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Polytope of Type {14,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,4}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 4
Schlafli Type : {14,6,4}
Number of vertices, edges, etc : 14, 84, 24, 8
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {14,6,4}*672b
   4-fold quotients : {14,6,2}*336
   7-fold quotients : {2,6,4}*192
   12-fold quotients : {14,2,2}*112
   14-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
   24-fold quotients : {7,2,2}*56
   28-fold quotients : {2,3,4}*48, {2,6,2}*48
   56-fold quotients : {2,3,2}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  5, 25)(  6, 26)(  7, 27)(  8, 28)(  9, 21)( 10, 22)( 11, 23)( 12, 24)
( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 33, 53)( 34, 54)( 35, 55)( 36, 56)
( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 77)( 66, 78)( 67, 79)( 68, 80)
( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 89,109)( 90,110)( 91,111)( 92,112)
( 93,105)( 94,106)( 95,107)( 96,108)( 97,101)( 98,102)( 99,103)(100,104)
(117,137)(118,138)(119,139)(120,140)(121,133)(122,134)(123,135)(124,136)
(125,129)(126,130)(127,131)(128,132)(145,165)(146,166)(147,167)(148,168)
(149,161)(150,162)(151,163)(152,164)(153,157)(154,158)(155,159)(156,160)
(173,193)(174,194)(175,195)(176,196)(177,189)(178,190)(179,191)(180,192)
(181,185)(182,186)(183,187)(184,188)(201,221)(202,222)(203,223)(204,224)
(205,217)(206,218)(207,219)(208,220)(209,213)(210,214)(211,215)(212,216)
(229,249)(230,250)(231,251)(232,252)(233,245)(234,246)(235,247)(236,248)
(237,241)(238,242)(239,243)(240,244)(257,277)(258,278)(259,279)(260,280)
(261,273)(262,274)(263,275)(264,276)(265,269)(266,270)(267,271)(268,272)
(285,305)(286,306)(287,307)(288,308)(289,301)(290,302)(291,303)(292,304)
(293,297)(294,298)(295,299)(296,300)(313,333)(314,334)(315,335)(316,336)
(317,329)(318,330)(319,331)(320,332)(321,325)(322,326)(323,327)(324,328);;
s1 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 25)( 10, 26)( 11, 28)( 12, 27)
( 13, 21)( 14, 22)( 15, 24)( 16, 23)( 19, 20)( 29, 61)( 30, 62)( 31, 64)
( 32, 63)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 81)( 38, 82)( 39, 84)
( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 73)( 46, 74)( 47, 76)
( 48, 75)( 49, 69)( 50, 70)( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)
( 56, 67)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 93,109)( 94,110)( 95,112)
( 96,111)( 97,105)( 98,106)( 99,108)(100,107)(103,104)(113,145)(114,146)
(115,148)(116,147)(117,141)(118,142)(119,144)(120,143)(121,165)(122,166)
(123,168)(124,167)(125,161)(126,162)(127,164)(128,163)(129,157)(130,158)
(131,160)(132,159)(133,153)(134,154)(135,156)(136,155)(137,149)(138,150)
(139,152)(140,151)(169,173)(170,174)(171,176)(172,175)(177,193)(178,194)
(179,196)(180,195)(181,189)(182,190)(183,192)(184,191)(187,188)(197,229)
(198,230)(199,232)(200,231)(201,225)(202,226)(203,228)(204,227)(205,249)
(206,250)(207,252)(208,251)(209,245)(210,246)(211,248)(212,247)(213,241)
(214,242)(215,244)(216,243)(217,237)(218,238)(219,240)(220,239)(221,233)
(222,234)(223,236)(224,235)(253,257)(254,258)(255,260)(256,259)(261,277)
