Polytope of Type {4,4,21}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,21}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11399)
Rank : 4
Schlafli Type : {4,4,21}
Number of vertices, edges, etc : 4, 16, 84, 42
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 : {2,4,21}*672
4-fold quotients : {4,2,21}*336, {2,4,21}*336
7-fold quotients : {4,4,3}*192b
8-fold quotients : {2,2,21}*168
12-fold quotients : {4,2,7}*112
14-fold quotients : {2,4,3}*96
24-fold quotients : {2,2,7}*56
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 := (169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)
(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)
(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)
(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)(200,284)
(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)(208,292)
(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)(216,300)
(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)(224,308)
(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,316)
(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)(240,324)
(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)(248,332)
(249,333)(250,334)(251,335)(252,336);;
s1 := ( 1,171)( 2,172)( 3,169)( 4,170)( 5,175)( 6,176)( 7,173)( 8,174)
( 9,179)( 10,180)( 11,177)( 12,178)( 13,183)( 14,184)( 15,181)( 16,182)
( 17,187)( 18,188)( 19,185)( 20,186)( 21,191)( 22,192)( 23,189)( 24,190)
( 25,195)( 26,196)( 27,193)( 28,194)( 29,199)( 30,200)( 31,197)( 32,198)
( 33,203)( 34,204)( 35,201)( 36,202)( 37,207)( 38,208)( 39,205)( 40,206)
( 41,211)( 42,212)( 43,209)( 44,210)( 45,215)( 46,216)( 47,213)( 48,214)
( 49,219)( 50,220)( 51,217)( 52,218)( 53,223)( 54,224)( 55,221)( 56,222)
( 57,227)( 58,228)( 59,225)( 60,226)( 61,231)( 62,232)( 63,229)( 64,230)
( 65,235)( 66,236)( 67,233)( 68,234)( 69,239)( 70,240)( 71,237)( 72,238)
( 73,243)( 74,244)( 75,241)( 76,242)( 77,247)( 78,248)( 79,245)( 80,246)
( 81,251)( 82,252)( 83,249)( 84,250)( 85,255)( 86,256)( 87,253)( 88,254)
( 89,259)( 90,260)( 91,257)( 92,258)( 93,263)( 94,264)( 95,261)( 96,262)
( 97,267)( 98,268)( 99,265)(100,266)(101,271)(102,272)(103,269)(104,270)
(105,275)(106,276)(107,273)(108,274)(109,279)(110,280)(111,277)(112,278)
(113,283)(114,284)(115,281)(116,282)(117,287)(118,288)(119,285)(120,286)
(121,291)(122,292)(123,289)(124,290)(125,295)(126,296)(127,293)(128,294)
(129,299)(130,300)(131,297)(132,298)(133,303)(134,304)(135,301)(136,302)
(137,307)(138,308)(139,305)(140,306)(141,311)(142,312)(143,309)(144,310)
(145,315)(146,316)(147,313)(148,314)(149,319)(150,320)(151,317)(152,318)
(153,323)(154,324)(155,321)(156,322)(157,327)(158,328)(159,325)(160,326)
(161,331)(162,332)(163,329)(164,330)(165,335)(166,336)(167,333)(168,334);;
s2 := ( 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);;
s3 := ( 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);;
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
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)!(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)
(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)
(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)
(192,276)(193,277)(194,278)(195,279)(196,280)(197,281)(198,282)(199,283)
(200,284)(201,285)(202,286)(203,287)(204,288)(205,289)(206,290)(207,291)
(208,292)(209,293)(210,294)(211,295)(212,296)(213,297)(214,298)(215,299)
(216,300)(217,301)(218,302)(219,303)(220,304)(221,305)(222,306)(223,307)
(224,308)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)
(232,316)(233,317)(234,318)(235,319)(236,320)(237,321)(238,322)(239,323)
(240,324)(241,325)(242,326)(243,327)(244,328)(245,329)(246,330)(247,331)
(248,332)(249,333)(250,334)(251,335)(252,336);
s1 := Sym(336)!( 1,171)( 2,172)( 3,169)( 4,170)( 5,175)( 6,176)( 7,173)
( 8,174)( 9,179)( 10,180)( 11,177)( 12,178)( 13,183)( 14,184)( 15,181)
( 16,182)( 17,187)( 18,188)( 19,185)( 20,186)( 21,191)( 22,192)( 23,189)
( 24,190)( 25,195)( 26,196)( 27,193)( 28,194)( 29,199)( 30,200)( 31,197)
( 32,198)( 33,203)( 34,204)( 35,201)( 36,202)( 37,207)( 38,208)( 39,205)
( 40,206)( 41,211)( 42,212)( 43,209)( 44,210)( 45,215)( 46,216)( 47,213)
( 48,214)( 49,219)( 50,220)( 51,217)( 52,218)( 53,223)( 54,224)( 55,221)
( 56,222)( 57,227)( 58,228)( 59,225)( 60,226)( 61,231)( 62,232)( 63,229)
( 64,230)( 65,235)( 66,236)( 67,233)( 68,234)( 69,239)( 70,240)( 71,237)
( 72,238)( 73,243)( 74,244)( 75,241)( 76,242)( 77,247)( 78,248)( 79,245)
( 80,246)( 81,251)( 82,252)( 83,249)( 84,250)( 85,255)( 86,256)( 87,253)
( 88,254)( 89,259)( 90,260)( 91,257)( 92,258)( 93,263)( 94,264)( 95,261)
( 96,262)( 97,267)( 98,268)( 99,265)(100,266)(101,271)(102,272)(103,269)
(104,270)(105,275)(106,276)(107,273)(108,274)(109,279)(110,280)(111,277)
(112,278)(113,283)(114,284)(115,281)(116,282)(117,287)(118,288)(119,285)
(120,286)(121,291)(122,292)(123,289)(124,290)(125,295)(126,296)(127,293)
(128,294)(129,299)(130,300)(131,297)(132,298)(133,303)(134,304)(135,301)
(136,302)(137,307)(138,308)(139,305)(140,306)(141,311)(142,312)(143,309)
(144,310)(145,315)(146,316)(147,313)(148,314)(149,319)(150,320)(151,317)
(152,318)(153,323)(154,324)(155,321)(156,322)(157,327)(158,328)(159,325)
(160,326)(161,331)(162,332)(163,329)(164,330)(165,335)(166,336)(167,333)
(168,334);
s2 := 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);
s3 := 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);
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
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