Polytope of Type {3,4,28}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,28}*1344
Also Known As : {{3,4},{4,28|2}}. if this polytope has another name.
Group : SmallGroup(1344,11328)
Rank : 4
Schlafli Type : {3,4,28}
Number of vertices, edges, etc : 6, 12, 112, 28
Order of s0s1s2s3 : 84
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 : {3,4,14}*672
4-fold quotients : {3,2,28}*336
7-fold quotients : {3,4,4}*192b
8-fold quotients : {3,2,14}*168
14-fold quotients : {3,4,2}*96
16-fold quotients : {3,2,7}*84
28-fold quotients : {3,2,4}*48, {3,4,2}*48
56-fold quotients : {3,2,2}*24
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, 29)( 2, 32)( 3, 31)( 4, 30)( 5, 33)( 6, 36)( 7, 35)( 8, 34)
( 9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 41)( 14, 44)( 15, 43)( 16, 42)
( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 49)( 22, 52)( 23, 51)( 24, 50)
( 25, 53)( 26, 56)( 27, 55)( 28, 54)( 58, 60)( 62, 64)( 66, 68)( 70, 72)
( 74, 76)( 78, 80)( 82, 84)( 85,113)( 86,116)( 87,115)( 88,114)( 89,117)
( 90,120)( 91,119)( 92,118)( 93,121)( 94,124)( 95,123)( 96,122)( 97,125)
( 98,128)( 99,127)(100,126)(101,129)(102,132)(103,131)(104,130)(105,133)
(106,136)(107,135)(108,134)(109,137)(110,140)(111,139)(112,138)(142,144)
(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)(169,197)(170,200)
(171,199)(172,198)(173,201)(174,204)(175,203)(176,202)(177,205)(178,208)
(179,207)(180,206)(181,209)(182,212)(183,211)(184,210)(185,213)(186,216)
(187,215)(188,214)(189,217)(190,220)(191,219)(192,218)(193,221)(194,224)
(195,223)(196,222)(226,228)(230,232)(234,236)(238,240)(242,244)(246,248)
(250,252)(253,281)(254,284)(255,283)(256,282)(257,285)(258,288)(259,287)
(260,286)(261,289)(262,292)(263,291)(264,290)(265,293)(266,296)(267,295)
(268,294)(269,297)(270,300)(271,299)(272,298)(273,301)(274,304)(275,303)
(276,302)(277,305)(278,308)(279,307)(280,306)(310,312)(314,316)(318,320)
(322,324)(326,328)(330,332)(334,336);;
s2 := ( 1, 2)( 3, 4)( 5, 26)( 6, 25)( 7, 28)( 8, 27)( 9, 22)( 10, 21)
( 11, 24)( 12, 23)( 13, 18)( 14, 17)( 15, 20)( 16, 19)( 29, 30)( 31, 32)
( 33, 54)( 34, 53)( 35, 56)( 36, 55)( 37, 50)( 38, 49)( 39, 52)( 40, 51)
( 41, 46)( 42, 45)( 43, 48)( 44, 47)( 57, 58)( 59, 60)( 61, 82)( 62, 81)
( 63, 84)( 64, 83)( 65, 78)( 66, 77)( 67, 80)( 68, 79)( 69, 74)( 70, 73)
( 71, 76)( 72, 75)( 85, 86)( 87, 88)( 89,110)( 90,109)( 91,112)( 92,111)
( 93,106)( 94,105)( 95,108)( 96,107)( 97,102)( 98,101)( 99,104)(100,103)
(113,114)(115,116)(117,138)(118,137)(119,140)(120,139)(121,134)(122,133)
(123,136)(124,135)(125,130)(126,129)(127,132)(128,131)(141,142)(143,144)
(145,166)(146,165)(147,168)(148,167)(149,162)(150,161)(151,164)(152,163)
(153,158)(154,157)(155,160)(156,159)(169,254)(170,253)(171,256)(172,255)
(173,278)(174,277)(175,280)(176,279)(177,274)(178,273)(179,276)(180,275)
(181,270)(182,269)(183,272)(184,271)(185,266)(186,265)(187,268)(188,267)
(189,262)(190,261)(191,264)(192,263)(193,258)(194,257)(195,260)(196,259)
(197,282)(198,281)(199,284)(200,283)(201,306)(202,305)(203,308)(204,307)
(205,302)(206,301)(207,304)(208,303)(209,298)(210,297)(211,300)(212,299)
(213,294)(214,293)(215,296)(216,295)(217,290)(218,289)(219,292)(220,291)
(221,286)(222,285)(223,288)(224,287)(225,310)(226,309)(227,312)(228,311)
