Polytope of Type {84,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,4}*672b
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}*96b
14-fold quotients : {6,4}*48c
28-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {168,4}*1344c, {168,4}*1344d, {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, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239)
(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)(254,255)
(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)(270,271)
(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)(286,287)
(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)(302,303)
(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)(318,319)
(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)(334,335);;
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,
s2*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*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, 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, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)
(206,207)(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)
(222,223)(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)
(238,239)(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)
(254,255)(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)
(270,271)(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)
(286,287)(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)
(302,303)(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)
(318,319)(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)
(334,335);
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,
s2*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*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