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Polytope of Type {4,6,14}

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