Polytope of Type {28,4,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,4,3}*1344
Also Known As : {{28,4|2},{4,3}}. if this polytope has another name.
Group : SmallGroup(1344,11328)
Rank : 4
Schlafli Type : {28,4,3}
Number of vertices, edges, etc : 28, 112, 12, 6
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 : {14,4,3}*672
4-fold quotients : {28,2,3}*336
7-fold quotients : {4,4,3}*192b
8-fold quotients : {14,2,3}*168
14-fold quotients : {2,4,3}*96
16-fold quotients : {7,2,3}*84
28-fold quotients : {4,2,3}*48, {2,4,3}*48
56-fold quotients : {2,2,3}*24
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)
(169,253)(170,254)(171,255)(172,256)(173,277)(174,278)(175,279)(176,280)
(177,273)(178,274)(179,275)(180,276)(181,269)(182,270)(183,271)(184,272)
(185,265)(186,266)(187,267)(188,268)(189,261)(190,262)(191,263)(192,264)
(193,257)(194,258)(195,259)(196,260)(197,281)(198,282)(199,283)(200,284)
(201,305)(202,306)(203,307)(204,308)(205,301)(206,302)(207,303)(208,304)
(209,297)(210,298)(211,299)(212,300)(213,293)(214,294)(215,295)(216,296)
(217,289)(218,290)(219,291)(220,292)(221,285)(222,286)(223,287)(224,288)
(225,309)(226,310)(227,311)(228,312)(229,333)(230,334)(231,335)(232,336)
(233,329)(234,330)(235,331)(236,332)(237,325)(238,326)(239,327)(240,328)
(241,321)(242,322)(243,323)(244,324)(245,317)(246,318)(247,319)(248,320)
(249,313)(250,314)(251,315)(252,316);;
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, 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);;
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, s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*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*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)(169,253)(170,254)(171,255)(172,256)(173,277)(174,278)(175,279)
(176,280)(177,273)(178,274)(179,275)(180,276)(181,269)(182,270)(183,271)
(184,272)(185,265)(186,266)(187,267)(188,268)(189,261)(190,262)(191,263)
(192,264)(193,257)(194,258)(195,259)(196,260)(197,281)(198,282)(199,283)
(200,284)(201,305)(202,306)(203,307)(204,308)(205,301)(206,302)(207,303)
(208,304)(209,297)(210,298)(211,299)(212,300)(213,293)(214,294)(215,295)
(216,296)(217,289)(218,290)(219,291)(220,292)(221,285)(222,286)(223,287)
(224,288)(225,309)(226,310)(227,311)(228,312)(229,333)(230,334)(231,335)
(232,336)(233,329)(234,330)(235,331)(236,332)(237,325)(238,326)(239,327)
(240,328)(241,321)(242,322)(243,323)(244,324)(245,317)(246,318)(247,319)
(248,320)(249,313)(250,314)(251,315)(252,316);
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, 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);
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,
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*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*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