Polytope of Type {356}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {356}*712
Also Known As : 356-gon, {356}. if this polytope has another name.
Group : SmallGroup(712,6)
Rank : 2
Schlafli Type : {356}
Number of vertices, edges, etc : 356, 356
Order of s0s1 : 356
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{356,2} of size 1424
Vertex Figure Of :
{2,356} of size 1424
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {178}*356
4-fold quotients : {89}*178
89-fold quotients : {4}*8
178-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {712}*1424
Permutation Representation (GAP) :
s0 := ( 2, 89)( 3, 88)( 4, 87)( 5, 86)( 6, 85)( 7, 84)( 8, 83)( 9, 82)
( 10, 81)( 11, 80)( 12, 79)( 13, 78)( 14, 77)( 15, 76)( 16, 75)( 17, 74)
( 18, 73)( 19, 72)( 20, 71)( 21, 70)( 22, 69)( 23, 68)( 24, 67)( 25, 66)
( 26, 65)( 27, 64)( 28, 63)( 29, 62)( 30, 61)( 31, 60)( 32, 59)( 33, 58)
( 34, 57)( 35, 56)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)
( 42, 49)( 43, 48)( 44, 47)( 45, 46)( 91,178)( 92,177)( 93,176)( 94,175)
( 95,174)( 96,173)( 97,172)( 98,171)( 99,170)(100,169)(101,168)(102,167)
(103,166)(104,165)(105,164)(106,163)(107,162)(108,161)(109,160)(110,159)
(111,158)(112,157)(113,156)(114,155)(115,154)(116,153)(117,152)(118,151)
(119,150)(120,149)(121,148)(122,147)(123,146)(124,145)(125,144)(126,143)
(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)
(179,268)(180,356)(181,355)(182,354)(183,353)(184,352)(185,351)(186,350)
(187,349)(188,348)(189,347)(190,346)(191,345)(192,344)(193,343)(194,342)
(195,341)(196,340)(197,339)(198,338)(199,337)(200,336)(201,335)(202,334)
(203,333)(204,332)(205,331)(206,330)(207,329)(208,328)(209,327)(210,326)
(211,325)(212,324)(213,323)(214,322)(215,321)(216,320)(217,319)(218,318)
(219,317)(220,316)(221,315)(222,314)(223,313)(224,312)(225,311)(226,310)
(227,309)(228,308)(229,307)(230,306)(231,305)(232,304)(233,303)(234,302)
(235,301)(236,300)(237,299)(238,298)(239,297)(240,296)(241,295)(242,294)
(243,293)(244,292)(245,291)(246,290)(247,289)(248,288)(249,287)(250,286)
(251,285)(252,284)(253,283)(254,282)(255,281)(256,280)(257,279)(258,278)
(259,277)(260,276)(261,275)(262,274)(263,273)(264,272)(265,271)(266,270)
(267,269);;
s1 := ( 1,180)( 2,179)( 3,267)( 4,266)( 5,265)( 6,264)( 7,263)( 8,262)
( 9,261)( 10,260)( 11,259)( 12,258)( 13,257)( 14,256)( 15,255)( 16,254)
( 17,253)( 18,252)( 19,251)( 20,250)( 21,249)( 22,248)( 23,247)( 24,246)
( 25,245)( 26,244)( 27,243)( 28,242)( 29,241)( 30,240)( 31,239)( 32,238)
( 33,237)( 34,236)( 35,235)( 36,234)( 37,233)( 38,232)( 39,231)( 40,230)
( 41,229)( 42,228)( 43,227)( 44,226)( 45,225)( 46,224)( 47,223)( 48,222)
( 49,221)( 50,220)( 51,219)( 52,218)( 53,217)( 54,216)( 55,215)( 56,214)
( 57,213)( 58,212)( 59,211)( 60,210)( 61,209)( 62,208)( 63,207)( 64,206)
( 65,205)( 66,204)( 67,203)( 68,202)( 69,201)( 70,200)( 71,199)( 72,198)
( 73,197)( 74,196)( 75,195)( 76,194)( 77,193)( 78,192)( 79,191)( 80,190)
( 81,189)( 82,188)( 83,187)( 84,186)( 85,185)( 86,184)( 87,183)( 88,182)
( 89,181)( 90,269)( 91,268)( 92,356)( 93,355)( 94,354)( 95,353)( 96,352)
( 97,351)( 98,350)( 99,349)(100,348)(101,347)(102,346)(103,345)(104,344)
(105,343)(106,342)(107,341)(108,340)(109,339)(110,338)(111,337)(112,336)
(113,335)(114,334)(115,333)(116,332)(117,331)(118,330)(119,329)(120,328)
(121,327)(122,326)(123,325)(124,324)(125,323)(126,322)(127,321)(128,320)
(129,319)(130,318)(131,317)(132,316)(133,315)(134,314)(135,313)(136,312)
(137,311)(138,310)(139,309)(140,308)(141,307)(142,306)(143,305)(144,304)
(145,303)(146,302)(147,301)(148,300)(149,299)(150,298)(151,297)(152,296)
(153,295)(154,294)(155,293)(156,292)(157,291)(158,290)(159,289)(160,288)
(161,287)(162,286)(163,285)(164,284)(165,283)(166,282)(167,281)(168,280)
(169,279)(170,278)(171,277)(172,276)(173,275)(174,274)(175,273)(176,272)
(177,271)(178,270);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(356)!( 2, 89)( 3, 88)( 4, 87)( 5, 86)( 6, 85)( 7, 84)( 8, 83)
( 9, 82)( 10, 81)( 11, 80)( 12, 79)( 13, 78)( 14, 77)( 15, 76)( 16, 75)
( 17, 74)( 18, 73)( 19, 72)( 20, 71)( 21, 70)( 22, 69)( 23, 68)( 24, 67)
( 25, 66)( 26, 65)( 27, 64)( 28, 63)( 29, 62)( 30, 61)( 31, 60)( 32, 59)
( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)
( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 45, 46)( 91,178)( 92,177)( 93,176)
( 94,175)( 95,174)( 96,173)( 97,172)( 98,171)( 99,170)(100,169)(101,168)
(102,167)(103,166)(104,165)(105,164)(106,163)(107,162)(108,161)(109,160)
(110,159)(111,158)(112,157)(113,156)(114,155)(115,154)(116,153)(117,152)
(118,151)(119,150)(120,149)(121,148)(122,147)(123,146)(124,145)(125,144)
(126,143)(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)(133,136)
(134,135)(179,268)(180,356)(181,355)(182,354)(183,353)(184,352)(185,351)
(186,350)(187,349)(188,348)(189,347)(190,346)(191,345)(192,344)(193,343)
(194,342)(195,341)(196,340)(197,339)(198,338)(199,337)(200,336)(201,335)
(202,334)(203,333)(204,332)(205,331)(206,330)(207,329)(208,328)(209,327)
(210,326)(211,325)(212,324)(213,323)(214,322)(215,321)(216,320)(217,319)
(218,318)(219,317)(220,316)(221,315)(222,314)(223,313)(224,312)(225,311)
(226,310)(227,309)(228,308)(229,307)(230,306)(231,305)(232,304)(233,303)
(234,302)(235,301)(236,300)(237,299)(238,298)(239,297)(240,296)(241,295)
(242,294)(243,293)(244,292)(245,291)(246,290)(247,289)(248,288)(249,287)
(250,286)(251,285)(252,284)(253,283)(254,282)(255,281)(256,280)(257,279)
(258,278)(259,277)(260,276)(261,275)(262,274)(263,273)(264,272)(265,271)
(266,270)(267,269);
s1 := Sym(356)!( 1,180)( 2,179)( 3,267)( 4,266)( 5,265)( 6,264)( 7,263)
( 8,262)( 9,261)( 10,260)( 11,259)( 12,258)( 13,257)( 14,256)( 15,255)
( 16,254)( 17,253)( 18,252)( 19,251)( 20,250)( 21,249)( 22,248)( 23,247)
( 24,246)( 25,245)( 26,244)( 27,243)( 28,242)( 29,241)( 30,240)( 31,239)
( 32,238)( 33,237)( 34,236)( 35,235)( 36,234)( 37,233)( 38,232)( 39,231)
( 40,230)( 41,229)( 42,228)( 43,227)( 44,226)( 45,225)( 46,224)( 47,223)
( 48,222)( 49,221)( 50,220)( 51,219)( 52,218)( 53,217)( 54,216)( 55,215)
( 56,214)( 57,213)( 58,212)( 59,211)( 60,210)( 61,209)( 62,208)( 63,207)
( 64,206)( 65,205)( 66,204)( 67,203)( 68,202)( 69,201)( 70,200)( 71,199)
( 72,198)( 73,197)( 74,196)( 75,195)( 76,194)( 77,193)( 78,192)( 79,191)
( 80,190)( 81,189)( 82,188)( 83,187)( 84,186)( 85,185)( 86,184)( 87,183)
( 88,182)( 89,181)( 90,269)( 91,268)( 92,356)( 93,355)( 94,354)( 95,353)
( 96,352)( 97,351)( 98,350)( 99,349)(100,348)(101,347)(102,346)(103,345)
(104,344)(105,343)(106,342)(107,341)(108,340)(109,339)(110,338)(111,337)
(112,336)(113,335)(114,334)(115,333)(116,332)(117,331)(118,330)(119,329)
(120,328)(121,327)(122,326)(123,325)(124,324)(125,323)(126,322)(127,321)
(128,320)(129,319)(130,318)(131,317)(132,316)(133,315)(134,314)(135,313)
(136,312)(137,311)(138,310)(139,309)(140,308)(141,307)(142,306)(143,305)
(144,304)(145,303)(146,302)(147,301)(148,300)(149,299)(150,298)(151,297)
(152,296)(153,295)(154,294)(155,293)(156,292)(157,291)(158,290)(159,289)
(160,288)(161,287)(162,286)(163,285)(164,284)(165,283)(166,282)(167,281)
(168,280)(169,279)(170,278)(171,277)(172,276)(173,275)(174,274)(175,273)
(176,272)(177,271)(178,270);
poly := sub<Sym(356)|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 >;
References : None.
to this polytope