Polytope of Type {354}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {354}*708
Also Known As : 354-gon, {354}. if this polytope has another name.
Group : SmallGroup(708,9)
Rank : 2
Schlafli Type : {354}
Number of vertices, edges, etc : 354, 354
Order of s₀s₁ : 354
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {354,2} of size 1416
Vertex Figure Of :
   {2,354} of size 1416
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {177}*354
   3-fold quotients : {118}*236
   6-fold quotients : {59}*118
   59-fold quotients : {6}*12
   118-fold quotients : {3}*6
   177-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {708}*1416
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope