Polytope of Type {4,12,14}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,14}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11327)
Rank : 4
Schlafli Type : {4,12,14}
Number of vertices, edges, etc : 4, 24, 84, 14
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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
7-fold quotients : {4,12,2}*192b
14-fold quotients : {4,6,2}*96c
28-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336);;
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)(169,253)(170,254)
(171,256)(172,255)(173,257)(174,258)(175,260)(176,259)(177,261)(178,262)
(179,264)(180,263)(181,265)(182,266)(183,268)(184,267)(185,269)(186,270)
(187,272)(188,271)(189,273)(190,274)(191,276)(192,275)(193,277)(194,278)
(195,280)(196,279)(197,309)(198,310)(199,312)(200,311)(201,313)(202,314)
(203,316)(204,315)(205,317)(206,318)(207,320)(208,319)(209,321)(210,322)
(211,324)(212,323)(213,325)(214,326)(215,328)(216,327)(217,329)(218,330)
(219,332)(220,331)(221,333)(222,334)(223,336)(224,335)(225,281)(226,282)
(227,284)(228,283)(229,285)(230,286)(231,288)(232,287)(233,289)(234,290)
(235,292)(236,291)(237,293)(238,294)(239,296)(240,295)(241,297)(242,298)
(243,300)(244,299)(245,301)(246,302)(247,304)(248,303)(249,305)(250,306)
(251,308)(252,307);;
s2 := ( 1,197)( 2,200)( 3,199)( 4,198)( 5,221)( 6,224)( 7,223)( 8,222)
( 9,217)( 10,220)( 11,219)( 12,218)( 13,213)( 14,216)( 15,215)( 16,214)
( 17,209)( 18,212)( 19,211)( 20,210)( 21,205)( 22,208)( 23,207)( 24,206)
( 25,201)( 26,204)( 27,203)( 28,202)( 29,169)( 30,172)( 31,171)( 32,170)
( 33,193)( 34,196)( 35,195)( 36,194)( 37,189)( 38,192)( 39,191)( 40,190)
( 41,185)( 42,188)( 43,187)( 44,186)( 45,181)( 46,184)( 47,183)( 48,182)
( 49,177)( 50,180)( 51,179)( 52,178)( 53,173)( 54,176)( 55,175)( 56,174)
( 57,225)( 58,228)( 59,227)( 60,226)( 61,249)( 62,252)( 63,251)( 64,250)
( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)( 72,242)
( 73,237)( 74,240)( 75,239)( 76,238)( 77,233)( 78,236)( 79,235)( 80,234)
( 81,229)( 82,232)( 83,231)( 84,230)( 85,281)( 86,284)( 87,283)( 88,282)
( 89,305)( 90,308)( 91,307)( 92,306)( 93,301)( 94,304)( 95,303)( 96,302)
( 97,297)( 98,300)( 99,299)(100,298)(101,293)(102,296)(103,295)(104,294)
(105,289)(106,292)(107,291)(108,290)(109,285)(110,288)(111,287)(112,286)
(113,253)(114,256)(115,255)(116,254)(117,277)(118,280)(119,279)(120,278)
(121,273)(122,276)(123,275)(124,274)(125,269)(126,272)(127,271)(128,270)
(129,265)(130,268)(131,267)(132,266)(133,261)(134,264)(135,263)(136,262)
(137,257)(138,260)(139,259)(140,258)(141,309)(142,312)(143,311)(144,310)
(145,333)(146,336)(147,335)(148,334)(149,329)(150,332)(151,331)(152,330)
(153,325)(154,328)(155,327)(156,326)(157,321)(158,324)(159,323)(160,322)
(161,317)(162,320)(163,319)(164,318)(165,313)(166,316)(167,315)(168,314);;
s3 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 25)( 10, 26)( 11, 27)( 12, 28)
( 13, 21)( 14, 22)( 15, 23)( 16, 24)( 29, 33)( 30, 34)( 31, 35)( 32, 36)
( 37, 53)( 38, 54)( 39, 55)( 40, 56)( 41, 49)( 42, 50)( 43, 51)( 44, 52)
( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 81)( 66, 82)( 67, 83)( 68, 84)
( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 85, 89)( 86, 90)( 87, 91)( 88, 92)
( 93,109)( 94,110)( 95,111)( 96,112)( 97,105)( 98,106)( 99,107)(100,108)
(113,117)(114,118)(115,119)(116,120)(121,137)(122,138)(123,139)(124,140)
(125,133)(126,134)(127,135)(128,136)(141,145)(142,146)(143,147)(144,148)
(149,165)(150,166)(151,167)(152,168)(153,161)(154,162)(155,163)(156,164)
(169,173)(170,174)(171,175)(172,176)(177,193)(178,194)(179,195)(180,196)
(181,189)(182,190)(183,191)(184,192)(197,201)(198,202)(199,203)(200,204)
(205,221)(206,222)(207,223)(208,224)(209,217)(210,218)(211,219)(212,220)
(225,229)(226,230)(227,231)(228,232)(233,249)(234,250)(235,251)(236,252)
(237,245)(238,246)(239,247)(240,248)(253,257)(254,258)(255,259)(256,260)
(261,277)(262,278)(263,279)(264,280)(265,273)(266,274)(267,275)(268,276)
(281,285)(282,286)(283,287)(284,288)(289,305)(290,306)(291,307)(292,308)
(293,301)(294,302)(295,303)(296,304)(309,313)(310,314)(311,315)(312,316)
(317,333)(318,334)(319,335)(320,336)(321,329)(322,330)(323,331)(324,332);;
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*s0*s1*s2*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336);
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)(169,253)
(170,254)(171,256)(172,255)(173,257)(174,258)(175,260)(176,259)(177,261)
(178,262)(179,264)(180,263)(181,265)(182,266)(183,268)(184,267)(185,269)
(186,270)(187,272)(188,271)(189,273)(190,274)(191,276)(192,275)(193,277)
(194,278)(195,280)(196,279)(197,309)(198,310)(199,312)(200,311)(201,313)
(202,314)(203,316)(204,315)(205,317)(206,318)(207,320)(208,319)(209,321)
(210,322)(211,324)(212,323)(213,325)(214,326)(215,328)(216,327)(217,329)
(218,330)(219,332)(220,331)(221,333)(222,334)(223,336)(224,335)(225,281)
(226,282)(227,284)(228,283)(229,285)(230,286)(231,288)(232,287)(233,289)
(234,290)(235,292)(236,291)(237,293)(238,294)(239,296)(240,295)(241,297)
(242,298)(243,300)(244,299)(245,301)(246,302)(247,304)(248,303)(249,305)
(250,306)(251,308)(252,307);
s2 := Sym(336)!( 1,197)( 2,200)( 3,199)( 4,198)( 5,221)( 6,224)( 7,223)
( 8,222)( 9,217)( 10,220)( 11,219)( 12,218)( 13,213)( 14,216)( 15,215)
( 16,214)( 17,209)( 18,212)( 19,211)( 20,210)( 21,205)( 22,208)( 23,207)
( 24,206)( 25,201)( 26,204)( 27,203)( 28,202)( 29,169)( 30,172)( 31,171)
( 32,170)( 33,193)( 34,196)( 35,195)( 36,194)( 37,189)( 38,192)( 39,191)
( 40,190)( 41,185)( 42,188)( 43,187)( 44,186)( 45,181)( 46,184)( 47,183)
( 48,182)( 49,177)( 50,180)( 51,179)( 52,178)( 53,173)( 54,176)( 55,175)
( 56,174)( 57,225)( 58,228)( 59,227)( 60,226)( 61,249)( 62,252)( 63,251)
( 64,250)( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)
( 72,242)( 73,237)( 74,240)( 75,239)( 76,238)( 77,233)( 78,236)( 79,235)
( 80,234)( 81,229)( 82,232)( 83,231)( 84,230)( 85,281)( 86,284)( 87,283)
( 88,282)( 89,305)( 90,308)( 91,307)( 92,306)( 93,301)( 94,304)( 95,303)
( 96,302)( 97,297)( 98,300)( 99,299)(100,298)(101,293)(102,296)(103,295)
(104,294)(105,289)(106,292)(107,291)(108,290)(109,285)(110,288)(111,287)
(112,286)(113,253)(114,256)(115,255)(116,254)(117,277)(118,280)(119,279)
(120,278)(121,273)(122,276)(123,275)(124,274)(125,269)(126,272)(127,271)
(128,270)(129,265)(130,268)(131,267)(132,266)(133,261)(134,264)(135,263)
(136,262)(137,257)(138,260)(139,259)(140,258)(141,309)(142,312)(143,311)
(144,310)(145,333)(146,336)(147,335)(148,334)(149,329)(150,332)(151,331)
(152,330)(153,325)(154,328)(155,327)(156,326)(157,321)(158,324)(159,323)
(160,322)(161,317)(162,320)(163,319)(164,318)(165,313)(166,316)(167,315)
(168,314);
s3 := Sym(336)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 25)( 10, 26)( 11, 27)
( 12, 28)( 13, 21)( 14, 22)( 15, 23)( 16, 24)( 29, 33)( 30, 34)( 31, 35)
( 32, 36)( 37, 53)( 38, 54)( 39, 55)( 40, 56)( 41, 49)( 42, 50)( 43, 51)
( 44, 52)( 57, 61)( 58, 62)( 59, 63)( 60, 64)( 65, 81)( 66, 82)( 67, 83)
( 68, 84)( 69, 77)( 70, 78)( 71, 79)( 72, 80)( 85, 89)( 86, 90)( 87, 91)
( 88, 92)( 93,109)( 94,110)( 95,111)( 96,112)( 97,105)( 98,106)( 99,107)
(100,108)(113,117)(114,118)(115,119)(116,120)(121,137)(122,138)(123,139)
(124,140)(125,133)(126,134)(127,135)(128,136)(141,145)(142,146)(143,147)
(144,148)(149,165)(150,166)(151,167)(152,168)(153,161)(154,162)(155,163)
(156,164)(169,173)(170,174)(171,175)(172,176)(177,193)(178,194)(179,195)
(180,196)(181,189)(182,190)(183,191)(184,192)(197,201)(198,202)(199,203)
(200,204)(205,221)(206,222)(207,223)(208,224)(209,217)(210,218)(211,219)
(212,220)(225,229)(226,230)(227,231)(228,232)(233,249)(234,250)(235,251)
(236,252)(237,245)(238,246)(239,247)(240,248)(253,257)(254,258)(255,259)
(256,260)(261,277)(262,278)(263,279)(264,280)(265,273)(266,274)(267,275)
(268,276)(281,285)(282,286)(283,287)(284,288)(289,305)(290,306)(291,307)
(292,308)(293,301)(294,302)(295,303)(296,304)(309,313)(310,314)(311,315)
(312,316)(317,333)(318,334)(319,335)(320,336)(321,329)(322,330)(323,331)
(324,332);
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*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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