(229,334)(230,333)(231,336)(232,335)(233,330)(234,329)(235,332)(236,331)
(237,326)(238,325)(239,328)(240,327)(241,322)(242,321)(243,324)(244,323)
(245,318)(246,317)(247,320)(248,319)(249,314)(250,313)(251,316)(252,315);;
s3 := ( 1,173)( 2,174)( 3,175)( 4,176)( 5,169)( 6,170)( 7,171)( 8,172)
( 9,193)( 10,194)( 11,195)( 12,196)( 13,189)( 14,190)( 15,191)( 16,192)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,181)( 22,182)( 23,183)( 24,184)
( 25,177)( 26,178)( 27,179)( 28,180)( 29,201)( 30,202)( 31,203)( 32,204)
( 33,197)( 34,198)( 35,199)( 36,200)( 37,221)( 38,222)( 39,223)( 40,224)
( 41,217)( 42,218)( 43,219)( 44,220)( 45,213)( 46,214)( 47,215)( 48,216)
( 49,209)( 50,210)( 51,211)( 52,212)( 53,205)( 54,206)( 55,207)( 56,208)
( 57,229)( 58,230)( 59,231)( 60,232)( 61,225)( 62,226)( 63,227)( 64,228)
( 65,249)( 66,250)( 67,251)( 68,252)( 69,245)( 70,246)( 71,247)( 72,248)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,237)( 78,238)( 79,239)( 80,240)
( 81,233)( 82,234)( 83,235)( 84,236)( 85,257)( 86,258)( 87,259)( 88,260)
( 89,253)( 90,254)( 91,255)( 92,256)( 93,277)( 94,278)( 95,279)( 96,280)
( 97,273)( 98,274)( 99,275)(100,276)(101,269)(102,270)(103,271)(104,272)
(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)(112,264)
(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)(120,284)
(121,305)(122,306)(123,307)(124,308)(125,301)(126,302)(127,303)(128,304)
(129,297)(130,298)(131,299)(132,300)(133,293)(134,294)(135,295)(136,296)
(137,289)(138,290)(139,291)(140,292)(141,313)(142,314)(143,315)(144,316)
(145,309)(146,310)(147,311)(148,312)(149,333)(150,334)(151,335)(152,336)
(153,329)(154,330)(155,331)(156,332)(157,325)(158,326)(159,327)(160,328)
(161,321)(162,322)(163,323)(164,324)(165,317)(166,318)(167,319)(168,320);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*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*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, 29)( 2, 32)( 3, 31)( 4, 30)( 5, 33)( 6, 36)( 7, 35)
( 8, 34)( 9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 41)( 14, 44)( 15, 43)
( 16, 42)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 49)( 22, 52)( 23, 51)
( 24, 50)( 25, 53)( 26, 56)( 27, 55)( 28, 54)( 58, 60)( 62, 64)( 66, 68)
( 70, 72)( 74, 76)( 78, 80)( 82, 84)( 85,113)( 86,116)( 87,115)( 88,114)
( 89,117)( 90,120)( 91,119)( 92,118)( 93,121)( 94,124)( 95,123)( 96,122)
( 97,125)( 98,128)( 99,127)(100,126)(101,129)(102,132)(103,131)(104,130)
(105,133)(106,136)(107,135)(108,134)(109,137)(110,140)(111,139)(112,138)
(142,144)(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)(169,197)
(170,200)(171,199)(172,198)(173,201)(174,204)(175,203)(176,202)(177,205)
(178,208)(179,207)(180,206)(181,209)(182,212)(183,211)(184,210)(185,213)
(186,216)(187,215)(188,214)(189,217)(190,220)(191,219)(192,218)(193,221)
(194,224)(195,223)(196,222)(226,228)(230,232)(234,236)(238,240)(242,244)
(246,248)(250,252)(253,281)(254,284)(255,283)(256,282)(257,285)(258,288)
(259,287)(260,286)(261,289)(262,292)(263,291)(264,290)(265,293)(266,296)
(267,295)(268,294)(269,297)(270,300)(271,299)(272,298)(273,301)(274,304)
(275,303)(276,302)(277,305)(278,308)(279,307)(280,306)(310,312)(314,316)
(318,320)(322,324)(326,328)(330,332)(334,336);
