Polytope of Type {84,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,8}*1344a
Also Known As : {84,8|2}. if this polytope has another name.
Group : SmallGroup(1344,5698)
Rank : 3
Schlafli Type : {84,8}
Number of vertices, edges, etc : 84, 336, 8
Order of s₀s₁s₂ : 168
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {84,4}*672a, {42,8}*672
   3-fold quotients : {28,8}*448a
   4-fold quotients : {84,2}*336, {42,4}*336a
   6-fold quotients : {28,4}*224, {14,8}*224
   7-fold quotients : {12,8}*192a
   8-fold quotients : {42,2}*168
   12-fold quotients : {28,2}*112, {14,4}*112
   14-fold quotients : {12,4}*96a, {6,8}*96
   16-fold quotients : {21,2}*84
   21-fold quotients : {4,8}*64a
   24-fold quotients : {14,2}*56
   28-fold quotients : {12,2}*48, {6,4}*48a
   42-fold quotients : {4,4}*32, {2,8}*32
   48-fold quotients : {7,2}*28
   56-fold quotients : {6,2}*24
   84-fold quotients : {2,4}*16, {4,2}*16
   112-fold quotients : {3,2}*12
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

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