Polytope of Type {14,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,4,6}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 4
Schlafli Type : {14,4,6}
Number of vertices, edges, etc : 14, 56, 24, 12
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 : {14,4,3}*672
   4-fold quotients : {14,2,6}*336
   7-fold quotients : {2,4,6}*192
   8-fold quotients : {7,2,6}*168, {14,2,3}*168
   12-fold quotients : {14,2,2}*112
   14-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   16-fold quotients : {7,2,3}*84
   24-fold quotients : {7,2,2}*56
   28-fold quotients : {2,4,3}*48, {2,2,6}*48
   56-fold quotients : {2,2,3}*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 := (  1,175)(  2,176)(  3,173)(  4,174)(  5,171)(  6,172)(  7,169)(  8,170)
(  9,195)( 10,196)( 11,193)( 12,194)( 13,191)( 14,192)( 15,189)( 16,190)
( 17,187)( 18,188)( 19,185)( 20,186)( 21,183)( 22,184)( 23,181)( 24,182)
( 25,179)( 26,180)( 27,177)( 28,178)( 29,203)( 30,204)( 31,201)( 32,202)
( 33,199)( 34,200)( 35,197)( 36,198)( 37,223)( 38,224)( 39,221)( 40,222)
( 41,219)( 42,220)( 43,217)( 44,218)( 45,215)( 46,216)( 47,213)( 48,214)
( 49,211)( 50,212)( 51,209)( 52,210)( 53,207)( 54,208)( 55,205)( 56,206)
( 57,231)( 58,232)( 59,229)( 60,230)( 61,227)( 62,228)( 63,225)( 64,226)
( 65,251)( 66,252)( 67,249)( 68,250)( 69,247)( 70,248)( 71,245)( 72,246)
( 73,243)( 74,244)( 75,241)( 76,242)( 77,239)( 78,240)( 79,237)( 80,238)
( 81,235)( 82,236)( 83,233)( 84,234)( 85,259)( 86,260)( 87,257)( 88,258)
( 89,255)( 90,256)( 91,253)( 92,254)( 93,279)( 94,280)( 95,277)( 96,278)
( 97,275)( 98,276)( 99,273)(100,274)(101,271)(102,272)(103,269)(104,270)
(105,267)(106,268)(107,265)(108,266)(109,263)(110,264)(111,261)(112,262)
(113,287)(114,288)(115,285)(116,286)(117,283)(118,284)(119,281)(120,282)
(121,307)(122,308)(123,305)(124,306)(125,303)(126,304)(127,301)(128,302)
(129,299)(130,300)(131,297)(132,298)(133,295)(134,296)(135,293)(136,294)
(137,291)(138,292)(139,289)(140,290)(141,315)(142,316)(143,313)(144,314)
(145,311)(146,312)(147,309)(148,310)(149,335)(150,336)(151,333)(152,334)
(153,331)(154,332)(155,329)(156,330)(157,327)(158,328)(159,325)(160,326)
(161,323)(162,324)(163,321)(164,322)(165,319)(166,320)(167,317)(168,318);;
s2 := (  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);;
s3 := (  1,113)(  2,116)(  3,115)(  4,114)(  5,117)(  6,120)(  7,119)(  8,118)
(  9,121)( 10,124)( 11,123)( 12,122)( 13,125)( 14,128)( 15,127)( 16,126)
( 17,129)( 18,132)( 19,131)( 20,130)( 21,133)( 22,136)( 23,135)( 24,134)
( 25,137)( 26,140)( 27,139)( 28,138)( 29, 85)( 30, 88)( 31, 87)( 32, 86)
( 33, 89)( 34, 92)( 35, 91)( 36, 90)( 37, 93)( 38, 96)( 39, 95)( 40, 94)
( 41, 97)( 42,100)( 43, 99)( 44, 98)( 45,101)( 46,104)( 47,103)( 48,102)
( 49,105)( 50,108)( 51,107)( 52,106)( 53,109)( 54,112)( 55,111)( 56,110)
( 57,141)( 58,144)( 59,143)( 60,142)( 61,145)( 62,148)( 63,147)( 64,146)
( 65,149)( 66,152)( 67,151)( 68,150)( 69,153)( 70,156)( 71,155)( 72,154)
( 73,157)( 74,160)( 75,159)( 76,158)( 77,161)( 78,164)( 79,163)( 80,162)
( 81,165)( 82,168)( 83,167)( 84,166)(169,281)(170,284)(171,283)(172,282)
(173,285)(174,288)(175,287)(176,286)(177,289)(178,292)(179,291)(180,290)
(181,293)(182,296)(183,295)(184,294)(185,297)(186,300)(187,299)(188,298)
(189,301)(190,304)(191,303)(192,302)(193,305)(194,308)(195,307)(196,306)
(197,253)(198,256)(199,255)(200,254)(201,257)(202,260)(203,259)(204,258)
(205,261)(206,264)(207,263)(208,262)(209,265)(210,268)(211,267)(212,266)
(213,269)(214,272)(215,271)(216,270)(217,273)(218,276)(219,275)(220,274)
(221,277)(222,280)(223,279)(224,278)(225,309)(226,312)(227,311)(228,310)
(229,313)(230,316)(231,315)(232,314)(233,317)(234,320)(235,319)(236,318)
(237,321)(238,324)(239,323)(240,322)(241,325)(242,328)(243,327)(244,326)
(245,329)(246,332)(247,331)(248,330)(249,333)(250,336)(251,335)(252,334);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)!(  1,175)(  2,176)(  3,173)(  4,174)(  5,171)(  6,172)(  7,169)
(  8,170)(  9,195)( 10,196)( 11,193)( 12,194)( 13,191)( 14,192)( 15,189)
( 16,190)( 17,187)( 18,188)( 19,185)( 20,186)( 21,183)( 22,184)( 23,181)
( 24,182)( 25,179)( 26,180)( 27,177)( 28,178)( 29,203)( 30,204)( 31,201)
( 32,202)( 33,199)( 34,200)( 35,197)( 36,198)( 37,223)( 38,224)( 39,221)
( 40,222)( 41,219)( 42,220)( 43,217)( 44,218)( 45,215)( 46,216)( 47,213)
( 48,214)( 49,211)( 50,212)( 51,209)( 52,210)( 53,207)( 54,208)( 55,205)
( 56,206)( 57,231)( 58,232)( 59,229)( 60,230)( 61,227)( 62,228)( 63,225)
( 64,226)( 65,251)( 66,252)( 67,249)( 68,250)( 69,247)( 70,248)( 71,245)
( 72,246)( 73,243)( 74,244)( 75,241)( 76,242)( 77,239)( 78,240)( 79,237)
( 80,238)( 81,235)( 82,236)( 83,233)( 84,234)( 85,259)( 86,260)( 87,257)
( 88,258)( 89,255)( 90,256)( 91,253)( 92,254)( 93,279)( 94,280)( 95,277)
( 96,278)( 97,275)( 98,276)( 99,273)(100,274)(101,271)(102,272)(103,269)
(104,270)(105,267)(106,268)(107,265)(108,266)(109,263)(110,264)(111,261)
(112,262)(113,287)(114,288)(115,285)(116,286)(117,283)(118,284)(119,281)
(120,282)(121,307)(122,308)(123,305)(124,306)(125,303)(126,304)(127,301)
(128,302)(129,299)(130,300)(131,297)(132,298)(133,295)(134,296)(135,293)
(136,294)(137,291)(138,292)(139,289)(140,290)(141,315)(142,316)(143,313)
(144,314)(145,311)(146,312)(147,309)(148,310)(149,335)(150,336)(151,333)
(152,334)(153,331)(154,332)(155,329)(156,330)(157,327)(158,328)(159,325)
(160,326)(161,323)(162,324)(163,321)(164,322)(165,319)(166,320)(167,317)
(168,318);
s2 := 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);
s3 := Sym(336)!(  1,113)(  2,116)(  3,115)(  4,114)(  5,117)(  6,120)(  7,119)
(  8,118)(  9,121)( 10,124)( 11,123)( 12,122)( 13,125)( 14,128)( 15,127)
( 16,126)( 17,129)( 18,132)( 19,131)( 20,130)( 21,133)( 22,136)( 23,135)
( 24,134)( 25,137)( 26,140)( 27,139)( 28,138)( 29, 85)( 30, 88)( 31, 87)
( 32, 86)( 33, 89)( 34, 92)( 35, 91)( 36, 90)( 37, 93)( 38, 96)( 39, 95)
( 40, 94)( 41, 97)( 42,100)( 43, 99)( 44, 98)( 45,101)( 46,104)( 47,103)
( 48,102)( 49,105)( 50,108)( 51,107)( 52,106)( 53,109)( 54,112)( 55,111)
( 56,110)( 57,141)( 58,144)( 59,143)( 60,142)( 61,145)( 62,148)( 63,147)
( 64,146)( 65,149)( 66,152)( 67,151)( 68,150)( 69,153)( 70,156)( 71,155)
( 72,154)( 73,157)( 74,160)( 75,159)( 76,158)( 77,161)( 78,164)( 79,163)
( 80,162)( 81,165)( 82,168)( 83,167)( 84,166)(169,281)(170,284)(171,283)
(172,282)(173,285)(174,288)(175,287)(176,286)(177,289)(178,292)(179,291)
(180,290)(181,293)(182,296)(183,295)(184,294)(185,297)(186,300)(187,299)
(188,298)(189,301)(190,304)(191,303)(192,302)(193,305)(194,308)(195,307)
(196,306)(197,253)(198,256)(199,255)(200,254)(201,257)(202,260)(203,259)
(204,258)(205,261)(206,264)(207,263)(208,262)(209,265)(210,268)(211,267)
(212,266)(213,269)(214,272)(215,271)(216,270)(217,273)(218,276)(219,275)
(220,274)(221,277)(222,280)(223,279)(224,278)(225,309)(226,312)(227,311)
(228,310)(229,313)(230,316)(231,315)(232,314)(233,317)(234,320)(235,319)
(236,318)(237,321)(238,324)(239,323)(240,322)(241,325)(242,328)(243,327)
(244,326)(245,329)(246,332)(247,331)(248,330)(249,333)(250,336)(251,335)
(252,334);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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