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