Polytope of Type {2,27,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,27,8}*1728
if this polytope has a name.
Group : SmallGroup(1728,11367)
Rank : 4
Schlafli Type : {2,27,8}
Number of vertices, edges, etc : 2, 54, 216, 16
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 : {2,27,4}*864
3-fold quotients : {2,9,8}*576
4-fold quotients : {2,27,4}*432
6-fold quotients : {2,9,4}*288
8-fold quotients : {2,27,2}*216
9-fold quotients : {2,3,8}*192
12-fold quotients : {2,9,4}*144
18-fold quotients : {2,3,4}*96
24-fold quotients : {2,9,2}*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 := (1,2);;
s1 := ( 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);;
s2 := ( 3, 75)( 4, 76)( 5, 78)( 6, 77)( 7, 81)( 8, 82)( 9, 79)( 10, 80)
( 11, 91)( 12, 92)( 13, 94)( 14, 93)( 15, 97)( 16, 98)( 17, 95)( 18, 96)
( 19, 83)( 20, 84)( 21, 86)( 22, 85)( 23, 89)( 24, 90)( 25, 87)( 26, 88)
( 27,139)( 28,140)( 29,142)( 30,141)( 31,145)( 32,146)( 33,143)( 34,144)
( 35,131)( 36,132)( 37,134)( 38,133)( 39,137)( 40,138)( 41,135)( 42,136)
( 43,123)( 44,124)( 45,126)( 46,125)( 47,129)( 48,130)( 49,127)( 50,128)
( 51,115)( 52,116)( 53,118)( 54,117)( 55,121)( 56,122)( 57,119)( 58,120)
( 59,107)( 60,108)( 61,110)( 62,109)( 63,113)( 64,114)( 65,111)( 66,112)
( 67, 99)( 68,100)( 69,102)( 70,101)( 71,105)( 72,106)( 73,103)( 74,104)
(147,211)(148,212)(149,214)(150,213)(151,217)(152,218)(153,215)(154,216)
(155,203)(156,204)(157,206)(158,205)(159,209)(160,210)(161,207)(162,208)
(163,195)(164,196)(165,198)(166,197)(167,201)(168,202)(169,199)(170,200)
(171,187)(172,188)(173,190)(174,189)(175,193)(176,194)(177,191)(178,192)
(181,182)(183,185)(184,186)(219,292)(220,291)(221,293)(222,294)(223,298)
(224,297)(225,296)(226,295)(227,308)(228,307)(229,309)(230,310)(231,314)
(232,313)(233,312)(234,311)(235,300)(236,299)(237,301)(238,302)(239,306)
(240,305)(241,304)(242,303)(243,356)(244,355)(245,357)(246,358)(247,362)
(248,361)(249,360)(250,359)(251,348)(252,347)(253,349)(254,350)(255,354)
(256,353)(257,352)(258,351)(259,340)(260,339)(261,341)(262,342)(263,346)
(264,345)(265,344)(266,343)(267,332)(268,331)(269,333)(270,334)(271,338)
(272,337)(273,336)(274,335)(275,324)(276,323)(277,325)(278,326)(279,330)
(280,329)(281,328)(282,327)(283,316)(284,315)(285,317)(286,318)(287,322)
(288,321)(289,320)(290,319)(363,428)(364,427)(365,429)(366,430)(367,434)
(368,433)(369,432)(370,431)(371,420)(372,419)(373,421)(374,422)(375,426)
(376,425)(377,424)(378,423)(379,412)(380,411)(381,413)(382,414)(383,418)
(384,417)(385,416)(386,415)(387,404)(388,403)(389,405)(390,406)(391,410)
(392,409)(393,408)(394,407)(395,396)(399,402)(400,401);;
s3 := ( 3,225)( 4,226)( 5,223)( 6,224)( 7,222)( 8,221)( 9,220)( 10,219)
( 11,233)( 12,234)( 13,231)( 14,232)( 15,230)( 16,229)( 17,228)( 18,227)
( 19,241)( 20,242)( 21,239)( 22,240)( 23,238)( 24,237)( 25,236)( 26,235)
( 27,249)( 28,250)( 29,247)( 30,248)( 31,246)( 32,245)( 33,244)( 34,243)
