Polytope of Type {2,84,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,84,4}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11397)
Rank : 4
Schlafli Type : {2,84,4}
Number of vertices, edges, etc : 2, 84, 168, 4
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,42,4}*672b
   4-fold quotients : {2,21,4}*336
   7-fold quotients : {2,12,4}*192c
   14-fold quotients : {2,6,4}*96c
   28-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  7, 27)(  8, 29)(  9, 28)( 10, 30)( 11, 23)( 12, 25)( 13, 24)
( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)( 33, 60)
( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)( 41, 80)
( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)( 49, 72)
( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)
( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)( 96,109)
( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)(116,145)
(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)(124,165)
(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)(132,157)
(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)(140,149)
(141,148)(142,150)(171,255)(172,257)(173,256)(174,258)(175,279)(176,281)
(177,280)(178,282)(179,275)(180,277)(181,276)(182,278)(183,271)(184,273)
(185,272)(186,274)(187,267)(188,269)(189,268)(190,270)(191,263)(192,265)
(193,264)(194,266)(195,259)(196,261)(197,260)(198,262)(199,311)(200,313)
(201,312)(202,314)(203,335)(204,337)(205,336)(206,338)(207,331)(208,333)
(209,332)(210,334)(211,327)(212,329)(213,328)(214,330)(215,323)(216,325)
(217,324)(218,326)(219,319)(220,321)(221,320)(222,322)(223,315)(224,317)
(225,316)(226,318)(227,283)(228,285)(229,284)(230,286)(231,307)(232,309)
(233,308)(234,310)(235,303)(236,305)(237,304)(238,306)(239,299)(240,301)
(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)(248,293)
(249,292)(250,294)(251,287)(252,289)(253,288)(254,290);;
s2 := (  3,203)(  4,204)(  5,206)(  6,205)(  7,199)(  8,200)(  9,202)( 10,201)
( 11,223)( 12,224)( 13,226)( 14,225)( 15,219)( 16,220)( 17,222)( 18,221)
( 19,215)( 20,216)( 21,218)( 22,217)( 23,211)( 24,212)( 25,214)( 26,213)
( 27,207)( 28,208)( 29,210)( 30,209)( 31,175)( 32,176)( 33,178)( 34,177)
( 35,171)( 36,172)( 37,174)( 38,173)( 39,195)( 40,196)( 41,198)( 42,197)
( 43,191)( 44,192)( 45,194)( 46,193)( 47,187)( 48,188)( 49,190)( 50,189)
( 51,183)( 52,184)( 53,186)( 54,185)( 55,179)( 56,180)( 57,182)( 58,181)
( 59,231)( 60,232)( 61,234)( 62,233)( 63,227)( 64,228)( 65,230)( 66,229)
( 67,251)( 68,252)( 69,254)( 70,253)( 71,247)( 72,248)( 73,250)( 74,249)
( 75,243)( 76,244)( 77,246)( 78,245)( 79,239)( 80,240)( 81,242)( 82,241)
( 83,235)( 84,236)( 85,238)( 86,237)( 87,287)( 88,288)( 89,290)( 90,289)
( 91,283)( 92,284)( 93,286)( 94,285)( 95,307)( 96,308)( 97,310)( 98,309)
( 99,303)(100,304)(101,306)(102,305)(103,299)(104,300)(105,302)(106,301)
(107,295)(108,296)(109,298)(110,297)(111,291)(112,292)(113,294)(114,293)
(115,259)(116,260)(117,262)(118,261)(119,255)(120,256)(121,258)(122,257)
(123,279)(124,280)(125,282)(126,281)(127,275)(128,276)(129,278)(130,277)
(131,271)(132,272)(133,274)(134,273)(135,267)(136,268)(137,270)(138,269)
(139,263)(140,264)(141,266)(142,265)(143,315)(144,316)(145,318)(146,317)
(147,311)(148,312)(149,314)(150,313)(151,335)(152,336)(153,338)(154,337)
(155,331)(156,332)(157,334)(158,333)(159,327)(160,328)(161,330)(162,329)
(163,323)(164,324)(165,326)(166,325)(167,319)(168,320)(169,322)(170,321);;
s3 := (  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 94)(  8, 93)(  9, 92)( 10, 91)
( 11, 98)( 12, 97)( 13, 96)( 14, 95)( 15,102)( 16,101)( 17,100)( 18, 99)
( 19,106)( 20,105)( 21,104)( 22,103)( 23,110)( 24,109)( 25,108)( 26,107)
( 27,114)( 28,113)( 29,112)( 30,111)( 31,118)( 32,117)( 33,116)( 34,115)
( 35,122)( 36,121)( 37,120)( 38,119)( 39,126)( 40,125)( 41,124)( 42,123)
( 43,130)( 44,129)( 45,128)( 46,127)( 47,134)( 48,133)( 49,132)( 50,131)
( 51,138)( 52,137)( 53,136)( 54,135)( 55,142)( 56,141)( 57,140)( 58,139)
( 59,146)( 60,145)( 61,144)( 62,143)( 63,150)( 64,149)( 65,148)( 66,147)
( 67,154)( 68,153)( 69,152)( 70,151)( 71,158)( 72,157)( 73,156)( 74,155)
( 75,162)( 76,161)( 77,160)( 78,159)( 79,166)( 80,165)( 81,164)( 82,163)
( 83,170)( 84,169)( 85,168)( 86,167)(171,258)(172,257)(173,256)(174,255)
(175,262)(176,261)(177,260)(178,259)(179,266)(180,265)(181,264)(182,263)
(183,270)(184,269)(185,268)(186,267)(187,274)(188,273)(189,272)(190,271)
(191,278)(192,277)(193,276)(194,275)(195,282)(196,281)(197,280)(198,279)
(199,286)(200,285)(201,284)(202,283)(203,290)(204,289)(205,288)(206,287)
(207,294)(208,293)(209,292)(210,291)(211,298)(212,297)(213,296)(214,295)
(215,302)(216,301)(217,300)(218,299)(219,306)(220,305)(221,304)(222,303)
(223,310)(224,309)(225,308)(226,307)(227,314)(228,313)(229,312)(230,311)
(231,318)(232,317)(233,316)(234,315)(235,322)(236,321)(237,320)(238,319)
(239,326)(240,325)(241,324)(242,323)(243,330)(244,329)(245,328)(246,327)
(247,334)(248,333)(249,332)(250,331)(251,338)(252,337)(253,336)(254,335);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(338)!(1,2);
s1 := Sym(338)!(  4,  5)(  7, 27)(  8, 29)(  9, 28)( 10, 30)( 11, 23)( 12, 25)
( 13, 24)( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)
( 33, 60)( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)
( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)
( 49, 72)( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)
( 57, 64)( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)
( 96,109)( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)
(116,145)(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)
(124,165)(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)
(132,157)(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)
(140,149)(141,148)(142,150)(171,255)(172,257)(173,256)(174,258)(175,279)
(176,281)(177,280)(178,282)(179,275)(180,277)(181,276)(182,278)(183,271)
(184,273)(185,272)(186,274)(187,267)(188,269)(189,268)(190,270)(191,263)
(192,265)(193,264)(194,266)(195,259)(196,261)(197,260)(198,262)(199,311)
(200,313)(201,312)(202,314)(203,335)(204,337)(205,336)(206,338)(207,331)
(208,333)(209,332)(210,334)(211,327)(212,329)(213,328)(214,330)(215,323)
(216,325)(217,324)(218,326)(219,319)(220,321)(221,320)(222,322)(223,315)
(224,317)(225,316)(226,318)(227,283)(228,285)(229,284)(230,286)(231,307)
(232,309)(233,308)(234,310)(235,303)(236,305)(237,304)(238,306)(239,299)
(240,301)(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)
(248,293)(249,292)(250,294)(251,287)(252,289)(253,288)(254,290);
s2 := Sym(338)!(  3,203)(  4,204)(  5,206)(  6,205)(  7,199)(  8,200)(  9,202)
( 10,201)( 11,223)( 12,224)( 13,226)( 14,225)( 15,219)( 16,220)( 17,222)
( 18,221)( 19,215)( 20,216)( 21,218)( 22,217)( 23,211)( 24,212)( 25,214)
( 26,213)( 27,207)( 28,208)( 29,210)( 30,209)( 31,175)( 32,176)( 33,178)
( 34,177)( 35,171)( 36,172)( 37,174)( 38,173)( 39,195)( 40,196)( 41,198)
( 42,197)( 43,191)( 44,192)( 45,194)( 46,193)( 47,187)( 48,188)( 49,190)
( 50,189)( 51,183)( 52,184)( 53,186)( 54,185)( 55,179)( 56,180)( 57,182)
( 58,181)( 59,231)( 60,232)( 61,234)( 62,233)( 63,227)( 64,228)( 65,230)
( 66,229)( 67,251)( 68,252)( 69,254)( 70,253)( 71,247)( 72,248)( 73,250)
( 74,249)( 75,243)( 76,244)( 77,246)( 78,245)( 79,239)( 80,240)( 81,242)
( 82,241)( 83,235)( 84,236)( 85,238)( 86,237)( 87,287)( 88,288)( 89,290)
( 90,289)( 91,283)( 92,284)( 93,286)( 94,285)( 95,307)( 96,308)( 97,310)
( 98,309)( 99,303)(100,304)(101,306)(102,305)(103,299)(104,300)(105,302)
(106,301)(107,295)(108,296)(109,298)(110,297)(111,291)(112,292)(113,294)
(114,293)(115,259)(116,260)(117,262)(118,261)(119,255)(120,256)(121,258)
(122,257)(123,279)(124,280)(125,282)(126,281)(127,275)(128,276)(129,278)
(130,277)(131,271)(132,272)(133,274)(134,273)(135,267)(136,268)(137,270)
(138,269)(139,263)(140,264)(141,266)(142,265)(143,315)(144,316)(145,318)
(146,317)(147,311)(148,312)(149,314)(150,313)(151,335)(152,336)(153,338)
(154,337)(155,331)(156,332)(157,334)(158,333)(159,327)(160,328)(161,330)
(162,329)(163,323)(164,324)(165,326)(166,325)(167,319)(168,320)(169,322)
(170,321);
s3 := Sym(338)!(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 94)(  8, 93)(  9, 92)
( 10, 91)( 11, 98)( 12, 97)( 13, 96)( 14, 95)( 15,102)( 16,101)( 17,100)
( 18, 99)( 19,106)( 20,105)( 21,104)( 22,103)( 23,110)( 24,109)( 25,108)
( 26,107)( 27,114)( 28,113)( 29,112)( 30,111)( 31,118)( 32,117)( 33,116)
( 34,115)( 35,122)( 36,121)( 37,120)( 38,119)( 39,126)( 40,125)( 41,124)
( 42,123)( 43,130)( 44,129)( 45,128)( 46,127)( 47,134)( 48,133)( 49,132)
( 50,131)( 51,138)( 52,137)( 53,136)( 54,135)( 55,142)( 56,141)( 57,140)
( 58,139)( 59,146)( 60,145)( 61,144)( 62,143)( 63,150)( 64,149)( 65,148)
( 66,147)( 67,154)( 68,153)( 69,152)( 70,151)( 71,158)( 72,157)( 73,156)
( 74,155)( 75,162)( 76,161)( 77,160)( 78,159)( 79,166)( 80,165)( 81,164)
( 82,163)( 83,170)( 84,169)( 85,168)( 86,167)(171,258)(172,257)(173,256)
(174,255)(175,262)(176,261)(177,260)(178,259)(179,266)(180,265)(181,264)
(182,263)(183,270)(184,269)(185,268)(186,267)(187,274)(188,273)(189,272)
(190,271)(191,278)(192,277)(193,276)(194,275)(195,282)(196,281)(197,280)
(198,279)(199,286)(200,285)(201,284)(202,283)(203,290)(204,289)(205,288)
(206,287)(207,294)(208,293)(209,292)(210,291)(211,298)(212,297)(213,296)
(214,295)(215,302)(216,301)(217,300)(218,299)(219,306)(220,305)(221,304)
(222,303)(223,310)(224,309)(225,308)(226,307)(227,314)(228,313)(229,312)
(230,311)(231,318)(232,317)(233,316)(234,315)(235,322)(236,321)(237,320)
(238,319)(239,326)(240,325)(241,324)(242,323)(243,330)(244,329)(245,328)
(246,327)(247,334)(248,333)(249,332)(250,331)(251,338)(252,337)(253,336)
(254,335);
poly := sub<Sym(338)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2*s1*s2 >;

to this polytope