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Polytope of Type {84,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,4}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11399)
Rank : 3
Schlafli Type : {84,4}
Number of vertices, edges, etc : 168, 336, 8
Order of s0s1s2 : 42
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {42,4}*672
   4-fold quotients : {21,4}*336, {42,4}*336b, {42,4}*336c
   7-fold quotients : {12,4}*192c
   8-fold quotients : {21,4}*168, {42,2}*168
   14-fold quotients : {6,4}*96
   16-fold quotients : {21,2}*84
   24-fold quotients : {14,2}*56
   28-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {7,2}*28
   56-fold quotients : {3,4}*24, {6,2}*24
   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 := (  3,  4)(  5, 25)(  6, 26)(  7, 28)(  8, 27)(  9, 21)( 10, 22)( 11, 24)
( 12, 23)( 13, 17)( 14, 18)( 15, 20)( 16, 19)( 29, 57)( 30, 58)( 31, 60)
( 32, 59)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 77)( 38, 78)( 39, 80)
( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)( 47, 72)
( 48, 71)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 61)( 54, 62)( 55, 64)
( 56, 63)( 87, 88)( 89,109)( 90,110)( 91,112)( 92,111)( 93,105)( 94,106)
( 95,108)( 96,107)( 97,101)( 98,102)( 99,104)(100,103)(113,141)(114,142)
(115,144)(116,143)(117,165)(118,166)(119,168)(120,167)(121,161)(122,162)
(123,164)(124,163)(125,157)(126,158)(127,160)(128,159)(129,153)(130,154)
(131,156)(132,155)(133,149)(134,150)(135,152)(136,151)(137,145)(138,146)
(139,148)(140,147)(169,253)(170,254)(171,256)(172,255)(173,277)(174,278)
(175,280)(176,279)(177,273)(178,274)(179,276)(180,275)(181,269)(182,270)
(183,272)(184,271)(185,265)(186,266)(187,268)(188,267)(189,261)(190,262)
(191,264)(192,263)(193,257)(194,258)(195,260)(196,259)(197,309)(198,310)
(199,312)(200,311)(201,333)(202,334)(203,336)(204,335)(205,329)(206,330)
(207,332)(208,331)(209,325)(210,326)(211,328)(212,327)(213,321)(214,322)
(215,324)(216,323)(217,317)(218,318)(219,320)(220,319)(221,313)(222,314)
(223,316)(224,315)(225,281)(226,282)(227,284)(228,283)(229,305)(230,306)
(231,308)(232,307)(233,301)(234,302)(235,304)(236,303)(237,297)(238,298)
(239,300)(240,299)(241,293)(242,294)(243,296)(244,295)(245,289)(246,290)
(247,292)(248,291)(249,285)(250,286)(251,288)(252,287);;
s1 := (  1,201)(  2,204)(  3,203)(  4,202)(  5,197)(  6,200)(  7,199)(  8,198)
(  9,221)( 10,224)( 11,223)( 12,222)( 13,217)( 14,220)( 15,219)( 16,218)
( 17,213)( 18,216)( 19,215)( 20,214)( 21,209)( 22,212)( 23,211)( 24,210)
( 25,205)( 26,208)( 27,207)( 28,206)( 29,173)( 30,176)( 31,175)( 32,174)
( 33,169)( 34,172)( 35,171)( 36,170)( 37,193)( 38,196)( 39,195)( 40,194)
( 41,189)( 42,192)( 43,191)( 44,190)( 45,185)( 46,188)( 47,187)( 48,186)
( 49,181)( 50,184)( 51,183)( 52,182)( 53,177)( 54,180)( 55,179)( 56,178)
( 57,229)( 58,232)( 59,231)( 60,230)( 61,225)( 62,228)( 63,227)( 64,226)