( 35,257)( 36,258)( 37,255)( 38,256)( 39,254)( 40,253)( 41,252)( 42,251)
( 43,265)( 44,266)( 45,263)( 46,264)( 47,262)( 48,261)( 49,260)( 50,259)
( 51,273)( 52,274)( 53,271)( 54,272)( 55,270)( 56,269)( 57,268)( 58,267)
( 59,281)( 60,282)( 61,279)( 62,280)( 63,278)( 64,277)( 65,276)( 66,275)
( 67,289)( 68,290)( 69,287)( 70,288)( 71,286)( 72,285)( 73,284)( 74,283)
( 75,297)( 76,298)( 77,295)( 78,296)( 79,294)( 80,293)( 81,292)( 82,291)
( 83,305)( 84,306)( 85,303)( 86,304)( 87,302)( 88,301)( 89,300)( 90,299)
( 91,313)( 92,314)( 93,311)( 94,312)( 95,310)( 96,309)( 97,308)( 98,307)
( 99,321)(100,322)(101,319)(102,320)(103,318)(104,317)(105,316)(106,315)
(107,329)(108,330)(109,327)(110,328)(111,326)(112,325)(113,324)(114,323)
(115,337)(116,338)(117,335)(118,336)(119,334)(120,333)(121,332)(122,331)
(123,345)(124,346)(125,343)(126,344)(127,342)(128,341)(129,340)(130,339)
(131,353)(132,354)(133,351)(134,352)(135,350)(136,349)(137,348)(138,347)
(139,361)(140,362)(141,359)(142,360)(143,358)(144,357)(145,356)(146,355)
(147,369)(148,370)(149,367)(150,368)(151,366)(152,365)(153,364)(154,363)
(155,377)(156,378)(157,375)(158,376)(159,374)(160,373)(161,372)(162,371)
(163,385)(164,386)(165,383)(166,384)(167,382)(168,381)(169,380)(170,379)
(171,393)(172,394)(173,391)(174,392)(175,390)(176,389)(177,388)(178,387)
(179,401)(180,402)(181,399)(182,400)(183,398)(184,397)(185,396)(186,395)
(187,409)(188,410)(189,407)(190,408)(191,406)(192,405)(193,404)(194,403)
(195,417)(196,418)(197,415)(198,416)(199,414)(200,413)(201,412)(202,411)
(203,425)(204,426)(205,423)(206,424)(207,422)(208,421)(209,420)(210,419)
(211,433)(212,434)(213,431)(214,432)(215,430)(216,429)(217,428)(218,427);;
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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(434)!(1,2);
s1 := 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);
s2 := Sym(434)!( 3, 75)( 4, 76)( 5, 78)( 6, 77)( 7, 81)( 8, 82)( 9, 79)
( 10, 80)( 11, 91)( 12, 92)( 13, 94)( 14, 93)( 15, 97)( 16, 98)( 17, 95)
( 18, 96)( 19, 83)( 20, 84)( 21, 86)( 22, 85)( 23, 89)( 24, 90)( 25, 87)
( 26, 88)( 27,139)( 28,140)( 29,142)( 30,141)( 31,145)( 32,146)( 33,143)
( 34,144)( 35,131)( 36,132)( 37,134)( 38,133)( 39,137)( 40,138)( 41,135)
( 42,136)( 43,123)( 44,124)( 45,126)( 46,125)( 47,129)( 48,130)( 49,127)
( 50,128)( 51,115)( 52,116)( 53,118)( 54,117)( 55,121)( 56,122)( 57,119)
( 58,120)( 59,107)( 60,108)( 61,110)( 62,109)( 63,113)( 64,114)( 65,111)
( 66,112)( 67, 99)( 68,100)( 69,102)( 70,101)( 71,105)( 72,106)( 73,103)
( 74,104)(147,211)(148,212)(149,214)(150,213)(151,217)(152,218)(153,215)
(154,216)(155,203)(156,204)(157,206)(158,205)(159,209)(160,210)(161,207)
(162,208)(163,195)(164,196)(165,198)(166,197)(167,201)(168,202)(169,199)
(170,200)(171,187)(172,188)(173,190)(174,189)(175,193)(176,194)(177,191)
(178,192)(181,182)(183,185)(184,186)(219,292)(220,291)(221,293)(222,294)
(223,298)(224,297)(225,296)(226,295)(227,308)(228,307)(229,309)(230,310)
(231,314)(232,313)(233,312)(234,311)(235,300)(236,299)(237,301)(238,302)
(239,306)(240,305)(241,304)(242,303)(243,356)(244,355)(245,357)(246,358)
(247,362)(248,361)(249,360)(250,359)(251,348)(252,347)(253,349)(254,350)
(255,354)(256,353)(257,352)(258,351)(259,340)(260,339)(261,341)(262,342)
(263,346)(264,345)(265,344)(266,343)(267,332)(268,331)(269,333)(270,334)
(271,338)(272,337)(273,336)(274,335)(275,324)(276,323)(277,325)(278,326)
(279,330)(280,329)(281,328)(282,327)(283,316)(284,315)(285,317)(286,318)
(287,322)(288,321)(289,320)(290,319)(363,428)(364,427)(365,429)(366,430)
(367,434)(368,433)(369,432)(370,431)(371,420)(372,419)(373,421)(374,422)
(375,426)(376,425)(377,424)(378,423)(379,412)(380,411)(381,413)(382,414)
(383,418)(384,417)(385,416)(386,415)(387,404)(388,403)(389,405)(390,406)
(391,410)(392,409)(393,408)(394,407)(395,396)(399,402)(400,401);
s3 := Sym(434)!( 3,225)( 4,226)( 5,223)( 6,224)( 7,222)( 8,221)( 9,220)
( 10,219)( 11,233)( 12,234)( 13,231)( 14,232)( 15,230)( 16,229)( 17,228)
( 18,227)( 19,241)( 20,242)( 21,239)( 22,240)( 23,238)( 24,237)( 25,236)
( 26,235)( 27,249)( 28,250)( 29,247)( 30,248)( 31,246)( 32,245)( 33,244)
( 34,243)( 35,257)( 36,258)( 37,255)( 38,256)( 39,254)( 40,253)( 41,252)
( 42,251)( 43,265)( 44,266)( 45,263)( 46,264)( 47,262)( 48,261)( 49,260)
( 50,259)( 51,273)( 52,274)( 53,271)( 54,272)( 55,270)( 56,269)( 57,268)
( 58,267)( 59,281)( 60,282)( 61,279)( 62,280)( 63,278)( 64,277)( 65,276)
( 66,275)( 67,289)( 68,290)( 69,287)( 70,288)( 71,286)( 72,285)( 73,284)
( 74,283)( 75,297)( 76,298)( 77,295)( 78,296)( 79,294)( 80,293)( 81,292)
( 82,291)( 83,305)( 84,306)( 85,303)( 86,304)( 87,302)( 88,301)( 89,300)
( 90,299)( 91,313)( 92,314)( 93,311)( 94,312)( 95,310)( 96,309)( 97,308)
( 98,307)( 99,321)(100,322)(101,319)(102,320)(103,318)(104,317)(105,316)
(106,315)(107,329)(108,330)(109,327)(110,328)(111,326)(112,325)(113,324)
(114,323)(115,337)(116,338)(117,335)(118,336)(119,334)(120,333)(121,332)
(122,331)(123,345)(124,346)(125,343)(126,344)(127,342)(128,341)(129,340)
(130,339)(131,353)(132,354)(133,351)(134,352)(135,350)(136,349)(137,348)
(138,347)(139,361)(140,362)(141,359)(142,360)(143,358)(144,357)(145,356)
(146,355)(147,369)(148,370)(149,367)(150,368)(151,366)(152,365)(153,364)
(154,363)(155,377)(156,378)(157,375)(158,376)(159,374)(160,373)(161,372)
(162,371)(163,385)(164,386)(165,383)(166,384)(167,382)(168,381)(169,380)
(170,379)(171,393)(172,394)(173,391)(174,392)(175,390)(176,389)(177,388)
(178,387)(179,401)(180,402)(181,399)(182,400)(183,398)(184,397)(185,396)
(186,395)(187,409)(188,410)(189,407)(190,408)(191,406)(192,405)(193,404)
(194,403)(195,417)(196,418)(197,415)(198,416)(199,414)(200,413)(201,412)
(202,411)(203,425)(204,426)(205,423)(206,424)(207,422)(208,421)(209,420)
(210,419)(211,433)(212,434)(213,431)(214,432)(215,430)(216,429)(217,428)
(218,427);
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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope