Polytope of Type {370}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {370}*740
Also Known As : 370-gon, {370}. if this polytope has another name.
Group : SmallGroup(740,14)
Rank : 2
Schlafli Type : {370}
Number of vertices, edges, etc : 370, 370
Order of s₀s₁ : 370
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {370,2} of size 1480
Vertex Figure Of :
   {2,370} of size 1480
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {185}*370
   5-fold quotients : {74}*148
   10-fold quotients : {37}*74
   37-fold quotients : {10}*20
   74-fold quotients : {5}*10
   185-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {740}*1480
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope