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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}*672c
if this polytope has a name.
Group : SmallGroup(672,1080)
Rank : 3
Schlafli Type : {84,4}
Number of vertices, edges, etc : 84, 168, 4
Order of s0s1s2 : 84
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {84,4,2} of size 1344
Vertex Figure Of :
   {2,84,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {42,4}*336b
   4-fold quotients : {21,4}*168
   7-fold quotients : {12,4}*96c
   14-fold quotients : {6,4}*48c
   28-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {84,4}*1344b
Permutation Representation (GAP) :
s0 := (  2,  3)(  5, 25)(  6, 27)(  7, 26)(  8, 28)(  9, 21)( 10, 23)( 11, 22)
( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)( 31, 58)
( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)( 39, 78)
( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)( 47, 70)
( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)
( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)( 94,107)
( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)(114,143)
(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)(122,163)
(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)(130,155)
(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)(138,147)
(139,146)(140,148)(169,253)(170,255)(171,254)(172,256)(173,277)(174,279)
(175,278)(176,280)(177,273)(178,275)(179,274)(180,276)(181,269)(182,271)
(183,270)(184,272)(185,265)(186,267)(187,266)(188,268)(189,261)(190,263)
(191,262)(192,264)(193,257)(194,259)(195,258)(196,260)(197,309)(198,311)
(199,310)(200,312)(201,333)(202,335)(203,334)(204,336)(205,329)(206,331)
(207,330)(208,332)(209,325)(210,327)(211,326)(212,328)(213,321)(214,323)
(215,322)(216,324)(217,317)(218,319)(219,318)(220,320)(221,313)(222,315)
(223,314)(224,316)(225,281)(226,283)(227,282)(228,284)(229,305)(230,307)
(231,306)(232,308)(233,301)(234,303)(235,302)(236,304)(237,297)(238,299)
(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)(246,291)
(247,290)(248,292)(249,285)(250,287)(251,286)(252,288);;
s1 := (  1,201)(  2,202)(  3,204)(  4,203)(  5,197)(  6,198)(  7,200)(  8,199)
(  9,221)( 10,222)( 11,224)( 12,223)( 13,217)( 14,218)( 15,220)( 16,219)
( 17,213)( 18,214)( 19,216)( 20,215)( 21,209)( 22,210)( 23,212)( 24,211)
( 25,205)( 26,206)( 27,208)( 28,207)( 29,173)( 30,174)( 31,176)( 32,175)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,193)( 38,194)( 39,196)( 40,195)
( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)( 48,187)
( 49,181)( 50,182)( 51,184)( 52,183)( 53,177)( 54,178)( 55,180)( 56,179)
( 57,229)( 58,230)( 59,232)( 60,231)( 61,225)( 62,226)( 63,228)( 64,227)
( 65,249)( 66,250)( 67,252)( 68,251)( 69,245)( 70,246)( 71,248)( 72,247)
( 73,241)( 74,242)( 75,244)( 76,243)( 77,237)( 78,238)( 79,240)( 80,239)
( 81,233)( 82,234)( 83,236)( 84,235)( 85,285)( 86,286)( 87,288)( 88,287)
( 89,281)( 90,282)( 91,284)( 92,283)( 93,305)( 94,306)( 95,308)( 96,307)
( 97,301)( 98,302)( 99,304)(100,303)(101,297)(102,298)(103,300)(104,299)
(105,293)(106,294)(107,296)(108,295)(109,289)(110,290)(111,292)(112,291)
(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)(120,255)
(121,277)(122,278)(123,280)(124,279)(125,273)(126,274)(127,276)(128,275)
(129,269)(130,270)(131,272)(132,271)(133,265)(134,266)(135,268)(136,267)
(137,261)(138,262)(139,264)(140,263)(141,313)(142,314)(143,316)(144,315)
(145,309)(146,310)(147,312)(148,311)(149,333)(150,334)(151,336)(152,335)
(153,329)(154,330)(155,332)(156,331)(157,325)(158,326)(159,328)(160,327)
(161,321)(162,322)(163,324)(164,323)(165,317)(166,318)(167,320)(168,319);;
s2 := (  1, 88)(  2, 87)(  3, 86)(  4, 85)(  5, 92)(  6, 91)(  7, 90)(  8, 89)
(  9, 96)( 10, 95)( 11, 94)( 12, 93)( 13,100)( 14, 99)( 15, 98)( 16, 97)
( 17,104)( 18,103)( 19,102)( 20,101)( 21,108)( 22,107)( 23,106)( 24,105)
( 25,112)( 26,111)( 27,110)( 28,109)( 29,116)( 30,115)( 31,114)( 32,113)
( 33,120)( 34,119)( 35,118)( 36,117)( 37,124)( 38,123)( 39,122)( 40,121)
( 41,128)( 42,127)( 43,126)( 44,125)( 45,132)( 46,131)( 47,130)( 48,129)
( 49,136)( 50,135)( 51,134)( 52,133)( 53,140)( 54,139)( 55,138)( 56,137)
( 57,144)( 58,143)( 59,142)( 60,141)( 61,148)( 62,147)( 63,146)( 64,145)
( 65,152)( 66,151)( 67,150)( 68,149)( 69,156)( 70,155)( 71,154)( 72,153)
( 73,160)( 74,159)( 75,158)( 76,157)( 77,164)( 78,163)( 79,162)( 80,161)
( 81,168)( 82,167)( 83,166)( 84,165)(169,256)(170,255)(171,254)(172,253)
(173,260)(174,259)(175,258)(176,257)(177,264)(178,263)(179,262)(180,261)
(181,268)(182,267)(183,266)(184,265)(185,272)(186,271)(187,270)(188,269)
(189,276)(190,275)(191,274)(192,273)(193,280)(194,279)(195,278)(196,277)
(197,284)(198,283)(199,282)(200,281)(201,288)(202,287)(203,286)(204,285)
(205,292)(206,291)(207,290)(208,289)(209,296)(210,295)(211,294)(212,293)
(213,300)(214,299)(215,298)(216,297)(217,304)(218,303)(219,302)(220,301)
(221,308)(222,307)(223,306)(224,305)(225,312)(226,311)(227,310)(228,309)
(229,316)(230,315)(231,314)(232,313)(233,320)(234,319)(235,318)(236,317)
(237,324)(238,323)(239,322)(240,321)(241,328)(242,327)(243,326)(244,325)
(245,332)(246,331)(247,330)(248,329)(249,336)(250,335)(251,334)(252,333);;
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*s0*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*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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(336)!(  2,  3)(  5, 25)(  6, 27)(  7, 26)(  8, 28)(  9, 21)( 10, 23)
( 11, 22)( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)
( 31, 58)( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)
( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)
( 47, 70)( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)
( 55, 62)( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)
( 94,107)( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)
(114,143)(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)
(122,163)(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)
(130,155)(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)
(138,147)(139,146)(140,148)(169,253)(170,255)(171,254)(172,256)(173,277)
(174,279)(175,278)(176,280)(177,273)(178,275)(179,274)(180,276)(181,269)
(182,271)(183,270)(184,272)(185,265)(186,267)(187,266)(188,268)(189,261)
(190,263)(191,262)(192,264)(193,257)(194,259)(195,258)(196,260)(197,309)
(198,311)(199,310)(200,312)(201,333)(202,335)(203,334)(204,336)(205,329)
(206,331)(207,330)(208,332)(209,325)(210,327)(211,326)(212,328)(213,321)
(214,323)(215,322)(216,324)(217,317)(218,319)(219,318)(220,320)(221,313)
(222,315)(223,314)(224,316)(225,281)(226,283)(227,282)(228,284)(229,305)
(230,307)(231,306)(232,308)(233,301)(234,303)(235,302)(236,304)(237,297)
(238,299)(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)
