Polytope of Type {4,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,84}*1344a
if this polytope has a name.
Group : SmallGroup(1344,5667)
Rank : 3
Schlafli Type : {4,84}
Number of vertices, edges, etc : 8, 336, 168
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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
   3-fold quotients : {4,28}*448
   4-fold quotients : {2,84}*336, {4,42}*336a
   6-fold quotients : {4,28}*224
   7-fold quotients : {4,12}*192a
   8-fold quotients : {2,42}*168
   12-fold quotients : {2,28}*112, {4,14}*112
   14-fold quotients : {4,12}*96a
   16-fold quotients : {2,21}*84
   21-fold quotients : {4,4}*64
   24-fold quotients : {2,14}*56
   28-fold quotients : {2,12}*48, {4,6}*48a
   42-fold quotients : {4,4}*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 := ( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 71)
( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)( 58, 79)
( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)(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)(211,316)(212,317)(213,318)(214,319)
(215,320)(216,321)(217,322)(218,323)(219,324)(220,325)(221,326)(222,327)
(223,328)(224,329)(225,330)(226,331)(227,332)(228,333)(229,334)(230,335)
(231,336)(232,295)(233,296)(234,297)(235,298)(236,299)(237,300)(238,301)
(239,302)(240,303)(241,304)(242,305)(243,306)(244,307)(245,308)(246,309)
(247,310)(248,311)(249,312)(250,313)(251,314)(252,315);;
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)( 85,253)( 86,259)( 87,258)( 88,257)
( 89,256)( 90,255)( 91,254)( 92,267)( 93,273)( 94,272)( 95,271)( 96,270)
( 97,269)( 98,268)( 99,260)(100,266)(101,265)(102,264)(103,263)(104,262)
(105,261)(106,274)(107,280)(108,279)(109,278)(110,277)(111,276)(112,275)
(113,288)(114,294)(115,293)(116,292)(117,291)(118,290)(119,289)(120,281)
(121,287)(122,286)(123,285)(124,284)(125,283)(126,282)(127,295)(128,301)
(129,300)(130,299)(131,298)(132,297)(133,296)(134,309)(135,315)(136,314)
(137,313)(138,312)(139,311)(140,310)(141,302)(142,308)(143,307)(144,306)
(145,305)(146,304)(147,303)(148,316)(149,322)(150,321)(151,320)(152,319)
(153,318)(154,317)(155,330)(156,336)(157,335)(158,334)(159,333)(160,332)
(161,331)(162,323)(163,329)(164,328)(165,327)(166,326)(167,325)(168,324);;
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)
( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)( 91,115)( 92,107)
( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,121)(100,120)
(101,126)(102,125)(103,124)(104,123)(105,122)(127,156)(128,155)(129,161)
(130,160)(131,159)(132,158)(133,157)(134,149)(135,148)(136,154)(137,153)
(138,152)(139,151)(140,150)(141,163)(142,162)(143,168)(144,167)(145,166)
(146,165)(147,164)(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)(253,324)(254,323)(255,329)(256,328)
(257,327)(258,326)(259,325)(260,317)(261,316)(262,322)(263,321)(264,320)
(265,319)(266,318)(267,331)(268,330)(269,336)(270,335)(271,334)(272,333)
(273,332)(274,303)(275,302)(276,308)(277,307)(278,306)(279,305)(280,304)
(281,296)(282,295)(283,301)(284,300)(285,299)(286,298)(287,297)(288,310)
(289,309)(290,315)(291,314)(292,313)(293,312)(294,311);;
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*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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)!( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)
( 50, 71)( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)
( 58, 79)( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)(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)(211,316)(212,317)(213,318)
(214,319)(215,320)(216,321)(217,322)(218,323)(219,324)(220,325)(221,326)
(222,327)(223,328)(224,329)(225,330)(226,331)(227,332)(228,333)(229,334)
(230,335)(231,336)(232,295)(233,296)(234,297)(235,298)(236,299)(237,300)
(238,301)(239,302)(240,303)(241,304)(242,305)(243,306)(244,307)(245,308)
(246,309)(247,310)(248,311)(249,312)(250,313)(251,314)(252,315);
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)( 85,253)( 86,259)( 87,258)
( 88,257)( 89,256)( 90,255)( 91,254)( 92,267)( 93,273)( 94,272)( 95,271)
( 96,270)( 97,269)( 98,268)( 99,260)(100,266)(101,265)(102,264)(103,263)
(104,262)(105,261)(106,274)(107,280)(108,279)(109,278)(110,277)(111,276)
(112,275)(113,288)(114,294)(115,293)(116,292)(117,291)(118,290)(119,289)
(120,281)(121,287)(122,286)(123,285)(124,284)(125,283)(126,282)(127,295)
(128,301)(129,300)(130,299)(131,298)(132,297)(133,296)(134,309)(135,315)
(136,314)(137,313)(138,312)(139,311)(140,310)(141,302)(142,308)(143,307)
(144,306)(145,305)(146,304)(147,303)(148,316)(149,322)(150,321)(151,320)
(152,319)(153,318)(154,317)(155,330)(156,336)(157,335)(158,334)(159,333)
(160,332)(161,331)(162,323)(163,329)(164,328)(165,327)(166,326)(167,325)
(168,324);
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)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)( 91,115)
( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,121)
(100,120)(101,126)(102,125)(103,124)(104,123)(105,122)(127,156)(128,155)
(129,161)(130,160)(131,159)(132,158)(133,157)(134,149)(135,148)(136,154)
(137,153)(138,152)(139,151)(140,150)(141,163)(142,162)(143,168)(144,167)
(145,166)(146,165)(147,164)(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)(253,324)(254,323)(255,329)
(256,328)(257,327)(258,326)(259,325)(260,317)(261,316)(262,322)(263,321)
(264,320)(265,319)(266,318)(267,331)(268,330)(269,336)(270,335)(271,334)
(272,333)(273,332)(274,303)(275,302)(276,308)(277,307)(278,306)(279,305)
(280,304)(281,296)(282,295)(283,301)(284,300)(285,299)(286,298)(287,297)
(288,310)(289,309)(290,315)(291,314)(292,313)(293,312)(294,311);
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*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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