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Polytope of Type {412}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {412}*824
Also Known As : 412-gon, {412}. if this polytope has another name.
Group : SmallGroup(824,5)
Rank : 2
Schlafli Type : {412}
Number of vertices, edges, etc : 412, 412
Order of s0s1 : 412
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {412,2} of size 1648
Vertex Figure Of :
   {2,412} of size 1648
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {206}*412
   4-fold quotients : {103}*206
   103-fold quotients : {4}*8
   206-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {824}*1648
Permutation Representation (GAP) :
s0 := (  2,103)(  3,102)(  4,101)(  5,100)(  6, 99)(  7, 98)(  8, 97)(  9, 96)
( 10, 95)( 11, 94)( 12, 93)( 13, 92)( 14, 91)( 15, 90)( 16, 89)( 17, 88)
( 18, 87)( 19, 86)( 20, 85)( 21, 84)( 22, 83)( 23, 82)( 24, 81)( 25, 80)
( 26, 79)( 27, 78)( 28, 77)( 29, 76)( 30, 75)( 31, 74)( 32, 73)( 33, 72)
( 34, 71)( 35, 70)( 36, 69)( 37, 68)( 38, 67)( 39, 66)( 40, 65)( 41, 64)
( 42, 63)( 43, 62)( 44, 61)( 45, 60)( 46, 59)( 47, 58)( 48, 57)( 49, 56)
( 50, 55)( 51, 54)( 52, 53)(105,206)(106,205)(107,204)(108,203)(109,202)
(110,201)(111,200)(112,199)(113,198)(114,197)(115,196)(116,195)(117,194)
(118,193)(119,192)(120,191)(121,190)(122,189)(123,188)(124,187)(125,186)
(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)(133,178)
(134,177)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)(141,170)
(142,169)(143,168)(144,167)(145,166)(146,165)(147,164)(148,163)(149,162)
(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)(207,310)(208,412)
(209,411)(210,410)(211,409)(212,408)(213,407)(214,406)(215,405)(216,404)
(217,403)(218,402)(219,401)(220,400)(221,399)(222,398)(223,397)(224,396)
(225,395)(226,394)(227,393)(228,392)(229,391)(230,390)(231,389)(232,388)
(233,387)(234,386)(235,385)(236,384)(237,383)(238,382)(239,381)(240,380)
(241,379)(242,378)(243,377)(244,376)(245,375)(246,374)(247,373)(248,372)
(249,371)(250,370)(251,369)(252,368)(253,367)(254,366)(255,365)(256,364)
(257,363)(258,362)(259,361)(260,360)(261,359)(262,358)(263,357)(264,356)
(265,355)(266,354)(267,353)(268,352)(269,351)(270,350)(271,349)(272,348)
(273,347)(274,346)(275,345)(276,344)(277,343)(278,342)(279,341)(280,340)
(281,339)(282,338)(283,337)(284,336)(285,335)(286,334)(287,333)(288,332)
(289,331)(290,330)(291,329)(292,328)(293,327)(294,326)(295,325)(296,324)
(297,323)(298,322)(299,321)(300,320)(301,319)(302,318)(303,317)(304,316)
(305,315)(306,314)(307,313)(308,312)(309,311);;
s1 := (  1,208)(  2,207)(  3,309)(  4,308)(  5,307)(  6,306)(  7,305)(  8,304)
(  9,303)( 10,302)( 11,301)( 12,300)( 13,299)( 14,298)( 15,297)( 16,296)
( 17,295)( 18,294)( 19,293)( 20,292)( 21,291)( 22,290)( 23,289)( 24,288)