(246,291)(247,290)(248,292)(249,285)(250,287)(251,286)(252,288);
s1 := Sym(336)!(  1,201)(  2,202)(  3,204)(  4,203)(  5,197)(  6,198)(  7,200)
(  8,199)(  9,221)( 10,222)( 11,224)( 12,223)( 13,217)( 14,218)( 15,220)
( 16,219)( 17,213)( 18,214)( 19,216)( 20,215)( 21,209)( 22,210)( 23,212)
( 24,211)( 25,205)( 26,206)( 27,208)( 28,207)( 29,173)( 30,174)( 31,176)
( 32,175)( 33,169)( 34,170)( 35,172)( 36,171)( 37,193)( 38,194)( 39,196)
( 40,195)( 41,189)( 42,190)( 43,192)( 44,191)( 45,185)( 46,186)( 47,188)
( 48,187)( 49,181)( 50,182)( 51,184)( 52,183)( 53,177)( 54,178)( 55,180)
( 56,179)( 57,229)( 58,230)( 59,232)( 60,231)( 61,225)( 62,226)( 63,228)
( 64,227)( 65,249)( 66,250)( 67,252)( 68,251)( 69,245)( 70,246)( 71,248)
( 72,247)( 73,241)( 74,242)( 75,244)( 76,243)( 77,237)( 78,238)( 79,240)
( 80,239)( 81,233)( 82,234)( 83,236)( 84,235)( 85,285)( 86,286)( 87,288)
( 88,287)( 89,281)( 90,282)( 91,284)( 92,283)( 93,305)( 94,306)( 95,308)
( 96,307)( 97,301)( 98,302)( 99,304)(100,303)(101,297)(102,298)(103,300)
(104,299)(105,293)(106,294)(107,296)(108,295)(109,289)(110,290)(111,292)
(112,291)(113,257)(114,258)(115,260)(116,259)(117,253)(118,254)(119,256)
(120,255)(121,277)(122,278)(123,280)(124,279)(125,273)(126,274)(127,276)
(128,275)(129,269)(130,270)(131,272)(132,271)(133,265)(134,266)(135,268)
(136,267)(137,261)(138,262)(139,264)(140,263)(141,313)(142,314)(143,316)
(144,315)(145,309)(146,310)(147,312)(148,311)(149,333)(150,334)(151,336)
(152,335)(153,329)(154,330)(155,332)(156,331)(157,325)(158,326)(159,328)
(160,327)(161,321)(162,322)(163,324)(164,323)(165,317)(166,318)(167,320)
(168,319);
s2 := Sym(336)!(  1, 88)(  2, 87)(  3, 86)(  4, 85)(  5, 92)(  6, 91)(  7, 90)
(  8, 89)(  9, 96)( 10, 95)( 11, 94)( 12, 93)( 13,100)( 14, 99)( 15, 98)
( 16, 97)( 17,104)( 18,103)( 19,102)( 20,101)( 21,108)( 22,107)( 23,106)
( 24,105)( 25,112)( 26,111)( 27,110)( 28,109)( 29,116)( 30,115)( 31,114)
( 32,113)( 33,120)( 34,119)( 35,118)( 36,117)( 37,124)( 38,123)( 39,122)
( 40,121)( 41,128)( 42,127)( 43,126)( 44,125)( 45,132)( 46,131)( 47,130)
( 48,129)( 49,136)( 50,135)( 51,134)( 52,133)( 53,140)( 54,139)( 55,138)
( 56,137)( 57,144)( 58,143)( 59,142)( 60,141)( 61,148)( 62,147)( 63,146)
( 64,145)( 65,152)( 66,151)( 67,150)( 68,149)( 69,156)( 70,155)( 71,154)
( 72,153)( 73,160)( 74,159)( 75,158)( 76,157)( 77,164)( 78,163)( 79,162)
( 80,161)( 81,168)( 82,167)( 83,166)( 84,165)(169,256)(170,255)(171,254)
(172,253)(173,260)(174,259)(175,258)(176,257)(177,264)(178,263)(179,262)
(180,261)(181,268)(182,267)(183,266)(184,265)(185,272)(186,271)(187,270)
(188,269)(189,276)(190,275)(191,274)(192,273)(193,280)(194,279)(195,278)
(196,277)(197,284)(198,283)(199,282)(200,281)(201,288)(202,287)(203,286)
(204,285)(205,292)(206,291)(207,290)(208,289)(209,296)(210,295)(211,294)
(212,293)(213,300)(214,299)(215,298)(216,297)(217,304)(218,303)(219,302)
(220,301)(221,308)(222,307)(223,306)(224,305)(225,312)(226,311)(227,310)
(228,309)(229,316)(230,315)(231,314)(232,313)(233,320)(234,319)(235,318)
(236,317)(237,324)(238,323)(239,322)(240,321)(241,328)(242,327)(243,326)
(244,325)(245,332)(246,331)(247,330)(248,329)(249,336)(250,335)(251,334)
(252,333);
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*s0*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*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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope