Polytope of Type {82,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {82,10}*1640
Also Known As : {82,10|2}. if this polytope has another name.
Group : SmallGroup(1640,64)
Rank : 3
Schlafli Type : {82,10}
Number of vertices, edges, etc : 82, 410, 10
Order of s₀s₁s₂ : 410
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   5-fold quotients : {82,2}*328
   10-fold quotients : {41,2}*164
   41-fold quotients : {2,10}*40
   82-fold quotients : {2,5}*20
   205-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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)( 43, 82)( 44, 81)( 45, 80)( 46, 79)
( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)( 54, 71)
( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)( 62, 63)
( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)( 91,116)
( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)( 99,108)
(100,107)(101,106)(102,105)(103,104)(125,164)(126,163)(127,162)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)
(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)
(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)
(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)
(182,189)(183,188)(184,187)(185,186)(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)
(248,287)(249,286)(250,285)(251,284)(252,283)(253,282)(254,281)(255,280)
(256,279)(257,278)(258,277)(259,276)(260,275)(261,274)(262,273)(263,272)
(264,271)(265,270)(266,269)(267,268)(289,328)(290,327)(291,326)(292,325)
(293,324)(294,323)(295,322)(296,321)(297,320)(298,319)(299,318)(300,317)
(301,316)(302,315)(303,314)(304,313)(305,312)(306,311)(307,310)(308,309)
(330,369)(331,368)(332,367)(333,366)(334,365)(335,364)(336,363)(337,362)
(338,361)(339,360)(340,359)(341,358)(342,357)(343,356)(344,355)(345,354)
(346,353)(347,352)(348,351)(349,350)(371,410)(372,409)(373,408)(374,407)
(375,406)(376,405)(377,404)(378,403)(379,402)(380,401)(381,400)(382,399)
(383,398)(384,397)(385,396)(386,395)(387,394)(388,393)(389,392)(390,391);;
s1 := (  1,  2)(  3, 41)(  4, 40)(  5, 39)(  6, 38)(  7, 37)(  8, 36)(  9, 35)
( 10, 34)( 11, 33)( 12, 32)( 13, 31)( 14, 30)( 15, 29)( 16, 28)( 17, 27)
( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 42,166)( 43,165)( 44,205)( 45,204)
( 46,203)( 47,202)( 48,201)( 49,200)( 50,199)( 51,198)( 52,197)( 53,196)
( 54,195)( 55,194)( 56,193)( 57,192)( 58,191)( 59,190)( 60,189)( 61,188)
( 62,187)( 63,186)( 64,185)( 65,184)( 66,183)( 67,182)( 68,181)( 69,180)
( 70,179)( 71,178)( 72,177)( 73,176)( 74,175)( 75,174)( 76,173)( 77,172)
( 78,171)( 79,170)( 80,169)( 81,168)( 82,167)( 83,125)( 84,124)( 85,164)
( 86,163)( 87,162)( 88,161)( 89,160)( 90,159)( 91,158)( 92,157)( 93,156)
( 94,155)( 95,154)( 96,153)( 97,152)( 98,151)( 99,150)(100,149)(101,148)
(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,140)
(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)(117,132)
(118,131)(119,130)(120,129)(121,128)(122,127)(123,126)(206,207)(208,246)
(209,245)(210,244)(211,243)(212,242)(213,241)(214,240)(215,239)(216,238)
(217,237)(218,236)(219,235)(220,234)(221,233)(222,232)(223,231)(224,230)
(225,229)(226,228)(247,371)(248,370)(249,410)(250,409)(251,408)(252,407)
(253,406)(254,405)(255,404)(256,403)(257,402)(258,401)(259,400)(260,399)
(261,398)(262,397)(263,396)(264,395)(265,394)(266,393)(267,392)(268,391)
(269,390)(270,389)(271,388)(272,387)(273,386)(274,385)(275,384)(276,383)
(277,382)(278,381)(279,380)(280,379)(281,378)(282,377)(283,376)(284,375)
(285,374)(286,373)(287,372)(288,330)(289,329)(290,369)(291,368)(292,367)
(293,366)(294,365)(295,364)(296,363)(297,362)(298,361)(299,360)(300,359)
(301,358)(302,357)(303,356)(304,355)(305,354)(306,353)(307,352)(308,351)