(262,278)(263,280)(264,279)(265,273)(266,274)(267,276)(268,275)(271,272)
(281,313)(282,314)(283,316)(284,315)(285,309)(286,310)(287,312)(288,311)
(289,333)(290,334)(291,336)(292,335)(293,329)(294,330)(295,332)(296,331)
(297,325)(298,326)(299,328)(300,327)(301,321)(302,322)(303,324)(304,323)
(305,317)(306,318)(307,320)(308,319);;
s2 := (  1,197)(  2,200)(  3,199)(  4,198)(  5,201)(  6,204)(  7,203)(  8,202)
(  9,205)( 10,208)( 11,207)( 12,206)( 13,209)( 14,212)( 15,211)( 16,210)
( 17,213)( 18,216)( 19,215)( 20,214)( 21,217)( 22,220)( 23,219)( 24,218)
( 25,221)( 26,224)( 27,223)( 28,222)( 29,169)( 30,172)( 31,171)( 32,170)
( 33,173)( 34,176)( 35,175)( 36,174)( 37,177)( 38,180)( 39,179)( 40,178)
( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)( 48,186)
( 49,189)( 50,192)( 51,191)( 52,190)( 53,193)( 54,196)( 55,195)( 56,194)
( 57,225)( 58,228)( 59,227)( 60,226)( 61,229)( 62,232)( 63,231)( 64,230)
( 65,233)( 66,236)( 67,235)( 68,234)( 69,237)( 70,240)( 71,239)( 72,238)
( 73,241)( 74,244)( 75,243)( 76,242)( 77,245)( 78,248)( 79,247)( 80,246)
( 81,249)( 82,252)( 83,251)( 84,250)( 85,281)( 86,284)( 87,283)( 88,282)
( 89,285)( 90,288)( 91,287)( 92,286)( 93,289)( 94,292)( 95,291)( 96,290)
( 97,293)( 98,296)( 99,295)(100,294)(101,297)(102,300)(103,299)(104,298)
(105,301)(106,304)(107,303)(108,302)(109,305)(110,308)(111,307)(112,306)
(113,253)(114,256)(115,255)(116,254)(117,257)(118,260)(119,259)(120,258)
(121,261)(122,264)(123,263)(124,262)(125,265)(126,268)(127,267)(128,266)
(129,269)(130,272)(131,271)(132,270)(133,273)(134,276)(135,275)(136,274)
(137,277)(138,280)(139,279)(140,278)(141,309)(142,312)(143,311)(144,310)
(145,313)(146,316)(147,315)(148,314)(149,317)(150,320)(151,319)(152,318)
(153,321)(154,324)(155,323)(156,322)(157,325)(158,328)(159,327)(160,326)
(161,329)(162,332)(163,331)(164,330)(165,333)(166,336)(167,335)(168,334);;
s3 := (  1, 86)(  2, 85)(  3, 88)(  4, 87)(  5, 90)(  6, 89)(  7, 92)(  8, 91)
(  9, 94)( 10, 93)( 11, 96)( 12, 95)( 13, 98)( 14, 97)( 15,100)( 16, 99)
( 17,102)( 18,101)( 19,104)( 20,103)( 21,106)( 22,105)( 23,108)( 24,107)
( 25,110)( 26,109)( 27,112)( 28,111)( 29,114)( 30,113)( 31,116)( 32,115)
( 33,118)( 34,117)( 35,120)( 36,119)( 37,122)( 38,121)( 39,124)( 40,123)
( 41,126)( 42,125)( 43,128)( 44,127)( 45,130)( 46,129)( 47,132)( 48,131)
( 49,134)( 50,133)( 51,136)( 52,135)( 53,138)( 54,137)( 55,140)( 56,139)
( 57,142)( 58,141)( 59,144)( 60,143)( 61,146)( 62,145)( 63,148)( 64,147)
( 65,150)( 66,149)( 67,152)( 68,151)( 69,154)( 70,153)( 71,156)( 72,155)
( 73,158)( 74,157)( 75,160)( 76,159)( 77,162)( 78,161)( 79,164)( 80,163)
( 81,166)( 82,165)( 83,168)( 84,167)(169,254)(170,253)(171,256)(172,255)
(173,258)(174,257)(175,260)(176,259)(177,262)(178,261)(179,264)(180,263)
(181,266)(182,265)(183,268)(184,267)(185,270)(186,269)(187,272)(188,271)
(189,274)(190,273)(191,276)(192,275)(193,278)(194,277)(195,280)(196,279)
(197,282)(198,281)(199,284)(200,283)(201,286)(202,285)(203,288)(204,287)
(205,290)(206,289)(207,292)(208,291)(209,294)(210,293)(211,296)(212,295)
(213,298)(214,297)(215,300)(216,299)(217,302)(218,301)(219,304)(220,303)
(221,306)(222,305)(223,308)(224,307)(225,310)(226,309)(227,312)(228,311)
(229,314)(230,313)(231,316)(232,315)(233,318)(234,317)(235,320)(236,319)
(237,322)(238,321)(239,324)(240,323)(241,326)(242,325)(243,328)(244,327)
(245,330)(246,329)(247,332)(248,331)(249,334)(250,333)(251,336)(252,335);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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)!(  5, 25)(  6, 26)(  7, 27)(  8, 28)(  9, 21)( 10, 22)( 11, 23)
( 12, 24)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 33, 53)( 34, 54)( 35, 55)
( 36, 56)( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 77)( 66, 78)( 67, 79)
( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 89,109)( 90,110)( 91,111)
( 92,112)( 93,105)( 94,106)( 95,107)( 96,108)( 97,101)( 98,102)( 99,103)
(100,104)(117,137)(118,138)(119,139)(120,140)(121,133)(122,134)(123,135)
(124,136)(125,129)(126,130)(127,131)(128,132)(145,165)(146,166)(147,167)
(148,168)(149,161)(150,162)(151,163)(152,164)(153,157)(154,158)(155,159)
(156,160)(173,193)(174,194)(175,195)(176,196)(177,189)(178,190)(179,191)
(180,192)(181,185)(182,186)(183,187)(184,188)(201,221)(202,222)(203,223)
(204,224)(205,217)(206,218)(207,219)(208,220)(209,213)(210,214)(211,215)
(212,216)(229,249)(230,250)(231,251)(232,252)(233,245)(234,246)(235,247)
(236,248)(237,241)(238,242)(239,243)(240,244)(257,277)(258,278)(259,279)
(260,280)(261,273)(262,274)(263,275)(264,276)(265,269)(266,270)(267,271)
(268,272)(285,305)(286,306)(287,307)(288,308)(289,301)(290,302)(291,303)
(292,304)(293,297)(294,298)(295,299)(296,300)(313,333)(314,334)(315,335)
(316,336)(317,329)(318,330)(319,331)(320,332)(321,325)(322,326)(323,327)
(324,328);
s1 := Sym(336)!(  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 25)( 10, 26)( 11, 28)
( 12, 27)( 13, 21)( 14, 22)( 15, 24)( 16, 23)( 19, 20)( 29, 61)( 30, 62)
( 31, 64)( 32, 63)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 81)( 38, 82)
( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 73)( 46, 74)
( 47, 76)( 48, 75)( 49, 69)( 50, 70)( 51, 72)( 52, 71)( 53, 65)( 54, 66)
( 55, 68)( 56, 67)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 93,109)( 94,110)
( 95,112)( 96,111)( 97,105)( 98,106)( 99,108)(100,107)(103,104)(113,145)
(114,146)(115,148)(116,147)(117,141)(118,142)(119,144)(120,143)(121,165)
(122,166)(123,168)(124,167)(125,161)(126,162)(127,164)(128,163)(129,157)
(130,158)(131,160)(132,159)(133,153)(134,154)(135,156)(136,155)(137,149)
(138,150)(139,152)(140,151)(169,173)(170,174)(171,176)(172,175)(177,193)
(178,194)(179,196)(180,195)(181,189)(182,190)(183,192)(184,191)(187,188)
(197,229)(198,230)(199,232)(200,231)(201,225)(202,226)(203,228)(204,227)
(205,249)(206,250)(207,252)(208,251)(209,245)(210,246)(211,248)(212,247)
(213,241)(214,242)(215,244)(216,243)(217,237)(218,238)(219,240)(220,239)
(221,233)(222,234)(223,236)(224,235)(253,257)(254,258)(255,260)(256,259)
(261,277)(262,278)(263,280)(264,279)(265,273)(266,274)(267,276)(268,275)
(271,272)(281,313)(282,314)(283,316)(284,315)(285,309)(286,310)(287,312)
(288,311)(289,333)(290,334)(291,336)(292,335)(293,329)(294,330)(295,332)
(296,331)(297,325)(298,326)(299,328)(300,327)(301,321)(302,322)(303,324)
(304,323)(305,317)(306,318)(307,320)(308,319);
s2 := Sym(336)!(  1,197)(  2,200)(  3,199)(  4,198)(  5,201)(  6,204)(  7,203)
(  8,202)(  9,205)( 10,208)( 11,207)( 12,206)( 13,209)( 14,212)( 15,211)
( 16,210)( 17,213)( 18,216)( 19,215)( 20,214)( 21,217)( 22,220)( 23,219)
( 24,218)( 25,221)( 26,224)( 27,223)( 28,222)( 29,169)( 30,172)( 31,171)
( 32,170)( 33,173)( 34,176)( 35,175)( 36,174)( 37,177)( 38,180)( 39,179)
( 40,178)( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)
( 48,186)( 49,189)( 50,192)( 51,191)( 52,190)( 53,193)( 54,196)( 55,195)
( 56,194)( 57,225)( 58,228)( 59,227)( 60,226)( 61,229)( 62,232)( 63,231)
( 64,230)( 65,233)( 66,236)( 67,235)( 68,234)( 69,237)( 70,240)( 71,239)
( 72,238)( 73,241)( 74,244)( 75,243)( 76,242)( 77,245)( 78,248)( 79,247)
( 80,246)( 81,249)( 82,252)( 83,251)( 84,250)( 85,281)( 86,284)( 87,283)
( 88,282)( 89,285)( 90,288)( 91,287)( 92,286)( 93,289)( 94,292)( 95,291)
( 96,290)( 97,293)( 98,296)( 99,295)(100,294)(101,297)(102,300)(103,299)
(104,298)(105,301)(106,304)(107,303)(108,302)(109,305)(110,308)(111,307)
(112,306)(113,253)(114,256)(115,255)(116,254)(117,257)(118,260)(119,259)
(120,258)(121,261)(122,264)(123,263)(124,262)(125,265)(126,268)(127,267)
(128,266)(129,269)(130,272)(131,271)(132,270)(133,273)(134,276)(135,275)
(136,274)(137,277)(138,280)(139,279)(140,278)(141,309)(142,312)(143,311)
(144,310)(145,313)(146,316)(147,315)(148,314)(149,317)(150,320)(151,319)
(152,318)(153,321)(154,324)(155,323)(156,322)(157,325)(158,328)(159,327)
(160,326)(161,329)(162,332)(163,331)(164,330)(165,333)(166,336)(167,335)
(168,334);
s3 := Sym(336)!(  1, 86)(  2, 85)(  3, 88)(  4, 87)(  5, 90)(  6, 89)(  7, 92)
(  8, 91)(  9, 94)( 10, 93)( 11, 96)( 12, 95)( 13, 98)( 14, 97)( 15,100)
( 16, 99)( 17,102)( 18,101)( 19,104)( 20,103)( 21,106)( 22,105)( 23,108)
( 24,107)( 25,110)( 26,109)( 27,112)( 28,111)( 29,114)( 30,113)( 31,116)
( 32,115)( 33,118)( 34,117)( 35,120)( 36,119)( 37,122)( 38,121)( 39,124)
( 40,123)( 41,126)( 42,125)( 43,128)( 44,127)( 45,130)( 46,129)( 47,132)
( 48,131)( 49,134)( 50,133)( 51,136)( 52,135)( 53,138)( 54,137)( 55,140)
( 56,139)( 57,142)( 58,141)( 59,144)( 60,143)( 61,146)( 62,145)( 63,148)
( 64,147)( 65,150)( 66,149)( 67,152)( 68,151)( 69,154)( 70,153)( 71,156)
( 72,155)( 73,158)( 74,157)( 75,160)( 76,159)( 77,162)( 78,161)( 79,164)
( 80,163)( 81,166)( 82,165)( 83,168)( 84,167)(169,254)(170,253)(171,256)
(172,255)(173,258)(174,257)(175,260)(176,259)(177,262)(178,261)(179,264)
(180,263)(181,266)(182,265)(183,268)(184,267)(185,270)(186,269)(187,272)
(188,271)(189,274)(190,273)(191,276)(192,275)(193,278)(194,277)(195,280)
(196,279)(197,282)(198,281)(199,284)(200,283)(201,286)(202,285)(203,288)
(204,287)(205,290)(206,289)(207,292)(208,291)(209,294)(210,293)(211,296)
(212,295)(213,298)(214,297)(215,300)(216,299)(217,302)(218,301)(219,304)
(220,303)(221,306)(222,305)(223,308)(224,307)(225,310)(226,309)(227,312)
(228,311)(229,314)(230,313)(231,316)(232,315)(233,318)(234,317)(235,320)
(236,319)(237,322)(238,321)(239,324)(240,323)(241,326)(242,325)(243,328)
(244,327)(245,330)(246,329)(247,332)(248,331)(249,334)(250,333)(251,336)
(252,335);
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*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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