Polytope of Type {404}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {404}*808
Also Known As : 404-gon, {404}. if this polytope has another name.
Group : SmallGroup(808,6)
Rank : 2
Schlafli Type : {404}
Number of vertices, edges, etc : 404, 404
Order of s₀s₁ : 404
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {404,2} of size 1616
Vertex Figure Of :
   {2,404} of size 1616
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {202}*404
   4-fold quotients : {101}*202
   101-fold quotients : {4}*8
   202-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {808}*1616
Permutation Representation (GAP) :

s0 := (  2,101)(  3,100)(  4, 99)(  5, 98)(  6, 97)(  7, 96)(  8, 95)(  9, 94)
( 10, 93)( 11, 92)( 12, 91)( 13, 90)( 14, 89)( 15, 88)( 16, 87)( 17, 86)
( 18, 85)( 19, 84)( 20, 83)( 21, 82)( 22, 81)( 23, 80)( 24, 79)( 25, 78)
( 26, 77)( 27, 76)( 28, 75)( 29, 74)( 30, 73)( 31, 72)( 32, 71)( 33, 70)
( 34, 69)( 35, 68)( 36, 67)( 37, 66)( 38, 65)( 39, 64)( 40, 63)( 41, 62)
( 42, 61)( 43, 60)( 44, 59)( 45, 58)( 46, 57)( 47, 56)( 48, 55)( 49, 54)
( 50, 53)( 51, 52)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)
(109,196)(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)(116,189)
(117,188)(118,187)(119,186)(120,185)(121,184)(122,183)(123,182)(124,181)
(125,180)(126,179)(127,178)(128,177)(129,176)(130,175)(131,174)(132,173)
(133,172)(134,171)(135,170)(136,169)(137,168)(138,167)(139,166)(140,165)
(141,164)(142,163)(143,162)(144,161)(145,160)(146,159)(147,158)(148,157)
(149,156)(150,155)(151,154)(152,153)(203,304)(204,404)(205,403)(206,402)
(207,401)(208,400)(209,399)(210,398)(211,397)(212,396)(213,395)(214,394)
(215,393)(216,392)(217,391)(218,390)(219,389)(220,388)(221,387)(222,386)
(223,385)(224,384)(225,383)(226,382)(227,381)(228,380)(229,379)(230,378)
(231,377)(232,376)(233,375)(234,374)(235,373)(236,372)(237,371)(238,370)
(239,369)(240,368)(241,367)(242,366)(243,365)(244,364)(245,363)(246,362)
(247,361)(248,360)(249,359)(250,358)(251,357)(252,356)(253,355)(254,354)
(255,353)(256,352)(257,351)(258,350)(259,349)(260,348)(261,347)(262,346)
(263,345)(264,344)(265,343)(266,342)(267,341)(268,340)(269,339)(270,338)
(271,337)(272,336)(273,335)(274,334)(275,333)(276,332)(277,331)(278,330)
(279,329)(280,328)(281,327)(282,326)(283,325)(284,324)(285,323)(286,322)
(287,321)(288,320)(289,319)(290,318)(291,317)(292,316)(293,315)(294,314)
(295,313)(296,312)(297,311)(298,310)(299,309)(300,308)(301,307)(302,306)
(303,305);;
s1 := (  1,204)(  2,203)(  3,303)(  4,302)(  5,301)(  6,300)(  7,299)(  8,298)
(  9,297)( 10,296)( 11,295)( 12,294)( 13,293)( 14,292)( 15,291)( 16,290)
( 17,289)( 18,288)( 19,287)( 20,286)( 21,285)( 22,284)( 23,283)( 24,282)
( 25,281)( 26,280)( 27,279)( 28,278)( 29,277)( 30,276)( 31,275)( 32,274)