(309,350)(310,349)(311,348)(312,347)(313,346)(314,345)(315,344)(316,343)
(317,342)(318,341)(319,340)(320,339)(321,338)(322,337)(323,336)(324,335)
(325,334)(326,333)(327,332)(328,331);;
s2 := (  1,247)(  2,248)(  3,249)(  4,250)(  5,251)(  6,252)(  7,253)(  8,254)
(  9,255)( 10,256)( 11,257)( 12,258)( 13,259)( 14,260)( 15,261)( 16,262)
( 17,263)( 18,264)( 19,265)( 20,266)( 21,267)( 22,268)( 23,269)( 24,270)
( 25,271)( 26,272)( 27,273)( 28,274)( 29,275)( 30,276)( 31,277)( 32,278)
( 33,279)( 34,280)( 35,281)( 36,282)( 37,283)( 38,284)( 39,285)( 40,286)
( 41,287)( 42,206)( 43,207)( 44,208)( 45,209)( 46,210)( 47,211)( 48,212)
( 49,213)( 50,214)( 51,215)( 52,216)( 53,217)( 54,218)( 55,219)( 56,220)
( 57,221)( 58,222)( 59,223)( 60,224)( 61,225)( 62,226)( 63,227)( 64,228)
( 65,229)( 66,230)( 67,231)( 68,232)( 69,233)( 70,234)( 71,235)( 72,236)
( 73,237)( 74,238)( 75,239)( 76,240)( 77,241)( 78,242)( 79,243)( 80,244)
( 81,245)( 82,246)( 83,370)( 84,371)( 85,372)( 86,373)( 87,374)( 88,375)
( 89,376)( 90,377)( 91,378)( 92,379)( 93,380)( 94,381)( 95,382)( 96,383)
( 97,384)( 98,385)( 99,386)(100,387)(101,388)(102,389)(103,390)(104,391)
(105,392)(106,393)(107,394)(108,395)(109,396)(110,397)(111,398)(112,399)
(113,400)(114,401)(115,402)(116,403)(117,404)(118,405)(119,406)(120,407)
(121,408)(122,409)(123,410)(124,329)(125,330)(126,331)(127,332)(128,333)
(129,334)(130,335)(131,336)(132,337)(133,338)(134,339)(135,340)(136,341)
(137,342)(138,343)(139,344)(140,345)(141,346)(142,347)(143,348)(144,349)
(145,350)(146,351)(147,352)(148,353)(149,354)(150,355)(151,356)(152,357)
(153,358)(154,359)(155,360)(156,361)(157,362)(158,363)(159,364)(160,365)
(161,366)(162,367)(163,368)(164,369)(165,288)(166,289)(167,290)(168,291)
(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)(176,299)
(177,300)(178,301)(179,302)(180,303)(181,304)(182,305)(183,306)(184,307)
(185,308)(186,309)(187,310)(188,311)(189,312)(190,313)(191,314)(192,315)
(193,316)(194,317)(195,318)(196,319)(197,320)(198,321)(199,322)(200,323)
(201,324)(202,325)(203,326)(204,327)(205,328);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)( 43, 82)( 44, 81)( 45, 80)
( 46, 79)( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)
( 54, 71)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)
( 62, 63)( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)
( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)
( 99,108)(100,107)(101,106)(102,105)(103,104)(125,164)(126,163)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)
(144,145)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)
(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)
(181,190)(182,189)(183,188)(184,187)(185,186)(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)(248,287)(249,286)(250,285)(251,284)(252,283)(253,282)(254,281)
(255,280)(256,279)(257,278)(258,277)(259,276)(260,275)(261,274)(262,273)
(263,272)(264,271)(265,270)(266,269)(267,268)(289,328)(290,327)(291,326)
(292,325)(293,324)(294,323)(295,322)(296,321)(297,320)(298,319)(299,318)
(300,317)(301,316)(302,315)(303,314)(304,313)(305,312)(306,311)(307,310)
(308,309)(330,369)(331,368)(332,367)(333,366)(334,365)(335,364)(336,363)
(337,362)(338,361)(339,360)(340,359)(341,358)(342,357)(343,356)(344,355)
(345,354)(346,353)(347,352)(348,351)(349,350)(371,410)(372,409)(373,408)
(374,407)(375,406)(376,405)(377,404)(378,403)(379,402)(380,401)(381,400)
(382,399)(383,398)(384,397)(385,396)(386,395)(387,394)(388,393)(389,392)
(390,391);
s1 := Sym(410)!(  1,  2)(  3, 41)(  4, 40)(  5, 39)(  6, 38)(  7, 37)(  8, 36)
(  9, 35)( 10, 34)( 11, 33)( 12, 32)( 13, 31)( 14, 30)( 15, 29)( 16, 28)
( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 42,166)( 43,165)( 44,205)
