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