( 33,273)( 34,272)( 35,271)( 36,270)( 37,269)( 38,268)( 39,267)( 40,266)
( 41,265)( 42,264)( 43,263)( 44,262)( 45,261)( 46,260)( 47,259)( 48,258)
( 49,257)( 50,256)( 51,255)( 52,254)( 53,253)( 54,252)( 55,251)( 56,250)
( 57,249)( 58,248)( 59,247)( 60,246)( 61,245)( 62,244)( 63,243)( 64,242)
( 65,241)( 66,240)( 67,239)( 68,238)( 69,237)( 70,236)( 71,235)( 72,234)
( 73,233)( 74,232)( 75,231)( 76,230)( 77,229)( 78,228)( 79,227)( 80,226)
( 81,225)( 82,224)( 83,223)( 84,222)( 85,221)( 86,220)( 87,219)( 88,218)
( 89,217)( 90,216)( 91,215)( 92,214)( 93,213)( 94,212)( 95,211)( 96,210)
( 97,209)( 98,208)( 99,207)(100,206)(101,205)(102,305)(103,304)(104,404)
(105,403)(106,402)(107,401)(108,400)(109,399)(110,398)(111,397)(112,396)
(113,395)(114,394)(115,393)(116,392)(117,391)(118,390)(119,389)(120,388)
(121,387)(122,386)(123,385)(124,384)(125,383)(126,382)(127,381)(128,380)
(129,379)(130,378)(131,377)(132,376)(133,375)(134,374)(135,373)(136,372)
(137,371)(138,370)(139,369)(140,368)(141,367)(142,366)(143,365)(144,364)
(145,363)(146,362)(147,361)(148,360)(149,359)(150,358)(151,357)(152,356)
(153,355)(154,354)(155,353)(156,352)(157,351)(158,350)(159,349)(160,348)
(161,347)(162,346)(163,345)(164,344)(165,343)(166,342)(167,341)(168,340)
(169,339)(170,338)(171,337)(172,336)(173,335)(174,334)(175,333)(176,332)
(177,331)(178,330)(179,329)(180,328)(181,327)(182,326)(183,325)(184,324)
(185,323)(186,322)(187,321)(188,320)(189,319)(190,318)(191,317)(192,316)
(193,315)(194,314)(195,313)(196,312)(197,311)(198,310)(199,309)(200,308)
(201,307)(202,306);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(404)!(  2,101)(  3,100)(  4, 99)(  5, 98)(  6, 97)(  7, 96)(  8, 95)
(  9, 94)( 10, 93)( 11, 92)( 12, 91)( 13, 90)( 14, 89)( 15, 88)( 16, 87)
( 17, 86)( 18, 85)( 19, 84)( 20, 83)( 21, 82)( 22, 81)( 23, 80)( 24, 79)
( 25, 78)( 26, 77)( 27, 76)( 28, 75)( 29, 74)( 30, 73)( 31, 72)( 32, 71)
( 33, 70)( 34, 69)( 35, 68)( 36, 67)( 37, 66)( 38, 65)( 39, 64)( 40, 63)
( 41, 62)( 42, 61)( 43, 60)( 44, 59)( 45, 58)( 46, 57)( 47, 56)( 48, 55)
( 49, 54)( 50, 53)( 51, 52)(103,202)(104,201)(105,200)(106,199)(107,198)
(108,197)(109,196)(110,195)(111,194)(112,193)(113,192)(114,191)(115,190)
(116,189)(117,188)(118,187)(119,186)(120,185)(121,184)(122,183)(123,182)
(124,181)(125,180)(126,179)(127,178)(128,177)(129,176)(130,175)(131,174)
(132,173)(133,172)(134,171)(135,170)(136,169)(137,168)(138,167)(139,166)
(140,165)(141,164)(142,163)(143,162)(144,161)(145,160)(146,159)(147,158)
(148,157)(149,156)(150,155)(151,154)(152,153)(203,304)(204,404)(205,403)
(206,402)(207,401)(208,400)(209,399)(210,398)(211,397)(212,396)(213,395)
(214,394)(215,393)(216,392)(217,391)(218,390)(219,389)(220,388)(221,387)
