Polytope of Type {2,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,8}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30374)
Rank : 4
Schlafli Type : {2,6,8}
Number of vertices, edges, etc : 2, 54, 216, 72
Order of s0s1s2s3 : 24
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 : {2,6,4}*864a
3-fold quotients : {2,6,8}*576
4-fold quotients : {2,6,4}*432
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 := ( 3,220)( 4,219)( 5,221)( 6,228)( 7,230)( 8,229)( 9,239)( 10,238)
( 11,237)( 12,222)( 13,224)( 14,223)( 15,231)( 16,233)( 17,232)( 18,240)
( 19,242)( 20,241)( 21,227)( 22,226)( 23,225)( 24,234)( 25,236)( 26,235)
( 27,244)( 28,243)( 29,245)( 30,247)( 31,246)( 32,248)( 33,255)( 34,257)
( 35,256)( 36,266)( 37,265)( 38,264)( 39,249)( 40,251)( 41,250)( 42,258)
( 43,260)( 44,259)( 45,267)( 46,269)( 47,268)( 48,254)( 49,253)( 50,252)
( 51,261)( 52,263)( 53,262)( 54,271)( 55,270)( 56,272)( 57,301)( 58,300)
( 59,302)( 60,309)( 61,311)( 62,310)( 63,320)( 64,319)( 65,318)( 66,303)
( 67,305)( 68,304)( 69,312)( 70,314)( 71,313)( 72,321)( 73,323)( 74,322)
( 75,308)( 76,307)( 77,306)( 78,315)( 79,317)( 80,316)( 81,325)( 82,324)
( 83,326)( 84,274)( 85,273)( 86,275)( 87,282)( 88,284)( 89,283)( 90,293)
( 91,292)( 92,291)( 93,276)( 94,278)( 95,277)( 96,285)( 97,287)( 98,286)
( 99,294)(100,296)(101,295)(102,281)(103,280)(104,279)(105,288)(106,290)
(107,289)(108,298)(109,297)(110,299)(111,355)(112,354)(113,356)(114,363)
(115,365)(116,364)(117,374)(118,373)(119,372)(120,357)(121,359)(122,358)
(123,366)(124,368)(125,367)(126,375)(127,377)(128,376)(129,362)(130,361)
(131,360)(132,369)(133,371)(134,370)(135,379)(136,378)(137,380)(138,328)
(139,327)(140,329)(141,336)(142,338)(143,337)(144,347)(145,346)(146,345)
(147,330)(148,332)(149,331)(150,339)(151,341)(152,340)(153,348)(154,350)
(155,349)(156,335)(157,334)(158,333)(159,342)(160,344)(161,343)(162,352)
(163,351)(164,353)(165,382)(166,381)(167,383)(168,390)(169,392)(170,391)
(171,401)(172,400)(173,399)(174,384)(175,386)(176,385)(177,393)(178,395)
(179,394)(180,402)(181,404)(182,403)(183,389)(184,388)(185,387)(186,396)
(187,398)(188,397)(189,406)(190,405)(191,407)(192,409)(193,408)(194,410)
(195,417)(196,419)(197,418)(198,428)(199,427)(200,426)(201,411)(202,413)
(203,412)(204,420)(205,422)(206,421)(207,429)(208,431)(209,430)(210,416)
(211,415)(212,414)(213,423)(214,425)(215,424)(216,433)(217,432)(218,434);;
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2 ];;
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)!( 3,220)( 4,219)( 5,221)( 6,228)( 7,230)( 8,229)( 9,239)
( 10,238)( 11,237)( 12,222)( 13,224)( 14,223)( 15,231)( 16,233)( 17,232)
( 18,240)( 19,242)( 20,241)( 21,227)( 22,226)( 23,225)( 24,234)( 25,236)
( 26,235)( 27,244)( 28,243)( 29,245)( 30,247)( 31,246)( 32,248)( 33,255)
( 34,257)( 35,256)( 36,266)( 37,265)( 38,264)( 39,249)( 40,251)( 41,250)
( 42,258)( 43,260)( 44,259)( 45,267)( 46,269)( 47,268)( 48,254)( 49,253)
( 50,252)( 51,261)( 52,263)( 53,262)( 54,271)( 55,270)( 56,272)( 57,301)
( 58,300)( 59,302)( 60,309)( 61,311)( 62,310)( 63,320)( 64,319)( 65,318)
( 66,303)( 67,305)( 68,304)( 69,312)( 70,314)( 71,313)( 72,321)( 73,323)
( 74,322)( 75,308)( 76,307)( 77,306)( 78,315)( 79,317)( 80,316)( 81,325)
( 82,324)( 83,326)( 84,274)( 85,273)( 86,275)( 87,282)( 88,284)( 89,283)
( 90,293)( 91,292)( 92,291)( 93,276)( 94,278)( 95,277)( 96,285)( 97,287)
( 98,286)( 99,294)(100,296)(101,295)(102,281)(103,280)(104,279)(105,288)
(106,290)(107,289)(108,298)(109,297)(110,299)(111,355)(112,354)(113,356)
(114,363)(115,365)(116,364)(117,374)(118,373)(119,372)(120,357)(121,359)
(122,358)(123,366)(124,368)(125,367)(126,375)(127,377)(128,376)(129,362)
(130,361)(131,360)(132,369)(133,371)(134,370)(135,379)(136,378)(137,380)
(138,328)(139,327)(140,329)(141,336)(142,338)(143,337)(144,347)(145,346)
(146,345)(147,330)(148,332)(149,331)(150,339)(151,341)(152,340)(153,348)
(154,350)(155,349)(156,335)(157,334)(158,333)(159,342)(160,344)(161,343)
(162,352)(163,351)(164,353)(165,382)(166,381)(167,383)(168,390)(169,392)
(170,391)(171,401)(172,400)(173,399)(174,384)(175,386)(176,385)(177,393)
(178,395)(179,394)(180,402)(181,404)(182,403)(183,389)(184,388)(185,387)
(186,396)(187,398)(188,397)(189,406)(190,405)(191,407)(192,409)(193,408)
(194,410)(195,417)(196,419)(197,418)(198,428)(199,427)(200,426)(201,411)
(202,413)(203,412)(204,420)(205,422)(206,421)(207,429)(208,431)(209,430)
(210,416)(211,415)(212,414)(213,423)(214,425)(215,424)(216,433)(217,432)
(218,434);
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2 >;
to this polytope