( 65,249)( 66,252)( 67,251)( 68,250)( 69,245)( 70,248)( 71,247)( 72,246)
( 73,241)( 74,244)( 75,243)( 76,242)( 77,237)( 78,240)( 79,239)( 80,238)
( 81,233)( 82,236)( 83,235)( 84,234)( 85,285)( 86,288)( 87,287)( 88,286)
( 89,281)( 90,284)( 91,283)( 92,282)( 93,305)( 94,308)( 95,307)( 96,306)
( 97,301)( 98,304)( 99,303)(100,302)(101,297)(102,300)(103,299)(104,298)
(105,293)(106,296)(107,295)(108,294)(109,289)(110,292)(111,291)(112,290)
(113,257)(114,260)(115,259)(116,258)(117,253)(118,256)(119,255)(120,254)
(121,277)(122,280)(123,279)(124,278)(125,273)(126,276)(127,275)(128,274)
(129,269)(130,272)(131,271)(132,270)(133,265)(134,268)(135,267)(136,266)
(137,261)(138,264)(139,263)(140,262)(141,313)(142,316)(143,315)(144,314)
(145,309)(146,312)(147,311)(148,310)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,325)(158,328)(159,327)(160,326)
(161,321)(162,324)(163,323)(164,322)(165,317)(166,320)(167,319)(168,318);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,254)(170,253)(171,256)(172,255)
(173,258)(174,257)(175,260)(176,259)(177,262)(178,261)(179,264)(180,263)
(181,266)(182,265)(183,268)(184,267)(185,270)(186,269)(187,272)(188,271)
(189,274)(190,273)(191,276)(192,275)(193,278)(194,277)(195,280)(196,279)
(197,282)(198,281)(199,284)(200,283)(201,286)(202,285)(203,288)(204,287)
(205,290)(206,289)(207,292)(208,291)(209,294)(210,293)(211,296)(212,295)
(213,298)(214,297)(215,300)(216,299)(217,302)(218,301)(219,304)(220,303)
(221,306)(222,305)(223,308)(224,307)(225,310)(226,309)(227,312)(228,311)
(229,314)(230,313)(231,316)(232,315)(233,318)(234,317)(235,320)(236,319)
(237,322)(238,321)(239,324)(240,323)(241,326)(242,325)(243,328)(244,327)
(245,330)(246,329)(247,332)(248,331)(249,334)(250,333)(251,336)(252,335);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!(  3,  4)(  5, 25)(  6, 26)(  7, 28)(  8, 27)(  9, 21)( 10, 22)
( 11, 24)( 12, 23)( 13, 17)( 14, 18)( 15, 20)( 16, 19)( 29, 57)( 30, 58)
( 31, 60)( 32, 59)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 77)( 38, 78)
( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)
( 47, 72)( 48, 71)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 61)( 54, 62)
( 55, 64)( 56, 63)( 87, 88)( 89,109)( 90,110)( 91,112)( 92,111)( 93,105)
( 94,106)( 95,108)( 96,107)( 97,101)( 98,102)( 99,104)(100,103)(113,141)
(114,142)(115,144)(116,143)(117,165)(118,166)(119,168)(120,167)(121,161)
(122,162)(123,164)(124,163)(125,157)(126,158)(127,160)(128,159)(129,153)
(130,154)(131,156)(132,155)(133,149)(134,150)(135,152)(136,151)(137,145)
(138,146)(139,148)(140,147)(169,253)(170,254)(171,256)(172,255)(173,277)
(174,278)(175,280)(176,279)(177,273)(178,274)(179,276)(180,275)(181,269)
(182,270)(183,272)(184,271)(185,265)(186,266)(187,268)(188,267)(189,261)
(190,262)(191,264)(192,263)(193,257)(194,258)(195,260)(196,259)(197,309)
(198,310)(199,312)(200,311)(201,333)(202,334)(203,336)(204,335)(205,329)
(206,330)(207,332)(208,331)(209,325)(210,326)(211,328)(212,327)(213,321)
(214,322)(215,324)(216,323)(217,317)(218,318)(219,320)(220,319)(221,313)
(222,314)(223,316)(224,315)(225,281)(226,282)(227,284)(228,283)(229,305)
(230,306)(231,308)(232,307)(233,301)(234,302)(235,304)(236,303)(237,297)
(238,298)(239,300)(240,299)(241,293)(242,294)(243,296)(244,295)(245,289)
(246,290)(247,292)(248,291)(249,285)(250,286)(251,288)(252,287);
s1 := Sym(336)!(  1,201)(  2,204)(  3,203)(  4,202)(  5,197)(  6,200)(  7,199)
(  8,198)(  9,221)( 10,224)( 11,223)( 12,222)( 13,217)( 14,220)( 15,219)
( 16,218)( 17,213)( 18,216)( 19,215)( 20,214)( 21,209)( 22,212)( 23,211)
( 24,210)( 25,205)( 26,208)( 27,207)( 28,206)( 29,173)( 30,176)( 31,175)
( 32,174)( 33,169)( 34,172)( 35,171)( 36,170)( 37,193)( 38,196)( 39,195)
( 40,194)( 41,189)( 42,192)( 43,191)( 44,190)( 45,185)( 46,188)( 47,187)
( 48,186)( 49,181)( 50,184)( 51,183)( 52,182)( 53,177)( 54,180)( 55,179)
( 56,178)( 57,229)( 58,232)( 59,231)( 60,230)( 61,225)( 62,228)( 63,227)
( 64,226)( 65,249)( 66,252)( 67,251)( 68,250)( 69,245)( 70,248)( 71,247)
( 72,246)( 73,241)( 74,244)( 75,243)( 76,242)( 77,237)( 78,240)( 79,239)
( 80,238)( 81,233)( 82,236)( 83,235)( 84,234)( 85,285)( 86,288)( 87,287)
( 88,286)( 89,281)( 90,284)( 91,283)( 92,282)( 93,305)( 94,308)( 95,307)
( 96,306)( 97,301)( 98,304)( 99,303)(100,302)(101,297)(102,300)(103,299)
(104,298)(105,293)(106,296)(107,295)(108,294)(109,289)(110,292)(111,291)
(112,290)(113,257)(114,260)(115,259)(116,258)(117,253)(118,256)(119,255)
(120,254)(121,277)(122,280)(123,279)(124,278)(125,273)(126,276)(127,275)
(128,274)(129,269)(130,272)(131,271)(132,270)(133,265)(134,268)(135,267)
(136,266)(137,261)(138,264)(139,263)(140,262)(141,313)(142,316)(143,315)
(144,314)(145,309)(146,312)(147,311)(148,310)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,325)(158,328)(159,327)
(160,326)(161,321)(162,324)(163,323)(164,322)(165,317)(166,320)(167,319)
(168,318);
s2 := Sym(336)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,254)(170,253)(171,256)
(172,255)(173,258)(174,257)(175,260)(176,259)(177,262)(178,261)(179,264)
(180,263)(181,266)(182,265)(183,268)(184,267)(185,270)(186,269)(187,272)
(188,271)(189,274)(190,273)(191,276)(192,275)(193,278)(194,277)(195,280)
(196,279)(197,282)(198,281)(199,284)(200,283)(201,286)(202,285)(203,288)
(204,287)(205,290)(206,289)(207,292)(208,291)(209,294)(210,293)(211,296)
(212,295)(213,298)(214,297)(215,300)(216,299)(217,302)(218,301)(219,304)
(220,303)(221,306)(222,305)(223,308)(224,307)(225,310)(226,309)(227,312)
(228,311)(229,314)(230,313)(231,316)(232,315)(233,318)(234,317)(235,320)
(236,319)(237,322)(238,321)(239,324)(240,323)(241,326)(242,325)(243,328)
(244,327)(245,330)(246,329)(247,332)(248,331)(249,334)(250,333)(251,336)
(252,335);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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