Polytope of Type {21,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,6,4}*1344
if this polytope has a name.
Group : SmallGroup(1344,11370)
Rank : 4
Schlafli Type : {21,6,4}
Number of vertices, edges, etc : 28, 84, 16, 4
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 : {21,6,2}*672
7-fold quotients : {3,6,4}*192
12-fold quotients : {7,2,4}*112
14-fold quotients : {3,6,2}*96
24-fold quotients : {7,2,2}*56
28-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 25)( 6, 26)( 7, 28)( 8, 27)( 9, 21)( 10, 22)( 11, 24)
( 12, 23)( 13, 17)( 14, 18)( 15, 20)( 16, 19)( 29, 57)( 30, 58)( 31, 60)
( 32, 59)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 77)( 38, 78)( 39, 80)
( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)( 47, 72)
( 48, 71)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 61)( 54, 62)( 55, 64)
( 56, 63)( 87, 88)( 89,109)( 90,110)( 91,112)( 92,111)( 93,105)( 94,106)
( 95,108)( 96,107)( 97,101)( 98,102)( 99,104)(100,103)(113,141)(114,142)
(115,144)(116,143)(117,165)(118,166)(119,168)(120,167)(121,161)(122,162)
(123,164)(124,163)(125,157)(126,158)(127,160)(128,159)(129,153)(130,154)
(131,156)(132,155)(133,149)(134,150)(135,152)(136,151)(137,145)(138,146)
(139,148)(140,147)(171,172)(173,193)(174,194)(175,196)(176,195)(177,189)
(178,190)(179,192)(180,191)(181,185)(182,186)(183,188)(184,187)(197,225)
(198,226)(199,228)(200,227)(201,249)(202,250)(203,252)(204,251)(205,245)
(206,246)(207,248)(208,247)(209,241)(210,242)(211,244)(212,243)(213,237)
(214,238)(215,240)(216,239)(217,233)(218,234)(219,236)(220,235)(221,229)
(222,230)(223,232)(224,231)(255,256)(257,277)(258,278)(259,280)(260,279)
(261,273)(262,274)(263,276)(264,275)(265,269)(266,270)(267,272)(268,271)
(281,309)(282,310)(283,312)(284,311)(285,333)(286,334)(287,336)(288,335)
(289,329)(290,330)(291,332)(292,331)(293,325)(294,326)(295,328)(296,327)
(297,321)(298,322)(299,324)(300,323)(301,317)(302,318)(303,320)(304,319)
(305,313)(306,314)(307,316)(308,315);;
s1 := ( 1, 33)( 2, 36)( 3, 35)( 4, 34)( 5, 29)( 6, 32)( 7, 31)( 8, 30)
( 9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)( 24, 42)
( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 57, 61)( 58, 64)( 59, 63)( 60, 62)
( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)
( 74, 76)( 85,117)( 86,120)( 87,119)( 88,118)( 89,113)( 90,116)( 91,115)
( 92,114)( 93,137)( 94,140)( 95,139)( 96,138)( 97,133)( 98,136)( 99,135)
(100,134)(101,129)(102,132)(103,131)(104,130)(105,125)(106,128)(107,127)
(108,126)(109,121)(110,124)(111,123)(112,122)(141,145)(142,148)(143,147)
(144,146)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)(155,163)
(156,162)(158,160)(169,201)(170,204)(171,203)(172,202)(173,197)(174,200)
(175,199)(176,198)(177,221)(178,224)(179,223)(180,222)(181,217)(182,220)
(183,219)(184,218)(185,213)(186,216)(187,215)(188,214)(189,209)(190,212)
(191,211)(192,210)(193,205)(194,208)(195,207)(196,206)(225,229)(226,232)
(227,231)(228,230)(233,249)(234,252)(235,251)(236,250)(237,245)(238,248)
(239,247)(240,246)(242,244)(253,285)(254,288)(255,287)(256,286)(257,281)
(258,284)(259,283)(260,282)(261,305)(262,308)(263,307)(264,306)(265,301)
(266,304)(267,303)(268,302)(269,297)(270,300)(271,299)(272,298)(273,293)
(274,296)(275,295)(276,294)(277,289)(278,292)(279,291)(280,290)(309,313)
(310,316)(311,315)(312,314)(317,333)(318,336)(319,335)(320,334)(321,329)
(322,332)(323,331)(324,330)(326,328);;
s2 := ( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 58)
( 30, 57)( 31, 59)( 32, 60)( 33, 62)( 34, 61)( 35, 63)( 36, 64)( 37, 66)
( 38, 65)( 39, 67)( 40, 68)( 41, 70)( 42, 69)( 43, 71)( 44, 72)( 45, 74)
( 46, 73)( 47, 75)( 48, 76)( 49, 78)( 50, 77)( 51, 79)( 52, 80)( 53, 82)
( 54, 81)( 55, 83)( 56, 84)( 85, 86)( 89, 90)( 93, 94)( 97, 98)(101,102)
(105,106)(109,110)(113,142)(114,141)(115,143)(116,144)(117,146)(118,145)
(119,147)(120,148)(121,150)(122,149)(123,151)(124,152)(125,154)(126,153)
(127,155)(128,156)(129,158)(130,157)(131,159)(132,160)(133,162)(134,161)
(135,163)(136,164)(137,166)(138,165)(139,167)(140,168)(169,254)(170,253)
(171,255)(172,256)(173,258)(174,257)(175,259)(176,260)(177,262)(178,261)
(179,263)(180,264)(181,266)(182,265)(183,267)(184,268)(185,270)(186,269)
(187,271)(188,272)(189,274)(190,273)(191,275)(192,276)(193,278)(194,277)
(195,279)(196,280)(197,310)(198,309)(199,311)(200,312)(201,314)(202,313)
(203,315)(204,316)(205,318)(206,317)(207,319)(208,320)(209,322)(210,321)
(211,323)(212,324)(213,326)(214,325)(215,327)(216,328)(217,330)(218,329)
(219,331)(220,332)(221,334)(222,333)(223,335)(224,336)(225,282)(226,281)
(227,283)(228,284)(229,286)(230,285)(231,287)(232,288)(233,290)(234,289)
(235,291)(236,292)(237,294)(238,293)(239,295)(240,296)(241,298)(242,297)
(243,299)(244,300)(245,302)(246,301)(247,303)(248,304)(249,306)(250,305)
(251,307)(252,308);;
s3 := ( 1,169)( 2,170)( 3,171)( 4,172)( 5,173)( 6,174)( 7,175)( 8,176)
( 9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)( 16,184)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)
( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)( 32,200)
( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)
( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)( 48,216)
( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)( 56,224)
( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)( 64,232)
( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)( 72,240)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)( 80,248)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)( 88,256)
( 89,257)( 90,258)( 91,259)( 92,260)( 93,261)( 94,262)( 95,263)( 96,264)
( 97,265)( 98,266)( 99,267)(100,268)(101,269)(102,270)(103,271)(104,272)
(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)(112,280)
(113,281)(114,282)(115,283)(116,284)(117,285)(118,286)(119,287)(120,288)
(121,289)(122,290)(123,291)(124,292)(125,293)(126,294)(127,295)(128,296)
(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)
(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)(144,312)
(145,313)(146,314)(147,315)(148,316)(149,317)(150,318)(151,319)(152,320)
(153,321)(154,322)(155,323)(156,324)(157,325)(158,326)(159,327)(160,328)
(161,329)(162,330)(163,331)(164,332)(165,333)(166,334)(167,335)(168,336);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(336)!( 3, 4)( 5, 25)( 6, 26)( 7, 28)( 8, 27)( 9, 21)( 10, 22)
( 11, 24)( 12, 23)( 13, 17)( 14, 18)( 15, 20)( 16, 19)( 29, 57)( 30, 58)
( 31, 60)( 32, 59)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 77)( 38, 78)
( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)
( 47, 72)( 48, 71)( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 61)( 54, 62)
( 55, 64)( 56, 63)( 87, 88)( 89,109)( 90,110)( 91,112)( 92,111)( 93,105)
( 94,106)( 95,108)( 96,107)( 97,101)( 98,102)( 99,104)(100,103)(113,141)
(114,142)(115,144)(116,143)(117,165)(118,166)(119,168)(120,167)(121,161)
(122,162)(123,164)(124,163)(125,157)(126,158)(127,160)(128,159)(129,153)
(130,154)(131,156)(132,155)(133,149)(134,150)(135,152)(136,151)(137,145)
(138,146)(139,148)(140,147)(171,172)(173,193)(174,194)(175,196)(176,195)
(177,189)(178,190)(179,192)(180,191)(181,185)(182,186)(183,188)(184,187)
(197,225)(198,226)(199,228)(200,227)(201,249)(202,250)(203,252)(204,251)
(205,245)(206,246)(207,248)(208,247)(209,241)(210,242)(211,244)(212,243)
(213,237)(214,238)(215,240)(216,239)(217,233)(218,234)(219,236)(220,235)
(221,229)(222,230)(223,232)(224,231)(255,256)(257,277)(258,278)(259,280)
(260,279)(261,273)(262,274)(263,276)(264,275)(265,269)(266,270)(267,272)
(268,271)(281,309)(282,310)(283,312)(284,311)(285,333)(286,334)(287,336)
(288,335)(289,329)(290,330)(291,332)(292,331)(293,325)(294,326)(295,328)
(296,327)(297,321)(298,322)(299,324)(300,323)(301,317)(302,318)(303,320)
(304,319)(305,313)(306,314)(307,316)(308,315);
s1 := Sym(336)!( 1, 33)( 2, 36)( 3, 35)( 4, 34)( 5, 29)( 6, 32)( 7, 31)
( 8, 30)( 9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)
( 24, 42)( 25, 37)( 26, 40)( 27, 39)( 28, 38)( 57, 61)( 58, 64)( 59, 63)
( 60, 62)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)
( 72, 78)( 74, 76)( 85,117)( 86,120)( 87,119)( 88,118)( 89,113)( 90,116)
