Polytope of Type {10,82}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,82}*1640
Also Known As : {10,82|2}. if this polytope has another name.
Group : SmallGroup(1640,64)
Rank : 3
Schlafli Type : {10,82}
Number of vertices, edges, etc : 10, 410, 82
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 : {2,82}*328
   10-fold quotients : {2,41}*164
   41-fold quotients : {10,2}*40
   82-fold quotients : {5,2}*20
   205-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
to this polytope