(222,386)(223,385)(224,384)(225,383)(226,382)(227,381)(228,380)(229,379)
(230,378)(231,377)(232,376)(233,375)(234,374)(235,373)(236,372)(237,371)
(238,370)(239,369)(240,368)(241,367)(242,366)(243,365)(244,364)(245,363)
(246,362)(247,361)(248,360)(249,359)(250,358)(251,357)(252,356)(253,355)
(254,354)(255,353)(256,352)(257,351)(258,350)(259,349)(260,348)(261,347)
(262,346)(263,345)(264,344)(265,343)(266,342)(267,341)(268,340)(269,339)
(270,338)(271,337)(272,336)(273,335)(274,334)(275,333)(276,332)(277,331)
(278,330)(279,329)(280,328)(281,327)(282,326)(283,325)(284,324)(285,323)
(286,322)(287,321)(288,320)(289,319)(290,318)(291,317)(292,316)(293,315)
(294,314)(295,313)(296,312)(297,311)(298,310)(299,309)(300,308)(301,307)
(302,306)(303,305);
s1 := Sym(404)!(  1,204)(  2,203)(  3,303)(  4,302)(  5,301)(  6,300)(  7,299)
(  8,298)(  9,297)( 10,296)( 11,295)( 12,294)( 13,293)( 14,292)( 15,291)
( 16,290)( 17,289)( 18,288)( 19,287)( 20,286)( 21,285)( 22,284)( 23,283)
( 24,282)( 25,281)( 26,280)( 27,279)( 28,278)( 29,277)( 30,276)( 31,275)
( 32,274)( 33,273)( 34,272)( 35,271)( 36,270)( 37,269)( 38,268)( 39,267)
( 40,266)( 41,265)( 42,264)( 43,263)( 44,262)( 45,261)( 46,260)( 47,259)
( 48,258)( 49,257)( 50,256)( 51,255)( 52,254)( 53,253)( 54,252)( 55,251)
( 56,250)( 57,249)( 58,248)( 59,247)( 60,246)( 61,245)( 62,244)( 63,243)
( 64,242)( 65,241)( 66,240)( 67,239)( 68,238)( 69,237)( 70,236)( 71,235)
( 72,234)( 73,233)( 74,232)( 75,231)( 76,230)( 77,229)( 78,228)( 79,227)
( 80,226)( 81,225)( 82,224)( 83,223)( 84,222)( 85,221)( 86,220)( 87,219)
( 88,218)( 89,217)( 90,216)( 91,215)( 92,214)( 93,213)( 94,212)( 95,211)
( 96,210)( 97,209)( 98,208)( 99,207)(100,206)(101,205)(102,305)(103,304)
(104,404)(105,403)(106,402)(107,401)(108,400)(109,399)(110,398)(111,397)
(112,396)(113,395)(114,394)(115,393)(116,392)(117,391)(118,390)(119,389)
(120,388)(121,387)(122,386)(123,385)(124,384)(125,383)(126,382)(127,381)
(128,380)(129,379)(130,378)(131,377)(132,376)(133,375)(134,374)(135,373)
(136,372)(137,371)(138,370)(139,369)(140,368)(141,367)(142,366)(143,365)
(144,364)(145,363)(146,362)(147,361)(148,360)(149,359)(150,358)(151,357)
(152,356)(153,355)(154,354)(155,353)(156,352)(157,351)(158,350)(159,349)
(160,348)(161,347)(162,346)(163,345)(164,344)(165,343)(166,342)(167,341)
(168,340)(169,339)(170,338)(171,337)(172,336)(173,335)(174,334)(175,333)
(176,332)(177,331)(178,330)(179,329)(180,328)(181,327)(182,326)(183,325)
(184,324)(185,323)(186,322)(187,321)(188,320)(189,319)(190,318)(191,317)
(192,316)(193,315)(194,314)(195,313)(196,312)(197,311)(198,310)(199,309)
(200,308)(201,307)(202,306);
poly := sub<Sym(404)|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 >;

References : None.
to this polytope