Polytope of Type {410}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {410}*820
Also Known As : 410-gon, {410}. if this polytope has another name.
Group : SmallGroup(820,19)
Rank : 2
Schlafli Type : {410}
Number of vertices, edges, etc : 410, 410
Order of s0s1 : 410
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{410,2} of size 1640
Vertex Figure Of :
{2,410} of size 1640
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {205}*410
5-fold quotients : {82}*164
10-fold quotients : {41}*82
41-fold quotients : {10}*20
82-fold quotients : {5}*10
205-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {820}*1640
Permutation Representation (GAP) :
s0 := ( 2, 41)( 3, 40)( 4, 39)( 5, 38)( 6, 37)( 7, 36)( 8, 35)( 9, 34)
( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)( 17, 26)
( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 42,165)( 43,205)( 44,204)( 45,203)
( 46,202)( 47,201)( 48,200)( 49,199)( 50,198)( 51,197)( 52,196)( 53,195)
( 54,194)( 55,193)( 56,192)( 57,191)( 58,190)( 59,189)( 60,188)( 61,187)
( 62,186)( 63,185)( 64,184)( 65,183)( 66,182)( 67,181)( 68,180)( 69,179)
( 70,178)( 71,177)( 72,176)( 73,175)( 74,174)( 75,173)( 76,172)( 77,171)
( 78,170)( 79,169)( 80,168)( 81,167)( 82,166)( 83,124)( 84,164)( 85,163)
( 86,162)( 87,161)( 88,160)( 89,159)( 90,158)( 91,157)( 92,156)( 93,155)
( 94,154)( 95,153)( 96,152)( 97,151)( 98,150)( 99,149)(100,148)(101,147)
(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)
(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)
(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(207,246)(208,245)
(209,244)(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)
(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)
(225,228)(226,227)(247,370)(248,410)(249,409)(250,408)(251,407)(252,406)
(253,405)(254,404)(255,403)(256,402)(257,401)(258,400)(259,399)(260,398)
(261,397)(262,396)(263,395)(264,394)(265,393)(266,392)(267,391)(268,390)
(269,389)(270,388)(271,387)(272,386)(273,385)(274,384)(275,383)(276,382)
(277,381)(278,380)(279,379)(280,378)(281,377)(282,376)(283,375)(284,374)
(285,373)(286,372)(287,371)(288,329)(289,369)(290,368)(291,367)(292,366)
(293,365)(294,364)(295,363)(296,362)(297,361)(298,360)(299,359)(300,358)
(301,357)(302,356)(303,355)(304,354)(305,353)(306,352)(307,351)(308,350)
(309,349)(310,348)(311,347)(312,346)(313,345)(314,344)(315,343)(316,342)
(317,341)(318,340)(319,339)(320,338)(321,337)(322,336)(323,335)(324,334)
(325,333)(326,332)(327,331)(328,330);;
s1 := ( 1,248)( 2,247)( 3,287)( 4,286)( 5,285)( 6,284)( 7,283)( 8,282)
( 9,281)( 10,280)( 11,279)( 12,278)( 13,277)( 14,276)( 15,275)( 16,274)
( 17,273)( 18,272)( 19,271)( 20,270)( 21,269)( 22,268)( 23,267)( 24,266)
( 25,265)( 26,264)( 27,263)( 28,262)( 29,261)( 30,260)( 31,259)( 32,258)
( 33,257)( 34,256)( 35,255)( 36,254)( 37,253)( 38,252)( 39,251)( 40,250)
( 41,249)( 42,207)( 43,206)( 44,246)( 45,245)( 46,244)( 47,243)( 48,242)
( 49,241)( 50,240)( 51,239)( 52,238)( 53,237)( 54,236)( 55,235)( 56,234)
( 57,233)( 58,232)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)( 64,226)
( 65,225)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,219)( 72,218)
( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,212)( 79,211)( 80,210)
( 81,209)( 82,208)( 83,371)( 84,370)( 85,410)( 86,409)( 87,408)( 88,407)
( 89,406)( 90,405)( 91,404)( 92,403)( 93,402)( 94,401)( 95,400)( 96,399)
( 97,398)( 98,397)( 99,396)(100,395)(101,394)(102,393)(103,392)(104,391)
(105,390)(106,389)(107,388)(108,387)(109,386)(110,385)(111,384)(112,383)
(113,382)(114,381)(115,380)(116,379)(117,378)(118,377)(119,376)(120,375)
(121,374)(122,373)(123,372)(124,330)(125,329)(126,369)(127,368)(128,367)
(129,366)(130,365)(131,364)(132,363)(133,362)(134,361)(135,360)(136,359)
(137,358)(138,357)(139,356)(140,355)(141,354)(142,353)(143,352)(144,351)
(145,350)(146,349)(147,348)(148,347)(149,346)(150,345)(151,344)(152,343)
(153,342)(154,341)(155,340)(156,339)(157,338)(158,337)(159,336)(160,335)
(161,334)(162,333)(163,332)(164,331)(165,289)(166,288)(167,328)(168,327)
(169,326)(170,325)(171,324)(172,323)(173,322)(174,321)(175,320)(176,319)
(177,318)(178,317)(179,316)(180,315)(181,314)(182,313)(183,312)(184,311)
(185,310)(186,309)(187,308)(188,307)(189,306)(190,305)(191,304)(192,303)
(193,302)(194,301)(195,300)(196,299)(197,298)(198,297)(199,296)(200,295)
(201,294)(202,293)(203,292)(204,291)(205,290);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(410)!( 2, 41)( 3, 40)( 4, 39)( 5, 38)( 6, 37)( 7, 36)( 8, 35)
( 9, 34)( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)
( 17, 26)( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 42,165)( 43,205)( 44,204)
( 45,203)( 46,202)( 47,201)( 48,200)( 49,199)( 50,198)( 51,197)( 52,196)
( 53,195)( 54,194)( 55,193)( 56,192)( 57,191)( 58,190)( 59,189)( 60,188)
( 61,187)( 62,186)( 63,185)( 64,184)( 65,183)( 66,182)( 67,181)( 68,180)
( 69,179)( 70,178)( 71,177)( 72,176)( 73,175)( 74,174)( 75,173)( 76,172)
( 77,171)( 78,170)( 79,169)( 80,168)( 81,167)( 82,166)( 83,124)( 84,164)
( 85,163)( 86,162)( 87,161)( 88,160)( 89,159)( 90,158)( 91,157)( 92,156)
( 93,155)( 94,154)( 95,153)( 96,152)( 97,151)( 98,150)( 99,149)(100,148)
(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)
(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)(115,133)(116,132)
(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(207,246)
(208,245)(209,244)(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)
(216,237)(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)
(224,229)(225,228)(226,227)(247,370)(248,410)(249,409)(250,408)(251,407)
(252,406)(253,405)(254,404)(255,403)(256,402)(257,401)(258,400)(259,399)
(260,398)(261,397)(262,396)(263,395)(264,394)(265,393)(266,392)(267,391)
(268,390)(269,389)(270,388)(271,387)(272,386)(273,385)(274,384)(275,383)
(276,382)(277,381)(278,380)(279,379)(280,378)(281,377)(282,376)(283,375)
(284,374)(285,373)(286,372)(287,371)(288,329)(289,369)(290,368)(291,367)
(292,366)(293,365)(294,364)(295,363)(296,362)(297,361)(298,360)(299,359)
(300,358)(301,357)(302,356)(303,355)(304,354)(305,353)(306,352)(307,351)
(308,350)(309,349)(310,348)(311,347)(312,346)(313,345)(314,344)(315,343)
(316,342)(317,341)(318,340)(319,339)(320,338)(321,337)(322,336)(323,335)
(324,334)(325,333)(326,332)(327,331)(328,330);
s1 := Sym(410)!( 1,248)( 2,247)( 3,287)( 4,286)( 5,285)( 6,284)( 7,283)
( 8,282)( 9,281)( 10,280)( 11,279)( 12,278)( 13,277)( 14,276)( 15,275)
( 16,274)( 17,273)( 18,272)( 19,271)( 20,270)( 21,269)( 22,268)( 23,267)
( 24,266)( 25,265)( 26,264)( 27,263)( 28,262)( 29,261)( 30,260)( 31,259)
( 32,258)( 33,257)( 34,256)( 35,255)( 36,254)( 37,253)( 38,252)( 39,251)
( 40,250)( 41,249)( 42,207)( 43,206)( 44,246)( 45,245)( 46,244)( 47,243)
( 48,242)( 49,241)( 50,240)( 51,239)( 52,238)( 53,237)( 54,236)( 55,235)
( 56,234)( 57,233)( 58,232)( 59,231)( 60,230)( 61,229)( 62,228)( 63,227)
( 64,226)( 65,225)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,219)
( 72,218)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,212)( 79,211)
( 80,210)( 81,209)( 82,208)( 83,371)( 84,370)( 85,410)( 86,409)( 87,408)
( 88,407)( 89,406)( 90,405)( 91,404)( 92,403)( 93,402)( 94,401)( 95,400)
( 96,399)( 97,398)( 98,397)( 99,396)(100,395)(101,394)(102,393)(103,392)
(104,391)(105,390)(106,389)(107,388)(108,387)(109,386)(110,385)(111,384)
(112,383)(113,382)(114,381)(115,380)(116,379)(117,378)(118,377)(119,376)
(120,375)(121,374)(122,373)(123,372)(124,330)(125,329)(126,369)(127,368)
(128,367)(129,366)(130,365)(131,364)(132,363)(133,362)(134,361)(135,360)
(136,359)(137,358)(138,357)(139,356)(140,355)(141,354)(142,353)(143,352)
(144,351)(145,350)(146,349)(147,348)(148,347)(149,346)(150,345)(151,344)
(152,343)(153,342)(154,341)(155,340)(156,339)(157,338)(158,337)(159,336)
(160,335)(161,334)(162,333)(163,332)(164,331)(165,289)(166,288)(167,328)
(168,327)(169,326)(170,325)(171,324)(172,323)(173,322)(174,321)(175,320)
(176,319)(177,318)(178,317)(179,316)(180,315)(181,314)(182,313)(183,312)
(184,311)(185,310)(186,309)(187,308)(188,307)(189,306)(190,305)(191,304)
(192,303)(193,302)(194,301)(195,300)(196,299)(197,298)(198,297)(199,296)
(200,295)(201,294)(202,293)(203,292)(204,291)(205,290);
poly := sub<Sym(410)|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 >;
References : None.
to this polytope