s2 := Sym(336)!( 1, 2)( 3, 4)( 5, 26)( 6, 25)( 7, 28)( 8, 27)( 9, 22)
( 10, 21)( 11, 24)( 12, 23)( 13, 18)( 14, 17)( 15, 20)( 16, 19)( 29, 30)
( 31, 32)( 33, 54)( 34, 53)( 35, 56)( 36, 55)( 37, 50)( 38, 49)( 39, 52)
( 40, 51)( 41, 46)( 42, 45)( 43, 48)( 44, 47)( 57, 58)( 59, 60)( 61, 82)
( 62, 81)( 63, 84)( 64, 83)( 65, 78)( 66, 77)( 67, 80)( 68, 79)( 69, 74)
( 70, 73)( 71, 76)( 72, 75)( 85, 86)( 87, 88)( 89,110)( 90,109)( 91,112)
( 92,111)( 93,106)( 94,105)( 95,108)( 96,107)( 97,102)( 98,101)( 99,104)
(100,103)(113,114)(115,116)(117,138)(118,137)(119,140)(120,139)(121,134)
(122,133)(123,136)(124,135)(125,130)(126,129)(127,132)(128,131)(141,142)
(143,144)(145,166)(146,165)(147,168)(148,167)(149,162)(150,161)(151,164)
(152,163)(153,158)(154,157)(155,160)(156,159)(169,254)(170,253)(171,256)
(172,255)(173,278)(174,277)(175,280)(176,279)(177,274)(178,273)(179,276)
(180,275)(181,270)(182,269)(183,272)(184,271)(185,266)(186,265)(187,268)
(188,267)(189,262)(190,261)(191,264)(192,263)(193,258)(194,257)(195,260)
(196,259)(197,282)(198,281)(199,284)(200,283)(201,306)(202,305)(203,308)
(204,307)(205,302)(206,301)(207,304)(208,303)(209,298)(210,297)(211,300)
(212,299)(213,294)(214,293)(215,296)(216,295)(217,290)(218,289)(219,292)
(220,291)(221,286)(222,285)(223,288)(224,287)(225,310)(226,309)(227,312)
(228,311)(229,334)(230,333)(231,336)(232,335)(233,330)(234,329)(235,332)
(236,331)(237,326)(238,325)(239,328)(240,327)(241,322)(242,321)(243,324)
(244,323)(245,318)(246,317)(247,320)(248,319)(249,314)(250,313)(251,316)
(252,315);
s3 := Sym(336)!( 1,173)( 2,174)( 3,175)( 4,176)( 5,169)( 6,170)( 7,171)
( 8,172)( 9,193)( 10,194)( 11,195)( 12,196)( 13,189)( 14,190)( 15,191)
( 16,192)( 17,185)( 18,186)( 19,187)( 20,188)( 21,181)( 22,182)( 23,183)
( 24,184)( 25,177)( 26,178)( 27,179)( 28,180)( 29,201)( 30,202)( 31,203)
( 32,204)( 33,197)( 34,198)( 35,199)( 36,200)( 37,221)( 38,222)( 39,223)
( 40,224)( 41,217)( 42,218)( 43,219)( 44,220)( 45,213)( 46,214)( 47,215)
( 48,216)( 49,209)( 50,210)( 51,211)( 52,212)( 53,205)( 54,206)( 55,207)
( 56,208)( 57,229)( 58,230)( 59,231)( 60,232)( 61,225)( 62,226)( 63,227)
( 64,228)( 65,249)( 66,250)( 67,251)( 68,252)( 69,245)( 70,246)( 71,247)
( 72,248)( 73,241)( 74,242)( 75,243)( 76,244)( 77,237)( 78,238)( 79,239)
( 80,240)( 81,233)( 82,234)( 83,235)( 84,236)( 85,257)( 86,258)( 87,259)
( 88,260)( 89,253)( 90,254)( 91,255)( 92,256)( 93,277)( 94,278)( 95,279)
( 96,280)( 97,273)( 98,274)( 99,275)(100,276)(101,269)(102,270)(103,271)
(104,272)(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)
(112,264)(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)
(120,284)(121,305)(122,306)(123,307)(124,308)(125,301)(126,302)(127,303)
(128,304)(129,297)(130,298)(131,299)(132,300)(133,293)(134,294)(135,295)
(136,296)(137,289)(138,290)(139,291)(140,292)(141,313)(142,314)(143,315)
(144,316)(145,309)(146,310)(147,311)(148,312)(149,333)(150,334)(151,335)
(152,336)(153,329)(154,330)(155,331)(156,332)(157,325)(158,326)(159,327)
(160,328)(161,321)(162,322)(163,323)(164,324)(165,317)(166,318)(167,319)
(168,320);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*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*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