Polytope of Type {2,4,108}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,108}*1728b
if this polytope has a name.
Group : SmallGroup(1728,11356)
Rank : 4
Schlafli Type : {2,4,108}
Number of vertices, edges, etc : 2, 4, 216, 108
Order of s0s1s2s3 : 108
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,4,54}*864b
3-fold quotients : {2,4,36}*576b
4-fold quotients : {2,4,27}*432
6-fold quotients : {2,4,18}*288b
9-fold quotients : {2,4,12}*192b
12-fold quotients : {2,4,9}*144
18-fold quotients : {2,4,6}*96c
36-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)( 16, 18)
( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)( 32, 34)
( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)( 64, 66)
( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)( 96, 98)
( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)(128,130)
(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)(144,146)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)(160,162)
(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)(176,178)
(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)(208,210)
(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)(224,226)
(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)(240,242)
(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)(256,258)
(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)(272,274)
(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)(288,290)
(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)(304,306)
(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)(320,322)
(323,325)(324,326)(327,329)(328,330)(331,333)(332,334)(335,337)(336,338)
(339,341)(340,342)(343,345)(344,346)(347,349)(348,350)(351,353)(352,354)
(355,357)(356,358)(359,361)(360,362)(363,365)(364,366)(367,369)(368,370)
(371,373)(372,374)(375,377)(376,378)(379,381)(380,382)(383,385)(384,386)
(387,389)(388,390)(391,393)(392,394)(395,397)(396,398)(399,401)(400,402)
(403,405)(404,406)(407,409)(408,410)(411,413)(412,414)(415,417)(416,418)
(419,421)(420,422)(423,425)(424,426)(427,429)(428,430)(431,433)(432,434);;
s2 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 35)( 16, 37)( 17, 36)
( 18, 38)( 19, 31)( 20, 33)( 21, 32)( 22, 34)( 23, 27)( 24, 29)( 25, 28)
( 26, 30)( 39,107)( 40,109)( 41,108)( 42,110)( 43,103)( 44,105)( 45,104)
( 46,106)( 47, 99)( 48,101)( 49,100)( 50,102)( 51, 95)( 52, 97)( 53, 96)
( 54, 98)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 87)( 60, 89)( 61, 88)
( 62, 90)( 63, 83)( 64, 85)( 65, 84)( 66, 86)( 67, 79)( 68, 81)( 69, 80)
( 70, 82)( 71, 75)( 72, 77)( 73, 76)( 74, 78)(112,113)(115,119)(116,121)
(117,120)(118,122)(123,143)(124,145)(125,144)(126,146)(127,139)(128,141)
(129,140)(130,142)(131,135)(132,137)(133,136)(134,138)(147,215)(148,217)
(149,216)(150,218)(151,211)(152,213)(153,212)(154,214)(155,207)(156,209)
(157,208)(158,210)(159,203)(160,205)(161,204)(162,206)(163,199)(164,201)
(165,200)(166,202)(167,195)(168,197)(169,196)(170,198)(171,191)(172,193)
(173,192)(174,194)(175,187)(176,189)(177,188)(178,190)(179,183)(180,185)
(181,184)(182,186)(219,327)(220,329)(221,328)(222,330)(223,335)(224,337)
(225,336)(226,338)(227,331)(228,333)(229,332)(230,334)(231,359)(232,361)
(233,360)(234,362)(235,355)(236,357)(237,356)(238,358)(239,351)(240,353)
(241,352)(242,354)(243,347)(244,349)(245,348)(246,350)(247,343)(248,345)
(249,344)(250,346)(251,339)(252,341)(253,340)(254,342)(255,431)(256,433)
(257,432)(258,434)(259,427)(260,429)(261,428)(262,430)(263,423)(264,425)
(265,424)(266,426)(267,419)(268,421)(269,420)(270,422)(271,415)(272,417)
(273,416)(274,418)(275,411)(276,413)(277,412)(278,414)(279,407)(280,409)
(281,408)(282,410)(283,403)(284,405)(285,404)(286,406)(287,399)(288,401)
(289,400)(290,402)(291,395)(292,397)(293,396)(294,398)(295,391)(296,393)
(297,392)(298,394)(299,387)(300,389)(301,388)(302,390)(303,383)(304,385)
(305,384)(306,386)(307,379)(308,381)(309,380)(310,382)(311,375)(312,377)
(313,376)(314,378)(315,371)(316,373)(317,372)(318,374)(319,367)(320,369)
(321,368)(322,370)(323,363)(324,365)(325,364)(326,366);;