( 91,115)( 92,114)( 93,137)( 94,140)( 95,139)( 96,138)( 97,133)( 98,136)
( 99,135)(100,134)(101,129)(102,132)(103,131)(104,130)(105,125)(106,128)
(107,127)(108,126)(109,121)(110,124)(111,123)(112,122)(141,145)(142,148)
(143,147)(144,146)(149,165)(150,168)(151,167)(152,166)(153,161)(154,164)
(155,163)(156,162)(158,160)(169,201)(170,204)(171,203)(172,202)(173,197)
(174,200)(175,199)(176,198)(177,221)(178,224)(179,223)(180,222)(181,217)
(182,220)(183,219)(184,218)(185,213)(186,216)(187,215)(188,214)(189,209)
(190,212)(191,211)(192,210)(193,205)(194,208)(195,207)(196,206)(225,229)
(226,232)(227,231)(228,230)(233,249)(234,252)(235,251)(236,250)(237,245)
(238,248)(239,247)(240,246)(242,244)(253,285)(254,288)(255,287)(256,286)
(257,281)(258,284)(259,283)(260,282)(261,305)(262,308)(263,307)(264,306)
(265,301)(266,304)(267,303)(268,302)(269,297)(270,300)(271,299)(272,298)
(273,293)(274,296)(275,295)(276,294)(277,289)(278,292)(279,291)(280,290)
(309,313)(310,316)(311,315)(312,314)(317,333)(318,336)(319,335)(320,334)
(321,329)(322,332)(323,331)(324,330)(326,328);
s2 := Sym(336)!( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)
( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 62)( 34, 61)( 35, 63)( 36, 64)
( 37, 66)( 38, 65)( 39, 67)( 40, 68)( 41, 70)( 42, 69)( 43, 71)( 44, 72)
( 45, 74)( 46, 73)( 47, 75)( 48, 76)( 49, 78)( 50, 77)( 51, 79)( 52, 80)
( 53, 82)( 54, 81)( 55, 83)( 56, 84)( 85, 86)( 89, 90)( 93, 94)( 97, 98)
(101,102)(105,106)(109,110)(113,142)(114,141)(115,143)(116,144)(117,146)
(118,145)(119,147)(120,148)(121,150)(122,149)(123,151)(124,152)(125,154)
(126,153)(127,155)(128,156)(129,158)(130,157)(131,159)(132,160)(133,162)
(134,161)(135,163)(136,164)(137,166)(138,165)(139,167)(140,168)(169,254)
(170,253)(171,255)(172,256)(173,258)(174,257)(175,259)(176,260)(177,262)
(178,261)(179,263)(180,264)(181,266)(182,265)(183,267)(184,268)(185,270)
(186,269)(187,271)(188,272)(189,274)(190,273)(191,275)(192,276)(193,278)
(194,277)(195,279)(196,280)(197,310)(198,309)(199,311)(200,312)(201,314)
(202,313)(203,315)(204,316)(205,318)(206,317)(207,319)(208,320)(209,322)
(210,321)(211,323)(212,324)(213,326)(214,325)(215,327)(216,328)(217,330)
(218,329)(219,331)(220,332)(221,334)(222,333)(223,335)(224,336)(225,282)
(226,281)(227,283)(228,284)(229,286)(230,285)(231,287)(232,288)(233,290)
(234,289)(235,291)(236,292)(237,294)(238,293)(239,295)(240,296)(241,298)
(242,297)(243,299)(244,300)(245,302)(246,301)(247,303)(248,304)(249,306)
(250,305)(251,307)(252,308);
s3 := Sym(336)!( 1,169)( 2,170)( 3,171)( 4,172)( 5,173)( 6,174)( 7,175)
( 8,176)( 9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)
( 16,184)( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)
( 24,192)( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)
( 32,200)( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)
( 40,208)( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)
( 48,216)( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)
( 56,224)( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)
( 64,232)( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)
( 72,240)( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)
( 80,248)( 81,249)( 82,250)( 83,251)( 84,252)( 85,253)( 86,254)( 87,255)
( 88,256)( 89,257)( 90,258)( 91,259)( 92,260)( 93,261)( 94,262)( 95,263)
( 96,264)( 97,265)( 98,266)( 99,267)(100,268)(101,269)(102,270)(103,271)
(104,272)(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)
(112,280)(113,281)(114,282)(115,283)(116,284)(117,285)(118,286)(119,287)
(120,288)(121,289)(122,290)(123,291)(124,292)(125,293)(126,294)(127,295)
(128,296)(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)
(136,304)(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)
(144,312)(145,313)(146,314)(147,315)(148,316)(149,317)(150,318)(151,319)
(152,320)(153,321)(154,322)(155,323)(156,324)(157,325)(158,326)(159,327)
(160,328)(161,329)(162,330)(163,331)(164,332)(165,333)(166,334)(167,335)
(168,336);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope