Polytope of Type {2,4,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,84}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11397)
Rank : 4
Schlafli Type : {2,4,84}
Number of vertices, edges, etc : 2, 4, 168, 84
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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) :
   2-fold quotients : {2,4,42}*672b
   4-fold quotients : {2,4,21}*336
   7-fold quotients : {2,4,12}*192b
   14-fold quotients : {2,4,6}*96c
   28-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 17)( 16, 18)
( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)( 32, 34)
( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)( 64, 66)
( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)( 96, 98)
( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)(128,130)
(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)(144,146)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)(160,162)
(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)(176,178)
(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)(208,210)
(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)(224,226)
(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)(240,242)
(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)(256,258)
(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)(272,274)
(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)(288,290)
(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)(304,306)
(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)(320,322)
(323,325)(324,326)(327,329)(328,330)(331,333)(332,334)(335,337)(336,338);;
s2 := (  4,  5)(  7, 27)(  8, 29)(  9, 28)( 10, 30)( 11, 23)( 12, 25)( 13, 24)
( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)( 33, 60)
( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)( 41, 80)
( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)( 49, 72)
( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)
( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)( 96,109)
( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)(116,145)
(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)(124,165)
(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)(132,157)
(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)(140,149)
(141,148)(142,150)(171,255)(172,257)(173,256)(174,258)(175,279)(176,281)
(177,280)(178,282)(179,275)(180,277)(181,276)(182,278)(183,271)(184,273)
(185,272)(186,274)(187,267)(188,269)(189,268)(190,270)(191,263)(192,265)
(193,264)(194,266)(195,259)(196,261)(197,260)(198,262)(199,311)(200,313)
(201,312)(202,314)(203,335)(204,337)(205,336)(206,338)(207,331)(208,333)
(209,332)(210,334)(211,327)(212,329)(213,328)(214,330)(215,323)(216,325)
(217,324)(218,326)(219,319)(220,321)(221,320)(222,322)(223,315)(224,317)
(225,316)(226,318)(227,283)(228,285)(229,284)(230,286)(231,307)(232,309)
(233,308)(234,310)(235,303)(236,305)(237,304)(238,306)(239,299)(240,301)
(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)(248,293)
(249,292)(250,294)(251,287)(252,289)(253,288)(254,290);;
s3 := (  3,231)(  4,234)(  5,233)(  6,232)(  7,227)(  8,230)(  9,229)( 10,228)
( 11,251)( 12,254)( 13,253)( 14,252)( 15,247)( 16,250)( 17,249)( 18,248)
( 19,243)( 20,246)( 21,245)( 22,244)( 23,239)( 24,242)( 25,241)( 26,240)
( 27,235)( 28,238)( 29,237)( 30,236)( 31,203)( 32,206)( 33,205)( 34,204)
( 35,199)( 36,202)( 37,201)( 38,200)( 39,223)( 40,226)( 41,225)( 42,224)
( 43,219)( 44,222)( 45,221)( 46,220)( 47,215)( 48,218)( 49,217)( 50,216)
( 51,211)( 52,214)( 53,213)( 54,212)( 55,207)( 56,210)( 57,209)( 58,208)
( 59,175)( 60,178)( 61,177)( 62,176)( 63,171)( 64,174)( 65,173)( 66,172)
( 67,195)( 68,198)( 69,197)( 70,196)( 71,191)( 72,194)( 73,193)( 74,192)
( 75,187)( 76,190)( 77,189)( 78,188)( 79,183)( 80,186)( 81,185)( 82,184)
( 83,179)( 84,182)( 85,181)( 86,180)( 87,315)( 88,318)( 89,317)( 90,316)
( 91,311)( 92,314)( 93,313)( 94,312)( 95,335)( 96,338)( 97,337)( 98,336)
( 99,331)(100,334)(101,333)(102,332)(103,327)(104,330)(105,329)(106,328)
(107,323)(108,326)(109,325)(110,324)(111,319)(112,322)(113,321)(114,320)
(115,287)(116,290)(117,289)(118,288)(119,283)(120,286)(121,285)(122,284)
(123,307)(124,310)(125,309)(126,308)(127,303)(128,306)(129,305)(130,304)
(131,299)(132,302)(133,301)(134,300)(135,295)(136,298)(137,297)(138,296)
(139,291)(140,294)(141,293)(142,292)(143,259)(144,262)(145,261)(146,260)
(147,255)(148,258)(149,257)(150,256)(151,279)(152,282)(153,281)(154,280)
(155,275)(156,278)(157,277)(158,276)(159,271)(160,274)(161,273)(162,272)
(163,267)(164,270)(165,269)(166,268)(167,263)(168,266)(169,265)(170,264);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(338)!(1,2);
s1 := Sym(338)!(  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 17)
( 16, 18)( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)
( 32, 34)( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)
( 64, 66)( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)
( 96, 98)( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)
(128,130)(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)
(144,146)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)
(160,162)(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)
(176,178)(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)
(208,210)(211,213)(212,214)(215,217)(216,218)(219,221)(220,222)(223,225)
(224,226)(227,229)(228,230)(231,233)(232,234)(235,237)(236,238)(239,241)
(240,242)(243,245)(244,246)(247,249)(248,250)(251,253)(252,254)(255,257)
(256,258)(259,261)(260,262)(263,265)(264,266)(267,269)(268,270)(271,273)
(272,274)(275,277)(276,278)(279,281)(280,282)(283,285)(284,286)(287,289)
(288,290)(291,293)(292,294)(295,297)(296,298)(299,301)(300,302)(303,305)
(304,306)(307,309)(308,310)(311,313)(312,314)(315,317)(316,318)(319,321)
(320,322)(323,325)(324,326)(327,329)(328,330)(331,333)(332,334)(335,337)
(336,338);
s2 := Sym(338)!(  4,  5)(  7, 27)(  8, 29)(  9, 28)( 10, 30)( 11, 23)( 12, 25)
( 13, 24)( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)
( 33, 60)( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)
( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)
( 49, 72)( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)
( 57, 64)( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)
( 96,109)( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)
(116,145)(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)
(124,165)(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)
(132,157)(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)
(140,149)(141,148)(142,150)(171,255)(172,257)(173,256)(174,258)(175,279)
(176,281)(177,280)(178,282)(179,275)(180,277)(181,276)(182,278)(183,271)
(184,273)(185,272)(186,274)(187,267)(188,269)(189,268)(190,270)(191,263)
(192,265)(193,264)(194,266)(195,259)(196,261)(197,260)(198,262)(199,311)
(200,313)(201,312)(202,314)(203,335)(204,337)(205,336)(206,338)(207,331)
(208,333)(209,332)(210,334)(211,327)(212,329)(213,328)(214,330)(215,323)
(216,325)(217,324)(218,326)(219,319)(220,321)(221,320)(222,322)(223,315)
(224,317)(225,316)(226,318)(227,283)(228,285)(229,284)(230,286)(231,307)
(232,309)(233,308)(234,310)(235,303)(236,305)(237,304)(238,306)(239,299)
(240,301)(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)
(248,293)(249,292)(250,294)(251,287)(252,289)(253,288)(254,290);
s3 := Sym(338)!(  3,231)(  4,234)(  5,233)(  6,232)(  7,227)(  8,230)(  9,229)
( 10,228)( 11,251)( 12,254)( 13,253)( 14,252)( 15,247)( 16,250)( 17,249)
( 18,248)( 19,243)( 20,246)( 21,245)( 22,244)( 23,239)( 24,242)( 25,241)
( 26,240)( 27,235)( 28,238)( 29,237)( 30,236)( 31,203)( 32,206)( 33,205)
( 34,204)( 35,199)( 36,202)( 37,201)( 38,200)( 39,223)( 40,226)( 41,225)
( 42,224)( 43,219)( 44,222)( 45,221)( 46,220)( 47,215)( 48,218)( 49,217)
( 50,216)( 51,211)( 52,214)( 53,213)( 54,212)( 55,207)( 56,210)( 57,209)
( 58,208)( 59,175)( 60,178)( 61,177)( 62,176)( 63,171)( 64,174)( 65,173)
( 66,172)( 67,195)( 68,198)( 69,197)( 70,196)( 71,191)( 72,194)( 73,193)
( 74,192)( 75,187)( 76,190)( 77,189)( 78,188)( 79,183)( 80,186)( 81,185)
( 82,184)( 83,179)( 84,182)( 85,181)( 86,180)( 87,315)( 88,318)( 89,317)
( 90,316)( 91,311)( 92,314)( 93,313)( 94,312)( 95,335)( 96,338)( 97,337)
( 98,336)( 99,331)(100,334)(101,333)(102,332)(103,327)(104,330)(105,329)
(106,328)(107,323)(108,326)(109,325)(110,324)(111,319)(112,322)(113,321)
(114,320)(115,287)(116,290)(117,289)(118,288)(119,283)(120,286)(121,285)
(122,284)(123,307)(124,310)(125,309)(126,308)(127,303)(128,306)(129,305)
(130,304)(131,299)(132,302)(133,301)(134,300)(135,295)(136,298)(137,297)
(138,296)(139,291)(140,294)(141,293)(142,292)(143,259)(144,262)(145,261)
(146,260)(147,255)(148,258)(149,257)(150,256)(151,279)(152,282)(153,281)
(154,280)(155,275)(156,278)(157,277)(158,276)(159,271)(160,274)(161,273)
(162,272)(163,267)(164,270)(165,269)(166,268)(167,263)(168,266)(169,265)
(170,264);
poly := sub<Sym(338)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope