Polytope of Type {2,4,42}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,42}*1344
if this polytope has a name.
Group : SmallGroup(1344,11701)
Rank : 4
Schlafli Type : {2,4,42}
Number of vertices, edges, etc : 2, 8, 168, 84
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
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 : {2,4,21}*672, {2,4,42}*672b, {2,4,42}*672c
4-fold quotients : {2,4,21}*336, {2,2,42}*336
7-fold quotients : {2,4,6}*192
8-fold quotients : {2,2,21}*168
12-fold quotients : {2,2,14}*112
14-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
24-fold quotients : {2,2,7}*56
28-fold quotients : {2,4,3}*48, {2,2,6}*48
56-fold quotients : {2,2,3}*24
84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,173)( 4,174)( 5,171)( 6,172)( 7,177)( 8,178)( 9,175)( 10,176)
( 11,181)( 12,182)( 13,179)( 14,180)( 15,185)( 16,186)( 17,183)( 18,184)
( 19,189)( 20,190)( 21,187)( 22,188)( 23,193)( 24,194)( 25,191)( 26,192)
( 27,197)( 28,198)( 29,195)( 30,196)( 31,201)( 32,202)( 33,199)( 34,200)
( 35,205)( 36,206)( 37,203)( 38,204)( 39,209)( 40,210)( 41,207)( 42,208)
( 43,213)( 44,214)( 45,211)( 46,212)( 47,217)( 48,218)( 49,215)( 50,216)
( 51,221)( 52,222)( 53,219)( 54,220)( 55,225)( 56,226)( 57,223)( 58,224)
( 59,229)( 60,230)( 61,227)( 62,228)( 63,233)( 64,234)( 65,231)( 66,232)
( 67,237)( 68,238)( 69,235)( 70,236)( 71,241)( 72,242)( 73,239)( 74,240)
( 75,245)( 76,246)( 77,243)( 78,244)( 79,249)( 80,250)( 81,247)( 82,248)
( 83,253)( 84,254)( 85,251)( 86,252)( 87,257)( 88,258)( 89,255)( 90,256)
( 91,261)( 92,262)( 93,259)( 94,260)( 95,265)( 96,266)( 97,263)( 98,264)
( 99,269)(100,270)(101,267)(102,268)(103,273)(104,274)(105,271)(106,272)
(107,277)(108,278)(109,275)(110,276)(111,281)(112,282)(113,279)(114,280)
(115,285)(116,286)(117,283)(118,284)(119,289)(120,290)(121,287)(122,288)
(123,293)(124,294)(125,291)(126,292)(127,297)(128,298)(129,295)(130,296)
(131,301)(132,302)(133,299)(134,300)(135,305)(136,306)(137,303)(138,304)
(139,309)(140,310)(141,307)(142,308)(143,313)(144,314)(145,311)(146,312)
(147,317)(148,318)(149,315)(150,316)(151,321)(152,322)(153,319)(154,320)
(155,325)(156,326)(157,323)(158,324)(159,329)(160,330)(161,327)(162,328)
(163,333)(164,334)(165,331)(166,332)(167,337)(168,338)(169,335)(170,336);;
s2 := ( 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)(172,173)(175,195)(176,197)(177,196)(178,198)(179,191)
(180,193)(181,192)(182,194)(183,187)(184,189)(185,188)(186,190)(199,227)
(200,229)(201,228)(202,230)(203,251)(204,253)(205,252)(206,254)(207,247)
(208,249)(209,248)(210,250)(211,243)(212,245)(213,244)(214,246)(215,239)
(216,241)(217,240)(218,242)(219,235)(220,237)(221,236)(222,238)(223,231)
(224,233)(225,232)(226,234)(256,257)(259,279)(260,281)(261,280)(262,282)
(263,275)(264,277)(265,276)(266,278)(267,271)(268,273)(269,272)(270,274)
(283,311)(284,313)(285,312)(286,314)(287,335)(288,337)(289,336)(290,338)
(291,331)(292,333)(293,332)(294,334)(295,327)(296,329)(297,328)(298,330)
(299,323)(300,325)(301,324)(302,326)(303,319)(304,321)(305,320)(306,322)
(307,315)(308,317)(309,316)(310,318);;
s3 := ( 3,147)( 4,150)( 5,149)( 6,148)( 7,143)( 8,146)( 9,145)( 10,144)
( 11,167)( 12,170)( 13,169)( 14,168)( 15,163)( 16,166)( 17,165)( 18,164)
( 19,159)( 20,162)( 21,161)( 22,160)( 23,155)( 24,158)( 25,157)( 26,156)
( 27,151)( 28,154)( 29,153)( 30,152)( 31,119)( 32,122)( 33,121)( 34,120)
( 35,115)( 36,118)( 37,117)( 38,116)( 39,139)( 40,142)( 41,141)( 42,140)
( 43,135)( 44,138)( 45,137)( 46,136)( 47,131)( 48,134)( 49,133)( 50,132)
( 51,127)( 52,130)( 53,129)( 54,128)( 55,123)( 56,126)( 57,125)( 58,124)
( 59, 91)( 60, 94)( 61, 93)( 62, 92)( 63, 87)( 64, 90)( 65, 89)( 66, 88)
( 67,111)( 68,114)( 69,113)( 70,112)( 71,107)( 72,110)( 73,109)( 74,108)
( 75,103)( 76,106)( 77,105)( 78,104)( 79, 99)( 80,102)( 81,101)( 82,100)
( 83, 95)( 84, 98)( 85, 97)( 86, 96)(171,315)(172,318)(173,317)(174,316)
(175,311)(176,314)(177,313)(178,312)(179,335)(180,338)(181,337)(182,336)
(183,331)(184,334)(185,333)(186,332)(187,327)(188,330)(189,329)(190,328)
(191,323)(192,326)(193,325)(194,324)(195,319)(196,322)(197,321)(198,320)
(199,287)(200,290)(201,289)(202,288)(203,283)(204,286)(205,285)(206,284)
(207,307)(208,310)(209,309)(210,308)(211,303)(212,306)(213,305)(214,304)
(215,299)(216,302)(217,301)(218,300)(219,295)(220,298)(221,297)(222,296)
(223,291)(224,294)(225,293)(226,292)(227,259)(228,262)(229,261)(230,260)
(231,255)(232,258)(233,257)(234,256)(235,279)(236,282)(237,281)(238,280)
(239,275)(240,278)(241,277)(242,276)(243,271)(244,274)(245,273)(246,272)
(247,267)(248,270)(249,269)(250,268)(251,263)(252,266)(253,265)(254,264);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(338)!(1,2);
s1 := Sym(338)!( 3,173)( 4,174)( 5,171)( 6,172)( 7,177)( 8,178)( 9,175)
( 10,176)( 11,181)( 12,182)( 13,179)( 14,180)( 15,185)( 16,186)( 17,183)
( 18,184)( 19,189)( 20,190)( 21,187)( 22,188)( 23,193)( 24,194)( 25,191)
( 26,192)( 27,197)( 28,198)( 29,195)( 30,196)( 31,201)( 32,202)( 33,199)
( 34,200)( 35,205)( 36,206)( 37,203)( 38,204)( 39,209)( 40,210)( 41,207)
( 42,208)( 43,213)( 44,214)( 45,211)( 46,212)( 47,217)( 48,218)( 49,215)
( 50,216)( 51,221)( 52,222)( 53,219)( 54,220)( 55,225)( 56,226)( 57,223)
( 58,224)( 59,229)( 60,230)( 61,227)( 62,228)( 63,233)( 64,234)( 65,231)
( 66,232)( 67,237)( 68,238)( 69,235)( 70,236)( 71,241)( 72,242)( 73,239)
( 74,240)( 75,245)( 76,246)( 77,243)( 78,244)( 79,249)( 80,250)( 81,247)
( 82,248)( 83,253)( 84,254)( 85,251)( 86,252)( 87,257)( 88,258)( 89,255)
( 90,256)( 91,261)( 92,262)( 93,259)( 94,260)( 95,265)( 96,266)( 97,263)
( 98,264)( 99,269)(100,270)(101,267)(102,268)(103,273)(104,274)(105,271)
(106,272)(107,277)(108,278)(109,275)(110,276)(111,281)(112,282)(113,279)
(114,280)(115,285)(116,286)(117,283)(118,284)(119,289)(120,290)(121,287)
(122,288)(123,293)(124,294)(125,291)(126,292)(127,297)(128,298)(129,295)
(130,296)(131,301)(132,302)(133,299)(134,300)(135,305)(136,306)(137,303)
(138,304)(139,309)(140,310)(141,307)(142,308)(143,313)(144,314)(145,311)
(146,312)(147,317)(148,318)(149,315)(150,316)(151,321)(152,322)(153,319)
(154,320)(155,325)(156,326)(157,323)(158,324)(159,329)(160,330)(161,327)
(162,328)(163,333)(164,334)(165,331)(166,332)(167,337)(168,338)(169,335)
(170,336);
s2 := 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)(172,173)(175,195)(176,197)(177,196)(178,198)
(179,191)(180,193)(181,192)(182,194)(183,187)(184,189)(185,188)(186,190)
(199,227)(200,229)(201,228)(202,230)(203,251)(204,253)(205,252)(206,254)
(207,247)(208,249)(209,248)(210,250)(211,243)(212,245)(213,244)(214,246)
(215,239)(216,241)(217,240)(218,242)(219,235)(220,237)(221,236)(222,238)
(223,231)(224,233)(225,232)(226,234)(256,257)(259,279)(260,281)(261,280)
(262,282)(263,275)(264,277)(265,276)(266,278)(267,271)(268,273)(269,272)
(270,274)(283,311)(284,313)(285,312)(286,314)(287,335)(288,337)(289,336)
(290,338)(291,331)(292,333)(293,332)(294,334)(295,327)(296,329)(297,328)
(298,330)(299,323)(300,325)(301,324)(302,326)(303,319)(304,321)(305,320)
(306,322)(307,315)(308,317)(309,316)(310,318);
s3 := Sym(338)!( 3,147)( 4,150)( 5,149)( 6,148)( 7,143)( 8,146)( 9,145)
( 10,144)( 11,167)( 12,170)( 13,169)( 14,168)( 15,163)( 16,166)( 17,165)
( 18,164)( 19,159)( 20,162)( 21,161)( 22,160)( 23,155)( 24,158)( 25,157)
( 26,156)( 27,151)( 28,154)( 29,153)( 30,152)( 31,119)( 32,122)( 33,121)
( 34,120)( 35,115)( 36,118)( 37,117)( 38,116)( 39,139)( 40,142)( 41,141)
( 42,140)( 43,135)( 44,138)( 45,137)( 46,136)( 47,131)( 48,134)( 49,133)
( 50,132)( 51,127)( 52,130)( 53,129)( 54,128)( 55,123)( 56,126)( 57,125)
( 58,124)( 59, 91)( 60, 94)( 61, 93)( 62, 92)( 63, 87)( 64, 90)( 65, 89)
( 66, 88)( 67,111)( 68,114)( 69,113)( 70,112)( 71,107)( 72,110)( 73,109)
( 74,108)( 75,103)( 76,106)( 77,105)( 78,104)( 79, 99)( 80,102)( 81,101)
( 82,100)( 83, 95)( 84, 98)( 85, 97)( 86, 96)(171,315)(172,318)(173,317)
(174,316)(175,311)(176,314)(177,313)(178,312)(179,335)(180,338)(181,337)
(182,336)(183,331)(184,334)(185,333)(186,332)(187,327)(188,330)(189,329)
(190,328)(191,323)(192,326)(193,325)(194,324)(195,319)(196,322)(197,321)
(198,320)(199,287)(200,290)(201,289)(202,288)(203,283)(204,286)(205,285)
(206,284)(207,307)(208,310)(209,309)(210,308)(211,303)(212,306)(213,305)
(214,304)(215,299)(216,302)(217,301)(218,300)(219,295)(220,298)(221,297)
(222,296)(223,291)(224,294)(225,293)(226,292)(227,259)(228,262)(229,261)
(230,260)(231,255)(232,258)(233,257)(234,256)(235,279)(236,282)(237,281)
(238,280)(239,275)(240,278)(241,277)(242,276)(243,271)(244,274)(245,273)
(246,272)(247,267)(248,270)(249,269)(250,268)(251,263)(252,266)(253,265)
(254,264);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope