Polytope of Type {14,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,6}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 4
Schlafli Type : {14,6,6}
Number of vertices, edges, etc : 14, 56, 24, 8
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 : {14,6,3}*672
   7-fold quotients : {2,6,6}*192
   12-fold quotients : {14,2,2}*112
   14-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {7,2,2}*56
   28-fold quotients : {2,3,3}*48
   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, 87)(  4, 88)(  5, 90)(  6, 89)(  7, 91)(  8, 92)
(  9, 94)( 10, 93)( 11, 95)( 12, 96)( 13, 98)( 14, 97)( 15, 99)( 16,100)
( 17,102)( 18,101)( 19,103)( 20,104)( 21,106)( 22,105)( 23,107)( 24,108)
( 25,110)( 26,109)( 27,111)( 28,112)( 29,142)( 30,141)( 31,143)( 32,144)
( 33,146)( 34,145)( 35,147)( 36,148)( 37,150)( 38,149)( 39,151)( 40,152)
( 41,154)( 42,153)( 43,155)( 44,156)( 45,158)( 46,157)( 47,159)( 48,160)
( 49,162)( 50,161)( 51,163)( 52,164)( 53,166)( 54,165)( 55,167)( 56,168)
( 57,114)( 58,113)( 59,115)( 60,116)( 61,118)( 62,117)( 63,119)( 64,120)
( 65,122)( 66,121)( 67,123)( 68,124)( 69,126)( 70,125)( 71,127)( 72,128)
( 73,130)( 74,129)( 75,131)( 76,132)( 77,134)( 78,133)( 79,135)( 80,136)
( 81,138)( 82,137)( 83,139)( 84,140)(169,254)(170,253)(171,255)(172,256)
(173,258)(174,257)(175,259)(176,260)(177,262)(178,261)(179,263)(180,264)
(181,266)(182,265)(183,267)(184,268)(185,270)(186,269)(187,271)(188,272)
(189,274)(190,273)(191,275)(192,276)(193,278)(194,277)(195,279)(196,280)
(197,310)(198,309)(199,311)(200,312)(201,314)(202,313)(203,315)(204,316)
(205,318)(206,317)(207,319)(208,320)(209,322)(210,321)(211,323)(212,324)
(213,326)(214,325)(215,327)(216,328)(217,330)(218,329)(219,331)(220,332)
(221,334)(222,333)(223,335)(224,336)(225,282)(226,281)(227,283)(228,284)
(229,286)(230,285)(231,287)(232,288)(233,290)(234,289)(235,291)(236,292)
(237,294)(238,293)(239,295)(240,296)(241,298)(242,297)(243,299)(244,300)
(245,302)(246,301)(247,303)(248,304)(249,306)(250,305)(251,307)(252,308);;
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, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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, 87)(  4, 88)(  5, 90)(  6, 89)(  7, 91)
(  8, 92)(  9, 94)( 10, 93)( 11, 95)( 12, 96)( 13, 98)( 14, 97)( 15, 99)
( 16,100)( 17,102)( 18,101)( 19,103)( 20,104)( 21,106)( 22,105)( 23,107)
( 24,108)( 25,110)( 26,109)( 27,111)( 28,112)( 29,142)( 30,141)( 31,143)
( 32,144)( 33,146)( 34,145)( 35,147)( 36,148)( 37,150)( 38,149)( 39,151)
( 40,152)( 41,154)( 42,153)( 43,155)( 44,156)( 45,158)( 46,157)( 47,159)
( 48,160)( 49,162)( 50,161)( 51,163)( 52,164)( 53,166)( 54,165)( 55,167)
( 56,168)( 57,114)( 58,113)( 59,115)( 60,116)( 61,118)( 62,117)( 63,119)
( 64,120)( 65,122)( 66,121)( 67,123)( 68,124)( 69,126)( 70,125)( 71,127)
( 72,128)( 73,130)( 74,129)( 75,131)( 76,132)( 77,134)( 78,133)( 79,135)
( 80,136)( 81,138)( 82,137)( 83,139)( 84,140)(169,254)(170,253)(171,255)
(172,256)(173,258)(174,257)(175,259)(176,260)(177,262)(178,261)(179,263)
(180,264)(181,266)(182,265)(183,267)(184,268)(185,270)(186,269)(187,271)
(188,272)(189,274)(190,273)(191,275)(192,276)(193,278)(194,277)(195,279)
(196,280)(197,310)(198,309)(199,311)(200,312)(201,314)(202,313)(203,315)
(204,316)(205,318)(206,317)(207,319)(208,320)(209,322)(210,321)(211,323)
(212,324)(213,326)(214,325)(215,327)(216,328)(217,330)(218,329)(219,331)
(220,332)(221,334)(222,333)(223,335)(224,336)(225,282)(226,281)(227,283)
(228,284)(229,286)(230,285)(231,287)(232,288)(233,290)(234,289)(235,291)
(236,292)(237,294)(238,293)(239,295)(240,296)(241,298)(242,297)(243,299)
(244,300)(245,302)(246,301)(247,303)(248,304)(249,306)(250,305)(251,307)
(252,308);
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, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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