Polytope of Type {8,84}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope