Polytope of Type {28,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,6,4}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11328)
Rank : 4
Schlafli Type : {28,6,4}
Number of vertices, edges, etc : 28, 84, 12, 4
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 : {14,6,4}*672b
7-fold quotients : {4,6,4}*192b
14-fold quotients : {2,6,4}*96c
28-fold quotients : {2,3,4}*48
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 := ( 1,173)( 2,174)( 3,176)( 4,175)( 5,169)( 6,170)( 7,172)( 8,171)
( 9,193)( 10,194)( 11,196)( 12,195)( 13,189)( 14,190)( 15,192)( 16,191)
( 17,185)( 18,186)( 19,188)( 20,187)( 21,181)( 22,182)( 23,184)( 24,183)
( 25,177)( 26,178)( 27,180)( 28,179)( 29,229)( 30,230)( 31,232)( 32,231)
( 33,225)( 34,226)( 35,228)( 36,227)( 37,249)( 38,250)( 39,252)( 40,251)
( 41,245)( 42,246)( 43,248)( 44,247)( 45,241)( 46,242)( 47,244)( 48,243)
( 49,237)( 50,238)( 51,240)( 52,239)( 53,233)( 54,234)( 55,236)( 56,235)
( 57,201)( 58,202)( 59,204)( 60,203)( 61,197)( 62,198)( 63,200)( 64,199)
( 65,221)( 66,222)( 67,224)( 68,223)( 69,217)( 70,218)( 71,220)( 72,219)
( 73,213)( 74,214)( 75,216)( 76,215)( 77,209)( 78,210)( 79,212)( 80,211)
( 81,205)( 82,206)( 83,208)( 84,207)( 85,257)( 86,258)( 87,260)( 88,259)
( 89,253)( 90,254)( 91,256)( 92,255)( 93,277)( 94,278)( 95,280)( 96,279)
( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)(104,271)
(105,265)(106,266)(107,268)(108,267)(109,261)(110,262)(111,264)(112,263)
(113,313)(114,314)(115,316)(116,315)(117,309)(118,310)(119,312)(120,311)
(121,333)(122,334)(123,336)(124,335)(125,329)(126,330)(127,332)(128,331)
(129,325)(130,326)(131,328)(132,327)(133,321)(134,322)(135,324)(136,323)
(137,317)(138,318)(139,320)(140,319)(141,285)(142,286)(143,288)(144,287)
(145,281)(146,282)(147,284)(148,283)(149,305)(150,306)(151,308)(152,307)
(153,301)(154,302)(155,304)(156,303)(157,297)(158,298)(159,300)(160,299)
(161,293)(162,294)(163,296)(164,295)(165,289)(166,290)(167,292)(168,291);;
s2 := ( 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);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2,
s1*s2*s1*s2*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)!( 1,173)( 2,174)( 3,176)( 4,175)( 5,169)( 6,170)( 7,172)
( 8,171)( 9,193)( 10,194)( 11,196)( 12,195)( 13,189)( 14,190)( 15,192)
( 16,191)( 17,185)( 18,186)( 19,188)( 20,187)( 21,181)( 22,182)( 23,184)
( 24,183)( 25,177)( 26,178)( 27,180)( 28,179)( 29,229)( 30,230)( 31,232)
( 32,231)( 33,225)( 34,226)( 35,228)( 36,227)( 37,249)( 38,250)( 39,252)
( 40,251)( 41,245)( 42,246)( 43,248)( 44,247)( 45,241)( 46,242)( 47,244)
( 48,243)( 49,237)( 50,238)( 51,240)( 52,239)( 53,233)( 54,234)( 55,236)
( 56,235)( 57,201)( 58,202)( 59,204)( 60,203)( 61,197)( 62,198)( 63,200)
( 64,199)( 65,221)( 66,222)( 67,224)( 68,223)( 69,217)( 70,218)( 71,220)
( 72,219)( 73,213)( 74,214)( 75,216)( 76,215)( 77,209)( 78,210)( 79,212)
( 80,211)( 81,205)( 82,206)( 83,208)( 84,207)( 85,257)( 86,258)( 87,260)
( 88,259)( 89,253)( 90,254)( 91,256)( 92,255)( 93,277)( 94,278)( 95,280)
( 96,279)( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)
(104,271)(105,265)(106,266)(107,268)(108,267)(109,261)(110,262)(111,264)
(112,263)(113,313)(114,314)(115,316)(116,315)(117,309)(118,310)(119,312)
(120,311)(121,333)(122,334)(123,336)(124,335)(125,329)(126,330)(127,332)
(128,331)(129,325)(130,326)(131,328)(132,327)(133,321)(134,322)(135,324)
(136,323)(137,317)(138,318)(139,320)(140,319)(141,285)(142,286)(143,288)
(144,287)(145,281)(146,282)(147,284)(148,283)(149,305)(150,306)(151,308)
(152,307)(153,301)(154,302)(155,304)(156,303)(157,297)(158,298)(159,300)
(160,299)(161,293)(162,294)(163,296)(164,295)(165,289)(166,290)(167,292)
(168,291);
s2 := 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);
s3 := Sym(336)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)
(335,336);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*s2*s1*s2, s1*s2*s1*s2*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