( 25,287)( 26,286)( 27,285)( 28,284)( 29,283)( 30,282)( 31,281)( 32,280)
( 33,279)( 34,278)( 35,277)( 36,276)( 37,275)( 38,274)( 39,273)( 40,272)
( 41,271)( 42,270)( 43,269)( 44,268)( 45,267)( 46,266)( 47,265)( 48,264)
( 49,263)( 50,262)( 51,261)( 52,260)( 53,259)( 54,258)( 55,257)( 56,256)
( 57,255)( 58,254)( 59,253)( 60,252)( 61,251)( 62,250)( 63,249)( 64,248)
( 65,247)( 66,246)( 67,245)( 68,244)( 69,243)( 70,242)( 71,241)( 72,240)
( 73,239)( 74,238)( 75,237)( 76,236)( 77,235)( 78,234)( 79,233)( 80,232)
( 81,231)( 82,230)( 83,229)( 84,228)( 85,227)( 86,226)( 87,225)( 88,224)
( 89,223)( 90,222)( 91,221)( 92,220)( 93,219)( 94,218)( 95,217)( 96,216)
( 97,215)( 98,214)( 99,213)(100,212)(101,211)(102,210)(103,209)(104,311)
(105,310)(106,412)(107,411)(108,410)(109,409)(110,408)(111,407)(112,406)
(113,405)(114,404)(115,403)(116,402)(117,401)(118,400)(119,399)(120,398)
(121,397)(122,396)(123,395)(124,394)(125,393)(126,392)(127,391)(128,390)
(129,389)(130,388)(131,387)(132,386)(133,385)(134,384)(135,383)(136,382)
(137,381)(138,380)(139,379)(140,378)(141,377)(142,376)(143,375)(144,374)
(145,373)(146,372)(147,371)(148,370)(149,369)(150,368)(151,367)(152,366)
(153,365)(154,364)(155,363)(156,362)(157,361)(158,360)(159,359)(160,358)
(161,357)(162,356)(163,355)(164,354)(165,353)(166,352)(167,351)(168,350)
(169,349)(170,348)(171,347)(172,346)(173,345)(174,344)(175,343)(176,342)
(177,341)(178,340)(179,339)(180,338)(181,337)(182,336)(183,335)(184,334)
(185,333)(186,332)(187,331)(188,330)(189,329)(190,328)(191,327)(192,326)
(193,325)(194,324)(195,323)(196,322)(197,321)(198,320)(199,319)(200,318)
(201,317)(202,316)(203,315)(204,314)(205,313)(206,312);;
poly := Group([s0,s1]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(412)!(  2,103)(  3,102)(  4,101)(  5,100)(  6, 99)(  7, 98)(  8, 97)
(  9, 96)( 10, 95)( 11, 94)( 12, 93)( 13, 92)( 14, 91)( 15, 90)( 16, 89)
( 17, 88)( 18, 87)( 19, 86)( 20, 85)( 21, 84)( 22, 83)( 23, 82)( 24, 81)
( 25, 80)( 26, 79)( 27, 78)( 28, 77)( 29, 76)( 30, 75)( 31, 74)( 32, 73)
( 33, 72)( 34, 71)( 35, 70)( 36, 69)( 37, 68)( 38, 67)( 39, 66)( 40, 65)
( 41, 64)( 42, 63)( 43, 62)( 44, 61)( 45, 60)( 46, 59)( 47, 58)( 48, 57)
( 49, 56)( 50, 55)( 51, 54)( 52, 53)(105,206)(106,205)(107,204)(108,203)
(109,202)(110,201)(111,200)(112,199)(113,198)(114,197)(115,196)(116,195)
(117,194)(118,193)(119,192)(120,191)(121,190)(122,189)(123,188)(124,187)
(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)
(133,178)(134,177)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)
(141,170)(142,169)(143,168)(144,167)(145,166)(146,165)(147,164)(148,163)
(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)(207,310)
(208,412)(209,411)(210,410)(211,409)(212,408)(213,407)(214,406)(215,405)
(216,404)(217,403)(218,402)(219,401)(220,400)(221,399)(222,398)(223,397)
(224,396)(225,395)(226,394)(227,393)(228,392)(229,391)(230,390)(231,389)
(232,388)(233,387)(234,386)(235,385)(236,384)(237,383)(238,382)(239,381)
(240,380)(241,379)(242,378)(243,377)(244,376)(245,375)(246,374)(247,373)
(248,372)(249,371)(250,370)(251,369)(252,368)(253,367)(254,366)(255,365)
(256,364)(257,363)(258,362)(259,361)(260,360)(261,359)(262,358)(263,357)
(264,356)(265,355)(266,354)(267,353)(268,352)(269,351)(270,350)(271,349)
(272,348)(273,347)(274,346)(275,345)(276,344)(277,343)(278,342)(279,341)
(280,340)(281,339)(282,338)(283,337)(284,336)(285,335)(286,334)(287,333)
(288,332)(289,331)(290,330)(291,329)(292,328)(293,327)(294,326)(295,325)
(296,324)(297,323)(298,322)(299,321)(300,320)(301,319)(302,318)(303,317)
(304,316)(305,315)(306,314)(307,313)(308,312)(309,311);
s1 := Sym(412)!(  1,208)(  2,207)(  3,309)(  4,308)(  5,307)(  6,306)(  7,305)
(  8,304)(  9,303)( 10,302)( 11,301)( 12,300)( 13,299)( 14,298)( 15,297)
( 16,296)( 17,295)( 18,294)( 19,293)( 20,292)( 21,291)( 22,290)( 23,289)
( 24,288)( 25,287)( 26,286)( 27,285)( 28,284)( 29,283)( 30,282)( 31,281)
( 32,280)( 33,279)( 34,278)( 35,277)( 36,276)( 37,275)( 38,274)( 39,273)
( 40,272)( 41,271)( 42,270)( 43,269)( 44,268)( 45,267)( 46,266)( 47,265)
( 48,264)( 49,263)( 50,262)( 51,261)( 52,260)( 53,259)( 54,258)( 55,257)
( 56,256)( 57,255)( 58,254)( 59,253)( 60,252)( 61,251)( 62,250)( 63,249)
( 64,248)( 65,247)( 66,246)( 67,245)( 68,244)( 69,243)( 70,242)( 71,241)
( 72,240)( 73,239)( 74,238)( 75,237)( 76,236)( 77,235)( 78,234)( 79,233)
( 80,232)( 81,231)( 82,230)( 83,229)( 84,228)( 85,227)( 86,226)( 87,225)
( 88,224)( 89,223)( 90,222)( 91,221)( 92,220)( 93,219)( 94,218)( 95,217)
( 96,216)( 97,215)( 98,214)( 99,213)(100,212)(101,211)(102,210)(103,209)
(104,311)(105,310)(106,412)(107,411)(108,410)(109,409)(110,408)(111,407)
(112,406)(113,405)(114,404)(115,403)(116,402)(117,401)(118,400)(119,399)
(120,398)(121,397)(122,396)(123,395)(124,394)(125,393)(126,392)(127,391)
(128,390)(129,389)(130,388)(131,387)(132,386)(133,385)(134,384)(135,383)
(136,382)(137,381)(138,380)(139,379)(140,378)(141,377)(142,376)(143,375)
(144,374)(145,373)(146,372)(147,371)(148,370)(149,369)(150,368)(151,367)
(152,366)(153,365)(154,364)(155,363)(156,362)(157,361)(158,360)(159,359)
(160,358)(161,357)(162,356)(163,355)(164,354)(165,353)(166,352)(167,351)
(168,350)(169,349)(170,348)(171,347)(172,346)(173,345)(174,344)(175,343)
(176,342)(177,341)(178,340)(179,339)(180,338)(181,337)(182,336)(183,335)
(184,334)(185,333)(186,332)(187,331)(188,330)(189,329)(190,328)(191,327)
(192,326)(193,325)(194,324)(195,323)(196,322)(197,321)(198,320)(199,319)
(200,318)(201,317)(202,316)(203,315)(204,314)(205,313)(206,312);
poly := sub<Sym(412)|s0,s1>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope