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Polytope of Type {4,42,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,42,4}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11399)
Rank : 4
Schlafli Type : {4,42,4}
Number of vertices, edges, etc : 4, 84, 84, 4
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {4,42,2}*672b
   4-fold quotients : {4,21,2}*336
   7-fold quotients : {4,6,4}*192c
   14-fold quotients : {4,6,2}*96c
   28-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336);;
s1 := (  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)(171,172)(173,193)(174,194)(175,196)(176,195)(177,189)
(178,190)(179,192)(180,191)(181,185)(182,186)(183,188)(184,187)(197,225)
(198,226)(199,228)(200,227)(201,249)(202,250)(203,252)(204,251)(205,245)
(206,246)(207,248)(208,247)(209,241)(210,242)(211,244)(212,243)(213,237)
(214,238)(215,240)(216,239)(217,233)(218,234)(219,236)(220,235)(221,229)
(222,230)(223,232)(224,231)(255,256)(257,277)(258,278)(259,280)(260,279)
(261,273)(262,274)(263,276)(264,275)(265,269)(266,270)(267,272)(268,271)
(281,309)(282,310)(283,312)(284,311)(285,333)(286,334)(287,336)(288,335)
(289,329)(290,330)(291,332)(292,331)(293,325)(294,326)(295,328)(296,327)
(297,321)(298,322)(299,324)(300,323)(301,317)(302,318)(303,320)(304,319)
(305,313)(306,314)(307,316)(308,315);;
s2 := (  1, 33)(  2, 36)(  3, 35)(  4, 34)(  5, 29)(  6, 32)(  7, 31)(  8, 30)
(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)( 24, 42)
( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 57, 61)( 58, 64)( 59, 63)( 60, 62)
( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)
( 74, 76)( 85,117)( 86,120)( 87,119)( 88,118)( 89,113)( 90,116)( 91,115)
( 92,114)( 93,137)( 94,140)( 95,139)( 96,138)( 97,133)( 98,136)( 99,135)
(100,134)(101,129)(102,132)(103,131)(104,130)(105,125)(106,128)(107,127)
(108,126)(109,121)(110,124)(111,123)(112,122)(141,145)(142,148)(143,147)
(144,146)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)
(156,162)(158,160)(169,285)(170,288)(171,287)(172,286)(173,281)(174,284)
(175,283)(176,282)(177,305)(178,308)(179,307)(180,306)(181,301)(182,304)
(183,303)(184,302)(185,297)(186,300)(187,299)(188,298)(189,293)(190,296)
(191,295)(192,294)(193,289)(194,292)(195,291)(196,290)(197,257)(198,260)
(199,259)(200,258)(201,253)(202,256)(203,255)(204,254)(205,277)(206,280)
(207,279)(208,278)(209,273)(210,276)(211,275)(212,274)(213,269)(214,272)
(215,271)(216,270)(217,265)(218,268)(219,267)(220,266)(221,261)(222,264)
(223,263)(224,262)(225,313)(226,316)(227,315)(228,314)(229,309)(230,312)
(231,311)(232,310)(233,333)(234,336)(235,335)(236,334)(237,329)(238,332)
(239,331)(240,330)(241,325)(242,328)(243,327)(244,326)(245,321)(246,324)
(247,323)(248,322)(249,317)(250,320)(251,319)(252,318);;
s3 := (  1,169)(  2,170)(  3,171)(  4,172)(  5,173)(  6,174)(  7,175)(  8,176)
(  9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)( 16,184)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)
( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)( 32,200)
( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)
( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)( 48,216)
( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)( 56,224)
( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)( 64,232)
( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)( 72,240)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)( 80,248)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)( 88,256)
( 89,257)( 90,258)( 91,259)( 92,260)( 93,261)( 94,262)( 95,263)( 96,264)
( 97,265)( 98,266)( 99,267)(100,268)(101,269)(102,270)(103,271)(104,272)
(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)(112,280)
(113,281)(114,282)(115,283)(116,284)(117,285)(118,286)(119,287)(120,288)
(121,289)(122,290)(123,291)(124,292)(125,293)(126,294)(127,295)(128,296)
(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)
(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)(144,312)
(145,313)(146,314)(147,315)(148,316)(149,317)(150,318)(151,319)(152,320)
(153,321)(154,322)(155,323)(156,324)(157,325)(158,326)(159,327)(160,328)
(161,329)(162,330)(163,331)(164,332)(165,333)(166,334)(167,335)(168,336);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(336)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336);
s1 := 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)(171,172)(173,193)(174,194)(175,196)(176,195)
(177,189)(178,190)(179,192)(180,191)(181,185)(182,186)(183,188)(184,187)
(197,225)(198,226)(199,228)(200,227)(201,249)(202,250)(203,252)(204,251)
(205,245)(206,246)(207,248)(208,247)(209,241)(210,242)(211,244)(212,243)
(213,237)(214,238)(215,240)(216,239)(217,233)(218,234)(219,236)(220,235)
(221,229)(222,230)(223,232)(224,231)(255,256)(257,277)(258,278)(259,280)
(260,279)(261,273)(262,274)(263,276)(264,275)(265,269)(266,270)(267,272)
(268,271)(281,309)(282,310)(283,312)(284,311)(285,333)(286,334)(287,336)
(288,335)(289,329)(290,330)(291,332)(292,331)(293,325)(294,326)(295,328)
(296,327)(297,321)(298,322)(299,324)(300,323)(301,317)(302,318)(303,320)
(304,319)(305,313)(306,314)(307,316)(308,315);
s2 := Sym(336)!(  1, 33)(  2, 36)(  3, 35)(  4, 34)(  5, 29)(  6, 32)(  7, 31)
(  8, 30)(  9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)
( 24, 42)( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 57, 61)( 58, 64)( 59, 63)
( 60, 62)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)
( 72, 78)( 74, 76)( 85,117)( 86,120)( 87,119)( 88,118)( 89,113)( 90,116)
( 91,115)( 92,114)( 93,137)( 94,140)( 95,139)( 96,138)( 97,133)( 98,136)
( 99,135)(100,134)(101,129)(102,132)(103,131)(104,130)(105,125)(106,128)
(107,127)(108,126)(109,121)(110,124)(111,123)(112,122)(141,145)(142,148)
(143,147)(144,146)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)
(155,163)(156,162)(158,160)(169,285)(170,288)(171,287)(172,286)(173,281)
(174,284)(175,283)(176,282)(177,305)(178,308)(179,307)(180,306)(181,301)
(182,304)(183,303)(184,302)(185,297)(186,300)(187,299)(188,298)(189,293)
(190,296)(191,295)(192,294)(193,289)(194,292)(195,291)(196,290)(197,257)
(198,260)(199,259)(200,258)(201,253)(202,256)(203,255)(204,254)(205,277)
(206,280)(207,279)(208,278)(209,273)(210,276)(211,275)(212,274)(213,269)
(214,272)(215,271)(216,270)(217,265)(218,268)(219,267)(220,266)(221,261)
(222,264)(223,263)(224,262)(225,313)(226,316)(227,315)(228,314)(229,309)
(230,312)(231,311)(232,310)(233,333)(234,336)(235,335)(236,334)(237,329)
(238,332)(239,331)(240,330)(241,325)(242,328)(243,327)(244,326)(245,321)
(246,324)(247,323)(248,322)(249,317)(250,320)(251,319)(252,318);
s3 := Sym(336)!(  1,169)(  2,170)(  3,171)(  4,172)(  5,173)(  6,174)(  7,175)
(  8,176)(  9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)
( 16,184)( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)
( 24,192)( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)
( 32,200)( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)
( 40,208)( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)
( 48,216)( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)
( 56,224)( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)
( 64,232)( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)
( 72,240)( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)
( 80,248)( 81,249)( 82,250)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)
( 88,256)( 89,257)( 90,258)( 91,259)( 92,260)( 93,261)( 94,262)( 95,263)
( 96,264)( 97,265)( 98,266)( 99,267)(100,268)(101,269)(102,270)(103,271)
(104,272)(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)
(112,280)(113,281)(114,282)(115,283)(116,284)(117,285)(118,286)(119,287)
(120,288)(121,289)(122,290)(123,291)(124,292)(125,293)(126,294)(127,295)
(128,296)(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)
(136,304)(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)
(144,312)(145,313)(146,314)(147,315)(148,316)(149,317)(150,318)(151,319)
(152,320)(153,321)(154,322)(155,323)(156,324)(157,325)(158,326)(159,327)
(160,328)(161,329)(162,330)(163,331)(164,332)(165,333)(166,334)(167,335)
(168,336);
poly := sub<Sym(336)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*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 >; 
 
References : None.
to this polytope