients : {2,4,9}*288
8-fold quotients : {2,2,27}*216
9-fold quotients : {2,8,3}*192
12-fold quotients : {2,4,9}*144
18-fold quotients : {2,4,3}*96
24-fold quotients : {2,2,9}*72
36-fold quotients : {2,4,3}*48
72-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,223)( 4,224)( 5,226)( 6,225)( 7,220)( 8,219)( 9,221)( 10,222)
( 11,231)( 12,232)( 13,234)( 14,233)( 15,228)( 16,227)( 17,229)( 18,230)
( 19,239)( 20,240)( 21,242)( 22,241)( 23,236)( 24,235)( 25,237)( 26,238)
( 27,247)( 28,248)( 29,250)( 30,249)( 31,244)( 32,243)( 33,245)( 34,246)
( 35,255)( 36,256)( 37,258)( 38,257)( 39,252)( 40,251)( 41,253)( 42,254)
( 43,263)( 44,264)( 45,266)( 46,265)( 47,260)( 48,259)( 49,261)( 50,262)
( 51,271)( 52,272)( 53,274)( 54,273)( 55,268)( 56,267)( 57,269)( 58,270)
( 59,279)( 60,280)( 61,282)( 62,281)( 63,276)( 64,275)( 65,277)( 66,278)
( 67,287)( 68,288)( 69,290)( 70,289)( 71,284)( 72,283)( 73,285)( 74,286)
( 75,295)( 76,296)( 77,298)( 78,297)( 79,292)( 80,291)( 81,293)( 82,294)
( 83,303)( 84,304)( 85,306)( 86,305)( 87,300)( 88,299)( 89,301)( 90,302)
( 91,311)( 92,312)( 93,314)( 94,313)( 95,308)( 96,307)( 97,309)( 98,310)
( 99,319)(100,320)(101,322)(102,321)(103,316)(104,315)(105,317)(106,318)
(107,327)(108,328)(109,330)(110,329)(111,324)(112,323)(113,325)(114,326)
(115,335)(116,336)(117,338)(118,337)(119,332)(120,331)(121,333)(122,334)
(123,343)(124,344)(125,346)(126,345)(127,340)(128,339)(129,341)(130,342)
(131,351)(132,352)(133,354)(134,353)(135,348)(136,347)(137,349)(138,350)
(139,359)(140,360)(141,362)(142,361)(143,356)(144,355)(145,357)(146,358)
(147,367)(148,368)(149,370)(150,369)(151,364)(152,363)(153,365)(154,366)
(155,375)(156,376)(157,378)(158,377)(159,372)(160,371)(161,373)(162,374)
(163,383)(164,384)(165,386)(166,385)(167,380)(168,379)(169,381)(170,382)
(171,391)(172,392)(173,394)(174,393)(175,388)(176,387)(177,389)(178,390)
(179,399)(180,400)(181,402)(182,401)(183,396)(184,395)(185,397)(186,398)
(187,407)(188,408)(189,410)(190,409)(191,404)(192,403)(193,405)(194,406)
(195,415)(196,416)(197,418)(198,417)(199,412)(200,411)(201,413)(202,414)
(203,423)(204,424)(205,426)(206,425)(207,420)(208,419)(209,421)(210,422)
(211,431)(212,432)(213,434)(214,433)(215,428)(216,427)(217,429)(218,430);;
s2 := ( 5, 8)( 6, 7)( 9, 10)( 11, 19)( 12, 20)( 13, 24)( 14, 23)( 15, 22)
( 16, 21)( 17, 26)( 18, 25)( 27, 67)( 28, 68)( 29, 72)( 30, 71)( 31, 70)
( 32, 69)( 33, 74)( 34, 73)( 35, 59)( 36, 60)( 37, 64)( 38, 63)( 39, 62)
( 40, 61)( 41, 66)( 42, 65)( 43, 51)( 44, 52)( 45, 56)( 46, 55)( 47, 54)
( 48, 53)( 49, 58)( 50, 57)( 75,211)( 76,212)( 77,216)( 78,215)( 79,214)
( 80,213)( 81,218)( 82,217)( 83,203)( 84,204)( 85,208)( 86,207)( 87,206)
( 88,205)( 89,210)( 90,209)( 91,195)( 92,196)( 93,200)( 94,199)( 95,198)
( 96,197)( 97,202)( 98,201)( 99,187)(100,188)(101,192)(102,191)(103,190)
(104,189)(105,194)(106,193)(107,179)(108,180)(109,184)(110,183)(111,182)
(112,181)(113,186)(114,185)(115,171)(116,172)(117,176)(118,175)(119,174)
(120,173)(121,178)(122,177)(123,163)(124,164)(125,168)(126,167)(127,166)
(128,165)(129,170)(130,169)(131,155)(132,156)(133,160)(134,159)(135,158)
