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Polytope of Type {6,6,14}

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