Polytope of Type {408}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {408}*816
Also Known As : 408-gon, {408}. if this polytope has another name.
Group : SmallGroup(816,68)
Rank : 2
Schlafli Type : {408}
Number of vertices, edges, etc : 408, 408
Order of s0s1 : 408
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{408,2} of size 1632
Vertex Figure Of :
{2,408} of size 1632
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {204}*408
3-fold quotients : {136}*272
4-fold quotients : {102}*204
6-fold quotients : {68}*136
8-fold quotients : {51}*102
12-fold quotients : {34}*68
17-fold quotients : {24}*48
24-fold quotients : {17}*34
34-fold quotients : {12}*24
51-fold quotients : {8}*16
68-fold quotients : {6}*12
102-fold quotients : {4}*8
136-fold quotients : {3}*6
204-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {816}*1632
Irregular Quotients (of which this is a minimal cover):
None.
Permutation Representation (GAP) :
s0 := ( 2, 17)( 3, 16)( 4, 15)( 5, 14)( 6, 13)( 7, 12)( 8, 11)( 9, 10)( 18, 35)( 19, 51)( 20, 50)( 21, 49)( 22, 48)( 23, 47)( 24, 46)( 25, 45)( 26, 44)( 27, 43)( 28, 42)( 29, 41)( 30, 40)( 31, 39)( 32, 38)( 33, 37)( 34, 36)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)( 59, 62)( 60, 61)( 69, 86)( 70,102)( 71,101)( 72,100)( 73, 99)( 74, 98)( 75, 97)( 76, 96)( 77, 95)( 78, 94)( 79, 93)( 80, 92)( 81, 91)( 82, 90)( 83, 89)( 84, 88)( 85, 87)(103,154)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,188)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)(132,193)(133,192)(134,191)(135,190)(136,189)(137,171)(138,187)(139,186)(140,185)(141,184)(142,183)(143,182)(144,181)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(205,307)(206,323)(207,322)(208,321)(209,320)(210,319)(211,318)(212,317)(213,316)(214,315)(215,314)(216,313)(217,312)(218,311)(219,310)(220,309)(221,308)(222,341)(223,357)(224,356)(225,355)(226,354)(227,353)(228,352)(229,351)(230,350)(231,349)(232,348)(233,347)(234,346)(235,345)(236,344)(237,343)(238,342)(239,324)(240,340)(241,339)(242,338)(243,337)(244,336)(245,335)(246,334)(247,333)(248,332)(249,331)(250,330)(251,329)(252,328)(253,327)(254,326)(255,325)(256,358)(257,374)(258,373)(259,372)(260,371)(261,370)(262,369)(263,368)(264,367)(265,366)(266,365)(267,364)(268,363)(269,362)(270,361)(271,360)(272,359)(273,392)(274,408)(275,407)(276,406)(277,405)(278,404)(279,403)(280,402)(281,401)(282,400)(283,399)(284,398)(285,397)(286,396)(287,395)(288,394)(289,393)(290,375)(291,391)(292,390)(293,389)(294,388)(295,387)(296,386)(297,385)(298,384)(299,383)(300,382)(301,381)(302,380)(303,379)(304,378)(305,377)(306,376);;
s1 := ( 1,223)( 2,222)( 3,238)( 4,237)( 5,236)( 6,235)( 7,234)( 8,233)( 9,232)( 10,231)( 11,230)( 12,229)( 13,228)( 14,227)( 15,226)( 16,225)( 17,224)( 18,206)( 19,205)( 20,221)( 21,220)( 22,219)( 23,218)( 24,217)( 25,216)( 26,215)( 27,214)( 28,213)( 29,212)( 30,211)( 31,210)( 32,209)( 33,208)( 34,207)( 35,240)( 36,239)( 37,255)( 38,254)( 39,253)( 40,252)( 41,251)( 42,250)( 43,249)( 44,248)( 45,247)( 46,246)( 47,245)( 48,244)( 49,243)( 50,242)( 51,241)( 52,274)( 53,273)( 54,289)( 55,288)( 56,287)( 57,286)( 58,285)( 59,284)( 60,283)( 61,282)( 62,281)( 63,280)( 64,279)( 65,278)( 66,277)( 67,276)( 68,275)( 69,257)( 70,256)( 71,272)( 72,271)( 73,270)( 74,269)( 75,268)( 76,267)( 77,266)( 78,265)( 79,264)( 80,263)( 81,262)( 82,261)( 83,260)( 84,259)( 85,258)( 86,291)( 87,290)( 88,306)( 89,305)( 90,304)( 91,303)( 92,302)( 93,301)( 94,300)( 95,299)( 96,298)( 97,297)( 98,296)( 99,295)(100,294)(101,293)(102,292)(103,376)(104,375)(105,391)(106,390)(107,389)(108,388)(109,387)(110,386)(111,385)(112,384)(113,383)(114,382)(115,381)(116,380)(117,379)(118,378)(119,377)(120,359)(121,358)(122,374)(123,373)(124,372)(125,371)(126,370)(127,369)(128,368)(129,367)(130,366)(131,365)(132,364)(133,363)(134,362)(135,361)(136,360)(137,393)(138,392)(139,408)(140,407)(141,406)(142,405)(143,404)(144,403)(145,402)(146,401)(147,400)(148,399)(149,398)(150,397)(151,396)(152,395)(153,394)(154,325)(155,324)(156,340)(157,339)(158,338)(159,337)(160,336)(161,335)(162,334)(163,333)(164,332)(165,331)(166,330)(167,329)(168,328)(169,327)(170,326)(171,308)(172,307)(173,323)(174,322)(175,321)(176,320)(177,319)(178,318)(179,317)(180,316)(181,315)(182,314)(183,313)(184,312)(185,311)(186,310)(187,309)(188,342)(189,341)(190,357)(191,356)(192,355)(193,354)(194,353)(195,352)(196,351)(197,350)(198,349)(199,348)(200,347)(201,346)(202,345)(203,344)(204,343);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(408)!( 2, 17)( 3, 16)( 4, 15)( 5, 14)( 6, 13)( 7, 12)( 8, 11)( 9, 10)( 18, 35)( 19, 51)( 20, 50)( 21, 49)( 22, 48)( 23, 47)( 24, 46)( 25, 45)( 26, 44)( 27, 43)( 28, 42)( 29, 41)( 30, 40)( 31, 39)( 32, 38)( 33, 37)( 34, 36)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)( 59, 62)( 60, 61)( 69, 86)( 70,102)( 71,101)( 72,100)( 73, 99)( 74, 98)( 75, 97)( 76, 96)( 77, 95)( 78, 94)( 79, 93)( 80, 92)( 81, 91)( 82, 90)( 83, 89)( 84, 88)( 85, 87)(103,154)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,188)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)(132,193)(133,192)(134,191)(135,190)(136,189)(137,171)(138,187)(139,186)(140,185)(141,184)(142,183)(143,182)(144,181)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(205,307)(206,323)(207,322)(208,321)(209,320)(210,319)(211,318)(212,317)(213,316)(214,315)(215,314)(216,313)(217,312)(218,311)(219,310)(220,309)(221,308)(222,341)(223,357)(224,356)(225,355)(226,354)(227,353)(228,352)(229,351)(230,350)(231,349)(232,348)(233,347)(234,346)(235,345)(236,344)(237,343)(238,342)(239,324)(240,340)(241,339)(242,338)(243,337)(244,336)(245,335)(246,334)(247,333)(248,332)(249,331)(250,330)(251,329)(252,328)(253,327)(254,326)(255,325)(256,358)(257,374)(258,373)(259,372)(260,371)(261,370)(262,369)(263,368)(264,367)(265,366)(266,365)(267,364)(268,363)(269,362)(270,361)(271,360)(272,359)(273,392)(274,408)(275,407)(276,406)(277,405)(278,404)(279,403)(280,402)(281,401)(282,400)(283,399)(284,398)(285,397)(286,396)(287,395)(288,394)(289,393)(290,375)(291,391)(292,390)(293,389)(294,388)(295,387)(296,386)(297,385)(298,384)(299,383)(300,382)(301,381)(302,380)(303,379)(304,378)(305,377)(306,376);
s1 := Sym(408)!( 1,223)( 2,222)( 3,238)( 4,237)( 5,236)( 6,235)( 7,234)( 8,233)( 9,232)( 10,231)( 11,230)( 12,229)( 13,228)( 14,227)( 15,226)( 16,225)( 17,224)( 18,206)( 19,205)( 20,221)( 21,220)( 22,219)( 23,218)( 24,217)( 25,216)( 26,215)( 27,214)( 28,213)( 29,212)( 30,211)( 31,210)( 32,209)( 33,208)( 34,207)( 35,240)( 36,239)( 37,255)( 38,254)( 39,253)( 40,252)( 41,251)( 42,250)( 43,249)( 44,248)( 45,247)( 46,246)( 47,245)( 48,244)( 49,243)( 50,242)( 51,241)( 52,274)( 53,273)( 54,289)( 55,288)( 56,287)( 57,286)( 58,285)( 59,284)( 60,283)( 61,282)( 62,281)( 63,280)( 64,279)( 65,278)( 66,277)( 67,276)( 68,275)( 69,257)( 70,256)( 71,272)( 72,271)( 73,270)( 74,269)( 75,268)( 76,267)( 77,266)( 78,265)( 79,264)( 80,263)( 81,262)( 82,261)( 83,260)( 84,259)( 85,258)( 86,291)( 87,290)( 88,306)( 89,305)( 90,304)( 91,303)( 92,302)( 93,301)( 94,300)( 95,299)( 96,298)( 97,297)( 98,296)( 99,295)(100,294)(101,293)(102,292)(103,376)(104,375)(105,391)(106,390)(107,389)(108,388)(109,387)(110,386)(111,385)(112,384)(113,383)(114,382)(115,381)(116,380)(117,379)(118,378)(119,377)(120,359)(121,358)(122,374)(123,373)(124,372)(125,371)(126,370)(127,369)(128,368)(129,367)(130,366)(131,365)(132,364)(133,363)(134,362)(135,361)(136,360)(137,393)(138,392)(139,408)(140,407)(141,406)(142,405)(143,404)(144,403)(145,402)(146,401)(147,400)(148,399)(149,398)(150,397)(151,396)(152,395)(153,394)(154,325)(155,324)(156,340)(157,339)(158,338)(159,337)(160,336)(161,335)(162,334)(163,333)(164,332)(165,331)(166,330)(167,329)(168,328)(169,327)(170,326)(171,308)(172,307)(173,323)(174,322)(175,321)(176,320)(177,319)(178,318)(179,317)(180,316)(181,315)(182,314)(183,313)(184,312)(185,311)(186,310)(187,309)(188,342)(189,341)(190,357)(191,356)(192,355)(193,354)(194,353)(195,352)(196,351)(197,350)(198,349)(199,348)(200,347)(201,346)(202,345)(203,344)(204,343);
poly := sub<Sym(408)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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