( 45,204)( 46,203)( 47,202)( 48,201)( 49,200)( 50,199)( 51,198)( 52,197)
( 53,196)( 54,195)( 55,194)( 56,193)( 57,192)( 58,191)( 59,190)( 60,189)
( 61,188)( 62,187)( 63,186)( 64,185)( 65,184)( 66,183)( 67,182)( 68,181)
( 69,180)( 70,179)( 71,178)( 72,177)( 73,176)( 74,175)( 75,174)( 76,173)
( 77,172)( 78,171)( 79,170)( 80,169)( 81,168)( 82,167)( 83,125)( 84,124)
( 85,164)( 86,163)( 87,162)( 88,161)( 89,160)( 90,159)( 91,158)( 92,157)
( 93,156)( 94,155)( 95,154)( 96,153)( 97,152)( 98,151)( 99,150)(100,149)
(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)
(109,140)(110,139)(111,138)(112,137)(113,136)(114,135)(115,134)(116,133)
(117,132)(118,131)(119,130)(120,129)(121,128)(122,127)(123,126)(206,207)
(208,246)(209,245)(210,244)(211,243)(212,242)(213,241)(214,240)(215,239)
(216,238)(217,237)(218,236)(219,235)(220,234)(221,233)(222,232)(223,231)
(224,230)(225,229)(226,228)(247,371)(248,370)(249,410)(250,409)(251,408)
(252,407)(253,406)(254,405)(255,404)(256,403)(257,402)(258,401)(259,400)
(260,399)(261,398)(262,397)(263,396)(264,395)(265,394)(266,393)(267,392)
(268,391)(269,390)(270,389)(271,388)(272,387)(273,386)(274,385)(275,384)
(276,383)(277,382)(278,381)(279,380)(280,379)(281,378)(282,377)(283,376)
(284,375)(285,374)(286,373)(287,372)(288,330)(289,329)(290,369)(291,368)
(292,367)(293,366)(294,365)(295,364)(296,363)(297,362)(298,361)(299,360)
(300,359)(301,358)(302,357)(303,356)(304,355)(305,354)(306,353)(307,352)
(308,351)(309,350)(310,349)(311,348)(312,347)(313,346)(314,345)(315,344)
(316,343)(317,342)(318,341)(319,340)(320,339)(321,338)(322,337)(323,336)
(324,335)(325,334)(326,333)(327,332)(328,331);
s2 := Sym(410)!(  1,247)(  2,248)(  3,249)(  4,250)(  5,251)(  6,252)(  7,253)
(  8,254)(  9,255)( 10,256)( 11,257)( 12,258)( 13,259)( 14,260)( 15,261)
( 16,262)( 17,263)( 18,264)( 19,265)( 20,266)( 21,267)( 22,268)( 23,269)
( 24,270)( 25,271)( 26,272)( 27,273)( 28,274)( 29,275)( 30,276)( 31,277)
( 32,278)( 33,279)( 34,280)( 35,281)( 36,282)( 37,283)( 38,284)( 39,285)
( 40,286)( 41,287)( 42,206)( 43,207)( 44,208)( 45,209)( 46,210)( 47,211)
( 48,212)( 49,213)( 50,214)( 51,215)( 52,216)( 53,217)( 54,218)( 55,219)
( 56,220)( 57,221)( 58,222)( 59,223)( 60,224)( 61,225)( 62,226)( 63,227)
( 64,228)( 65,229)( 66,230)( 67,231)( 68,232)( 69,233)( 70,234)( 71,235)
( 72,236)( 73,237)( 74,238)( 75,239)( 76,240)( 77,241)( 78,242)( 79,243)
( 80,244)( 81,245)( 82,246)( 83,370)( 84,371)( 85,372)( 86,373)( 87,374)
( 88,375)( 89,376)( 90,377)( 91,378)( 92,379)( 93,380)( 94,381)( 95,382)
( 96,383)( 97,384)( 98,385)( 99,386)(100,387)(101,388)(102,389)(103,390)
(104,391)(105,392)(106,393)(107,394)(108,395)(109,396)(110,397)(111,398)
(112,399)(113,400)(114,401)(115,402)(116,403)(117,404)(118,405)(119,406)
(120,407)(121,408)(122,409)(123,410)(124,329)(125,330)(126,331)(127,332)
(128,333)(129,334)(130,335)(131,336)(132,337)(133,338)(134,339)(135,340)
(136,341)(137,342)(138,343)(139,344)(140,345)(141,346)(142,347)(143,348)
(144,349)(145,350)(146,351)(147,352)(148,353)(149,354)(150,355)(151,356)
(152,357)(153,358)(154,359)(155,360)(156,361)(157,362)(158,363)(159,364)
(160,365)(161,366)(162,367)(163,368)(164,369)(165,288)(166,289)(167,290)
(168,291)(169,292)(170,293)(171,294)(172,295)(173,296)(174,297)(175,298)
(176,299)(177,300)(178,301)(179,302)(180,303)(181,304)(182,305)(183,306)
(184,307)(185,308)(186,309)(187,310)(188,311)(189,312)(190,313)(191,314)
(192,315)(193,316)(194,317)(195,318)(196,319)(197,320)(198,321)(199,322)
(200,323)(201,324)(202,325)(203,326)(204,327)(205,328);
poly := sub<Sym(410)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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