s3 := ( 3,291)( 4,294)( 5,293)( 6,292)( 7,299)( 8,302)( 9,301)( 10,300)
( 11,295)( 12,298)( 13,297)( 14,296)( 15,323)( 16,326)( 17,325)( 18,324)
( 19,319)( 20,322)( 21,321)( 22,320)( 23,315)( 24,318)( 25,317)( 26,316)
( 27,311)( 28,314)( 29,313)( 30,312)( 31,307)( 32,310)( 33,309)( 34,308)
( 35,303)( 36,306)( 37,305)( 38,304)( 39,255)( 40,258)( 41,257)( 42,256)
( 43,263)( 44,266)( 45,265)( 46,264)( 47,259)( 48,262)( 49,261)( 50,260)
( 51,287)( 52,290)( 53,289)( 54,288)( 55,283)( 56,286)( 57,285)( 58,284)
( 59,279)( 60,282)( 61,281)( 62,280)( 63,275)( 64,278)( 65,277)( 66,276)
( 67,271)( 68,274)( 69,273)( 70,272)( 71,267)( 72,270)( 73,269)( 74,268)
( 75,219)( 76,222)( 77,221)( 78,220)( 79,227)( 80,230)( 81,229)( 82,228)
( 83,223)( 84,226)( 85,225)( 86,224)( 87,251)( 88,254)( 89,253)( 90,252)
( 91,247)( 92,250)( 93,249)( 94,248)( 95,243)( 96,246)( 97,245)( 98,244)
( 99,239)(100,242)(101,241)(102,240)(103,235)(104,238)(105,237)(106,236)
(107,231)(108,234)(109,233)(110,232)(111,399)(112,402)(113,401)(114,400)
(115,407)(116,410)(117,409)(118,408)(119,403)(120,406)(121,405)(122,404)
(123,431)(124,434)(125,433)(126,432)(127,427)(128,430)(129,429)(130,428)
(131,423)(132,426)(133,425)(134,424)(135,419)(136,422)(137,421)(138,420)
(139,415)(140,418)(141,417)(142,416)(143,411)(144,414)(145,413)(146,412)
(147,363)(148,366)(149,365)(150,364)(151,371)(152,374)(153,373)(154,372)
(155,367)(156,370)(157,369)(158,368)(159,395)(160,398)(161,397)(162,396)
(163,391)(164,394)(165,393)(166,392)(167,387)(168,390)(169,389)(170,388)
(171,383)(172,386)(173,385)(174,384)(175,379)(176,382)(177,381)(178,380)
(179,375)(180,378)(181,377)(182,376)(183,327)(184,330)(185,329)(186,328)
(187,335)(188,338)(189,337)(190,336)(191,331)(192,334)(193,333)(194,332)
(195,359)(196,362)(197,361)(198,360)(199,355)(200,358)(201,357)(202,356)
(203,351)(204,354)(205,353)(206,352)(207,347)(208,350)(209,349)(210,348)
(211,343)(212,346)(213,345)(214,344)(215,339)(216,342)(217,341)(218,340);;
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*s3*s2*s1*s2*s3*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*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*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*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, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)
( 16, 18)( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)
( 32, 34)( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)
( 64, 66)( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)
( 96, 98)( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)
(128,130)(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)
(144,146)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)
(160,162)(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)
(176,178)(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)
(208,210)(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)
(224,226)(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)
(240,242)(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)
(256,258)(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)
(272,274)(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)
(288,290)(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)
(304,306)(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)
(320,322)(323,325)(324,326)(327,329)(328,330)(331,333)(332,334)(335,337)
(336,338)(339,341)(340,342)(343,345)(344,346)(347,349)(348,350)(351,353)
(352,354)(355,357)(356,358)(359,361)(360,362)(363,365)(364,366)(367,369)
(368,370)(371,373)(372,374)(375,377)(376,378)(379,381)(380,382)(383,385)
(384,386)(387,389)(388,390)(391,393)(392,394)(395,397)(396,398)(399,401)
(400,402)(403,405)(404,406)(407,409)(408,410)(411,413)(412,414)(415,417)
(416,418)(419,421)(420,422)(423,425)(424,426)(427,429)(428,430)(431,433)
(432,434);
s2 := Sym(434)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 15, 35)( 16, 37)
( 17, 36)( 18, 38)( 19, 31)( 20, 33)( 21, 32)( 22, 34)( 23, 27)( 24, 29)
( 25, 28)( 26, 30)( 39,107)( 40,109)( 41,108)( 42,110)( 43,103)( 44,105)
( 45,104)( 46,106)( 47, 99)( 48,101)( 49,100)( 50,102)( 51, 95)( 52, 97)
( 53, 96)( 54, 98)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 87)( 60, 89)
( 61, 88)( 62, 90)( 63, 83)( 64, 85)( 65, 84)( 66, 86)( 67, 79)( 68, 81)
( 69, 80)( 70, 82)( 71, 75)( 72, 77)( 73, 76)( 74, 78)(112,113)(115,119)
(116,121)(117,120)(118,122)(123,143)(124,145)(125,144)(126,146)(127,139)
(128,141)(129,140)(130,142)(131,135)(132,137)(133,136)(134,138)(147,215)
(148,217)(149,216)(150,218)(151,211)(152,213)(153,212)(154,214)(155,207)
(156,209)(157,208)(158,210)(159,203)(160,205)(161,204)(162,206)(163,199)
(164,201)(165,200)(166,202)(167,195)(168,197)(169,196)(170,198)(171,191)
(172,193)(173,192)(174,194)(175,187)(176,189)(177,188)(178,190)(179,183)
(180,185)(181,184)(182,186)(219,327)(220,329)(221,328)(222,330)(223,335)
(224,337)(225,336)(226,338)(227,331)(228,333)(229,332)(230,334)(231,359)
(232,361)(233,360)(234,362)(235,355)(236,357)(237,356)(238,358)(239,351)
(240,353)(241,352)(242,354)(243,347)(244,349)(245,348)(246,350)(247,343)
(248,345)(249,344)(250,346)(251,339)(252,341)(253,340)(254,342)(255,431)
(256,433)(257,432)(258,434)(259,427)(260,429)(261,428)(262,430)(263,423)
(264,425)(265,424)(266,426)(267,419)(268,421)(269,420)(270,422)(271,415)
(272,417)(273,416)(274,418)(275,411)(276,413)(277,412)(278,414)(279,407)
(280,409)(281,408)(282,410)(283,403)(284,405)(285,404)(286,406)(287,399)
(288,401)(289,400)(290,402)(291,395)(292,397)(293,396)(294,398)(295,391)
(296,393)(297,392)(298,394)(299,387)(300,389)(301,388)(302,390)(303,383)
(304,385)(305,384)(306,386)(307,379)(308,381)(309,380)(310,382)(311,375)
(312,377)(313,376)(314,378)(315,371)(316,373)(317,372)(318,374)(319,367)
(320,369)(321,368)(322,370)(323,363)(324,365)(325,364)(326,366);
s3 := Sym(434)!( 3,291)( 4,294)( 5,293)( 6,292)( 7,299)( 8,302)( 9,301)
( 10,300)( 11,295)( 12,298)( 13,297)( 14,296)( 15,323)( 16,326)( 17,325)
( 18,324)( 19,319)( 20,322)( 21,321)( 22,320)( 23,315)( 24,318)( 25,317)
( 26,316)( 27,311)( 28,314)( 29,313)( 30,312)( 31,307)( 32,310)( 33,309)
( 34,308)( 35,303)( 36,306)( 37,305)( 38,304)( 39,255)( 40,258)( 41,257)
( 42,256)( 43,263)( 44,266)( 45,265)( 46,264)( 47,259)( 48,262)( 49,261)
( 50,260)( 51,287)( 52,290)( 53,289)( 54,288)( 55,283)( 56,286)( 57,285)
( 58,284)( 59,279)( 60,282)( 61,281)( 62,280)( 63,275)( 64,278)( 65,277)
( 66,276)( 67,271)( 68,274)( 69,273)( 70,272)( 71,267)( 72,270)( 73,269)
( 74,268)( 75,219)( 76,222)( 77,221)( 78,220)( 79,227)( 80,230)( 81,229)
( 82,228)( 83,223)( 84,226)( 85,225)( 86,224)( 87,251)( 88,254)( 89,253)
( 90,252)( 91,247)( 92,250)( 93,249)( 94,248)( 95,243)( 96,246)( 97,245)
( 98,244)( 99,239)(100,242)(101,241)(102,240)(103,235)(104,238)(105,237)
(106,236)(107,231)(108,234)(109,233)(110,232)(111,399)(112,402)(113,401)
(114,400)(115,407)(116,410)(117,409)(118,408)(119,403)(120,406)(121,405)
(122,404)(123,431)(124,434)(125,433)(126,432)(127,427)(128,430)(129,429)
(130,428)(131,423)(132,426)(133,425)(134,424)(135,419)(136,422)(137,421)
(138,420)(139,415)(140,418)(141,417)(142,416)(143,411)(144,414)(145,413)
(146,412)(147,363)(148,366)(149,365)(150,364)(151,371)(152,374)(153,373)
(154,372)(155,367)(156,370)(157,369)(158,368)(159,395)(160,398)(161,397)
(162,396)(163,391)(164,394)(165,393)(166,392)(167,387)(168,390)(169,389)
(170,388)(171,383)(172,386)(173,385)(174,384)(175,379)(176,382)(177,381)
(178,380)(179,375)(180,378)(181,377)(182,376)(183,327)(184,330)(185,329)
(186,328)(187,335)(188,338)(189,337)(190,336)(191,331)(192,334)(193,333)
(194,332)(195,359)(196,362)(197,361)(198,360)(199,355)(200,358)(201,357)
(202,356)(203,351)(204,354)(205,353)(206,352)(207,347)(208,350)(209,349)
(210,348)(211,343)(212,346)(213,345)(214,344)(215,339)(216,342)(217,341)
(218,340);
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*s3*s2*s1*s2*s3*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*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*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*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