Polytope of Type {3,6,28}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,28}*1344
Also Known As : {{3,6}4,{6,28|2}}. if this polytope has another name.
Group : SmallGroup(1344,11328)
Rank : 4
Schlafli Type : {3,6,28}
Number of vertices, edges, etc : 4, 12, 112, 28
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 : {3,6,14}*672
7-fold quotients : {3,6,4}*192
14-fold quotients : {3,6,2}*96
28-fold quotients : {3,3,2}*48
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)( 5, 26)( 6, 25)( 7, 27)( 8, 28)( 9, 22)( 10, 21)( 11, 23)
( 12, 24)( 13, 18)( 14, 17)( 15, 19)( 16, 20)( 29, 58)( 30, 57)( 31, 59)
( 32, 60)( 33, 82)( 34, 81)( 35, 83)( 36, 84)( 37, 78)( 38, 77)( 39, 79)
( 40, 80)( 41, 74)( 42, 73)( 43, 75)( 44, 76)( 45, 70)( 46, 69)( 47, 71)
( 48, 72)( 49, 66)( 50, 65)( 51, 67)( 52, 68)( 53, 62)( 54, 61)( 55, 63)
( 56, 64)( 85, 86)( 89,110)( 90,109)( 91,111)( 92,112)( 93,106)( 94,105)
( 95,107)( 96,108)( 97,102)( 98,101)( 99,103)(100,104)(113,142)(114,141)
(115,143)(116,144)(117,166)(118,165)(119,167)(120,168)(121,162)(122,161)
(123,163)(124,164)(125,158)(126,157)(127,159)(128,160)(129,154)(130,153)
(131,155)(132,156)(133,150)(134,149)(135,151)(136,152)(137,146)(138,145)
(139,147)(140,148)(169,254)(170,253)(171,255)(172,256)(173,278)(174,277)
(175,279)(176,280)(177,274)(178,273)(179,275)(180,276)(181,270)(182,269)
(183,271)(184,272)(185,266)(186,265)(187,267)(188,268)(189,262)(190,261)
(191,263)(192,264)(193,258)(194,257)(195,259)(196,260)(197,310)(198,309)
(199,311)(200,312)(201,334)(202,333)(203,335)(204,336)(205,330)(206,329)
(207,331)(208,332)(209,326)(210,325)(211,327)(212,328)(213,322)(214,321)
(215,323)(216,324)(217,318)(218,317)(219,319)(220,320)(221,314)(222,313)
(223,315)(224,316)(225,282)(226,281)(227,283)(228,284)(229,306)(230,305)
(231,307)(232,308)(233,302)(234,301)(235,303)(236,304)(237,298)(238,297)
(239,299)(240,300)(241,294)(242,293)(243,295)(244,296)(245,290)(246,289)
(247,291)(248,292)(249,286)(250,285)(251,287)(252,288);;
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*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
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)( 5, 26)( 6, 25)( 7, 27)( 8, 28)( 9, 22)( 10, 21)
( 11, 23)( 12, 24)( 13, 18)( 14, 17)( 15, 19)( 16, 20)( 29, 58)( 30, 57)
( 31, 59)( 32, 60)( 33, 82)( 34, 81)( 35, 83)( 36, 84)( 37, 78)( 38, 77)
( 39, 79)( 40, 80)( 41, 74)( 42, 73)( 43, 75)( 44, 76)( 45, 70)( 46, 69)
( 47, 71)( 48, 72)( 49, 66)( 50, 65)( 51, 67)( 52, 68)( 53, 62)( 54, 61)
( 55, 63)( 56, 64)( 85, 86)( 89,110)( 90,109)( 91,111)( 92,112)( 93,106)
( 94,105)( 95,107)( 96,108)( 97,102)( 98,101)( 99,103)(100,104)(113,142)
(114,141)(115,143)(116,144)(117,166)(118,165)(119,167)(120,168)(121,162)
(122,161)(123,163)(124,164)(125,158)(126,157)(127,159)(128,160)(129,154)
(130,153)(131,155)(132,156)(133,150)(134,149)(135,151)(136,152)(137,146)
(138,145)(139,147)(140,148)(169,254)(170,253)(171,255)(172,256)(173,278)
(174,277)(175,279)(176,280)(177,274)(178,273)(179,275)(180,276)(181,270)
(182,269)(183,271)(184,272)(185,266)(186,265)(187,267)(188,268)(189,262)
(190,261)(191,263)(192,264)(193,258)(194,257)(195,259)(196,260)(197,310)
(198,309)(199,311)(200,312)(201,334)(202,333)(203,335)(204,336)(205,330)
(206,329)(207,331)(208,332)(209,326)(210,325)(211,327)(212,328)(213,322)
(214,321)(215,323)(216,324)(217,318)(218,317)(219,319)(220,320)(221,314)
(222,313)(223,315)(224,316)(225,282)(226,281)(227,283)(228,284)(229,306)
(230,305)(231,307)(232,308)(233,302)(234,301)(235,303)(236,304)(237,298)
(238,297)(239,299)(240,300)(241,294)(242,293)(243,295)(244,296)(245,290)
(246,289)(247,291)(248,292)(249,286)(250,285)(251,287)(252,288);
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*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
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