(136,157)(137,162)(138,161)(139,147)(140,148)(141,152)(142,151)(143,150)
(144,149)(145,154)(146,153)(219,220)(221,223)(222,224)(227,236)(228,235)
(229,239)(230,240)(231,237)(232,238)(233,241)(234,242)(243,284)(244,283)
(245,287)(246,288)(247,285)(248,286)(249,289)(250,290)(251,276)(252,275)
(253,279)(254,280)(255,277)(256,278)(257,281)(258,282)(259,268)(260,267)
(261,271)(262,272)(263,269)(264,270)(265,273)(266,274)(291,428)(292,427)
(293,431)(294,432)(295,429)(296,430)(297,433)(298,434)(299,420)(300,419)
(301,423)(302,424)(303,421)(304,422)(305,425)(306,426)(307,412)(308,411)
(309,415)(310,416)(311,413)(312,414)(313,417)(314,418)(315,404)(316,403)
(317,407)(318,408)(319,405)(320,406)(321,409)(322,410)(323,396)(324,395)
(325,399)(326,400)(327,397)(328,398)(329,401)(330,402)(331,388)(332,387)
(333,391)(334,392)(335,389)(336,390)(337,393)(338,394)(339,380)(340,379)
(341,383)(342,384)(343,381)(344,382)(345,385)(346,386)(347,372)(348,371)
(349,375)(350,376)(351,373)(352,374)(353,377)(354,378)(355,364)(356,363)
(357,367)(358,368)(359,365)(360,366)(361,369)(362,370);;
s3 := ( 3,147)( 4,148)( 5,153)( 6,154)( 7,152)( 8,151)( 9,149)( 10,150)
( 11,163)( 12,164)( 13,169)( 14,170)( 15,168)( 16,167)( 17,165)( 18,166)
( 19,155)( 20,156)( 21,161)( 22,162)( 23,160)( 24,159)( 25,157)( 26,158)
( 27,211)( 28,212)( 29,217)( 30,218)( 31,216)( 32,215)( 33,213)( 34,214)
( 35,203)( 36,204)( 37,209)( 38,210)( 39,208)( 40,207)( 41,205)( 42,206)
( 43,195)( 44,196)( 45,201)( 46,202)( 47,200)( 48,199)( 49,197)( 50,198)
( 51,187)( 52,188)( 53,193)( 54,194)( 55,192)( 56,191)( 57,189)( 58,190)
( 59,179)( 60,180)( 61,185)( 62,186)( 63,184)( 64,183)( 65,181)( 66,182)
( 67,171)( 68,172)( 69,177)( 70,178)( 71,176)( 72,175)( 73,173)( 74,174)
( 77, 81)( 78, 82)( 79, 80)( 83, 91)( 84, 92)( 85, 97)( 86, 98)( 87, 96)
( 88, 95)( 89, 93)( 90, 94)( 99,139)(100,140)(101,145)(102,146)(103,144)
(104,143)(105,141)(106,142)(107,131)(108,132)(109,137)(110,138)(111,136)
(112,135)(113,133)(114,134)(115,123)(116,124)(117,129)(118,130)(119,128)
(120,127)(121,125)(122,126)(219,364)(220,363)(221,370)(222,369)(223,367)
(224,368)(225,366)(226,365)(227,380)(228,379)(229,386)(230,385)(231,383)
(232,384)(233,382)(234,381)(235,372)(236,371)(237,378)(238,377)(239,375)
(240,376)(241,374)(242,373)(243,428)(244,427)(245,434)(246,433)(247,431)
(248,432)(249,430)(250,429)(251,420)(252,419)(253,426)(254,425)(255,423)
(256,424)(257,422)(258,421)(259,412)(260,411)(261,418)(262,417)(263,415)
(264,416)(265,414)(266,413)(267,404)(268,403)(269,410)(270,409)(271,407)
(272,408)(273,406)(274,405)(275,396)(276,395)(277,402)(278,401)(279,399)
(280,400)(281,398)(282,397)(283,388)(284,387)(285,394)(286,393)(287,391)
(288,392)(289,390)(290,389)(291,292)(293,298)(294,297)(299,308)(300,307)
(301,314)(302,313)(303,311)(304,312)(305,310)(306,309)(315,356)(316,355)
(317,362)(318,361)(319,359)(320,360)(321,358)(322,357)(323,348)(324,347)
(325,354)(326,353)(327,351)(328,352)(329,350)(330,349)(331,340)(332,339)
(333,346)(334,345)(335,343)(336,344)(337,342)(338,341);;
