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Polytope of Type {2,108,4}

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