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*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*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(434)!(1,2);
s1 := Sym(434)!( 3,223)( 4,224)( 5,226)( 6,225)( 7,220)( 8,219)( 9,221)
( 10,222)( 11,231)( 12,232)( 13,234)( 14,233)( 15,228)( 16,227)( 17,229)
( 18,230)( 19,239)( 20,240)( 21,242)( 22,241)( 23,236)( 24,235)( 25,237)
( 26,238)( 27,247)( 28,248)( 29,250)( 30,249)( 31,244)( 32,243)( 33,245)
( 34,246)( 35,255)( 36,256)( 37,258)( 38,257)( 39,252)( 40,251)( 41,253)
( 42,254)( 43,263)( 44,264)( 45,266)( 46,265)( 47,260)( 48,259)( 49,261)
( 50,262)( 51,271)( 52,272)( 53,274)( 54,273)( 55,268)( 56,267)( 57,269)
( 58,270)( 59,279)( 60,280)( 61,282)( 62,281)( 63,276)( 64,275)( 65,277)
( 66,278)( 67,287)( 68,288)( 69,290)( 70,289)( 71,284)( 72,283)( 73,285)
( 74,286)( 75,295)( 76,296)( 77,298)( 78,297)( 79,292)( 80,291)( 81,293)
( 82,294)( 83,303)( 84,304)( 85,306)( 86,305)( 87,300)( 88,299)( 89,301)
( 90,302)( 91,311)( 92,312)( 93,314)( 94,313)( 95,308)( 96,307)( 97,309)
( 98,310)( 99,319)(100,320)(101,322)(102,321)(103,316)(104,315)(105,317)
(106,318)(107,327)(108,328)(109,330)(110,329)(111,324)(112,323)(113,325)
(114,326)(115,335)(116,336)(117,338)(118,337)(119,332)(120,331)(121,333)
(122,334)(123,343)(124,344)(125,346)(126,345)(127,340)(128,339)(129,341)
(130,342)(131,351)(132,352)(133,354)(134,353)(135,348)(136,347)(137,349)
(138,350)(139,359)(140,360)(141,362)(142,361)(143,356)(144,355)(145,357)
(146,358)(147,367)(148,368)(149,370)(150,369)(151,364)(152,363)(153,365)
(154,366)(155,375)(156,376)(157,378)(158,377)(159,372)(160,371)(161,373)
(162,374)(163,383)(164,384)(165,386)(166,385)(167,380)(168,379)(169,381)
(170,382)(171,391)(172,392)(173,394)(174,393)(175,388)(176,387)(177,389)
(178,390)(179,399)(180,400)(181,402)(182,401)(183,396)(184,395)(185,397)
(186,398)(187,407)(188,408)(189,410)(190,409)(191,404)(192,403)(193,405)
(194,406)(195,415)(196,416)(197,418)(198,417)(199,412)(200,411)(201,413)
(202,414)(203,423)(204,424)(205,426)(206,425)(207,420)(208,419)(209,421)
(210,422)(211,431)(212,432)(213,434)(214,433)(215,428)(216,427)(217,429)
(218,430);
s2 := Sym(434)!( 5, 8)( 6, 7)( 9, 10)( 11, 19)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 21)( 17, 26)( 18, 25)( 27, 67)( 28, 68)( 29, 72)( 30, 71)
( 31, 70)( 32, 69)( 33, 74)( 34, 73)( 35, 59)( 36, 60)( 37, 64)( 38, 63)
( 39, 62)( 40, 61)( 41, 66)( 42, 65)( 43, 51)( 44, 52)( 45, 56)( 46, 55)
( 47, 54)( 48, 53)( 49, 58)( 50, 57)( 75,211)( 76,212)( 77,216)( 78,215)
( 79,214)( 80,213)( 81,218)( 82,217)( 83,203)( 84,204)( 85,208)( 86,207)
( 87,206)( 88,205)( 89,210)( 90,209)( 91,195)( 92,196)( 93,200)( 94,199)
( 95,198)( 96,197)( 97,202)( 98,201)( 99,187)(100,188)(101,192)(102,191)
(103,190)(104,189)(105,194)(106,193)(107,179)(108,180)(109,184)(110,183)
(111,182)(112,181)(113,186)(114,185)(115,171)(116,172)(117,176)(118,175)
(119,174)(120,173)(121,178)(122,177)(123,163)(124,164)(125,168)(126,167)
(127,166)(128,165)(129,170)(130,169)(131,155)(132,156)(133,160)(134,159)
(135,158)(136,157)(137,162)(138,161)(139,147)(140,148)(141,152)(142,151)
(143,150)(144,149)(145,154)(146,153)(219,220)(221,223)(222,224)(227,236)
(228,235)(229,239)(230,240)(231,237)(232,238)(233,241)(234,242)(243,284)
(244,283)(245,287)(246,288)(247,285)(248,286)(249,289)(250,290)(251,276)
(252,275)(253,279)(254,280)(255,277)(256,278)(257,281)(258,282)(259,268)
(260,267)(261,271)(262,272)(263,269)(264,270)(265,273)(266,274)(291,428)
(292,427)(293,431)(294,432)(295,429)(296,430)(297,433)(298,434)(299,420)
(300,419)(301,423)(302,424)(303,421)(304,422)(305,425)(306,426)(307,412)
(308,411)(309,415)(310,416)(311,413)(312,414)(313,417)(314,418)(315,404)
(316,403)(317,407)(318,408)(319,405)(320,406)(321,409)(322,410)(323,396)
(324,395)(325,399)(326,400)(327,397)(328,398)(329,401)(330,402)(331,388)
(332,387)(333,391)(334,392)(335,389)(336,390)(337,393)(338,394)(339,380)
(340,379)(341,383)(342,384)(343,381)(344,382)(345,385)(346,386)(347,372)
(348,371)(349,375)(350,376)(351,373)(352,374)(353,377)(354,378)(355,364)
(356,363)(357,367)(358,368)(359,365)(360,366)(361,369)(362,370);
s3 := Sym(434)!( 3,147)( 4,148)( 5,153)( 6,154)( 7,152)( 8,151)( 9,149)
( 10,150)( 11,163)( 12,164)( 13,169)( 14,170)( 15,168)( 16,167)( 17,165)
( 18,166)( 19,155)( 20,156)( 21,161)( 22,162)( 23,160)( 24,159)( 25,157)
( 26,158)( 27,211)( 28,212)( 29,217)( 30,218)( 31,216)( 32,215)( 33,213)
( 34,214)( 35,203)( 36,204)( 37,209)( 38,210)( 39,208)( 40,207)( 41,205)
( 42,206)( 43,195)( 44,196)( 45,201)( 46,202)( 47,200)( 48,199)( 49,197)
( 50,198)( 51,187)( 52,188)( 53,193)( 54,194)( 55,192)( 56,191)( 57,189)
( 58,190)( 59,179)( 60,180)( 61,185)( 62,186)( 63,184)( 64,183)( 65,181)
( 66,182)( 67,171)( 68,172)( 69,177)( 70,178)( 71,176)( 72,175)( 73,173)
( 74,174)( 77, 81)( 78, 82)( 79, 80)( 83, 91)( 84, 92)( 85, 97)( 86, 98)
( 87, 96)( 88, 95)( 89, 93)( 90, 94)( 99,139)(100,140)(101,145)(102,146)
(103,144)(104,143)(105,141)(106,142)(107,131)(108,132)(109,137)(110,138)
(111,136)(112,135)(113,133)(114,134)(115,123)(116,124)(117,129)(118,130)
(119,128)(120,127)(121,125)(122,126)(219,364)(220,363)(221,370)(222,369)
(223,367)(224,368)(225,366)(226,365)(227,380)(228,379)(229,386)(230,385)
(231,383)(232,384)(233,382)(234,381)(235,372)(236,371)(237,378)(238,377)
(239,375)(240,376)(241,374)(242,373)(243,428)(244,427)(245,434)(246,433)
(247,431)(248,432)(249,430)(250,429)(251,420)(252,419)(253,426)(254,425)
(255,423)(256,424)(257,422)(258,421)(259,412)(260,411)(261,418)(262,417)
(263,415)(264,416)(265,414)(266,413)(267,404)(268,403)(269,410)(270,409)
(271,407)(272,408)(273,406)(274,405)(275,396)(276,395)(277,402)(278,401)
(279,399)(280,400)(281,398)(282,397)(283,388)(284,387)(285,394)(286,393)
(287,391)(288,392)(289,390)(290,389)(291,292)(293,298)(294,297)(299,308)
(300,307)(301,314)(302,313)(303,311)(304,312)(305,310)(306,309)(315,356)
(316,355)(317,362)(318,361)(319,359)(320,360)(321,358)(322,357)(323,348)
(324,347)(325,354)(326,353)(327,351)(328,352)(329,350)(330,349)(331,340)
(332,339)(333,346)(334,345)(335,343)(336,344)(337,342)(338,341);
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*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*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 >;
to this polytope