Polytope of Type {21,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,6,6}*1512a
if this polytope has a name.
Group : SmallGroup(1512,561)
Rank : 4
Schlafli Type : {21,6,6}
Number of vertices, edges, etc : 21, 63, 18, 6
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 6
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 : {21,6,3}*756
   3-fold quotients : {21,2,6}*504
   6-fold quotients : {21,2,3}*252
   7-fold quotients : {3,6,6}*216a
   9-fold quotients : {7,2,6}*168, {21,2,2}*168
   14-fold quotients : {3,6,3}*108
   18-fold quotients : {7,2,3}*84
   21-fold quotients : {3,2,6}*72
   27-fold quotients : {7,2,2}*56
   42-fold quotients : {3,2,3}*36
   63-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)( 10, 13)
( 11, 15)( 12, 14)( 22, 43)( 23, 45)( 24, 44)( 25, 61)( 26, 63)( 27, 62)
( 28, 58)( 29, 60)( 30, 59)( 31, 55)( 32, 57)( 33, 56)( 34, 52)( 35, 54)
( 36, 53)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 65, 66)
( 67, 82)( 68, 84)( 69, 83)( 70, 79)( 71, 81)( 72, 80)( 73, 76)( 74, 78)
( 75, 77)( 85,106)( 86,108)( 87,107)( 88,124)( 89,126)( 90,125)( 91,121)
( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,115)( 98,117)( 99,116)
(100,112)(101,114)(102,113)(103,109)(104,111)(105,110)(128,129)(130,145)
(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)(137,141)(138,140)
(148,169)(149,171)(150,170)(151,187)(152,189)(153,188)(154,184)(155,186)
(156,185)(157,181)(158,183)(159,182)(160,178)(161,180)(162,179)(163,175)
(164,177)(165,176)(166,172)(167,174)(168,173)(191,192)(193,208)(194,210)
(195,209)(196,205)(197,207)(198,206)(199,202)(200,204)(201,203)(211,232)
(212,234)(213,233)(214,250)(215,252)(216,251)(217,247)(218,249)(219,248)
(220,244)(221,246)(222,245)(223,241)(224,243)(225,242)(226,238)(227,240)
(228,239)(229,235)(230,237)(231,236)(254,255)(256,271)(257,273)(258,272)
(259,268)(260,270)(261,269)(262,265)(263,267)(264,266)(274,295)(275,297)
(276,296)(277,313)(278,315)(279,314)(280,310)(281,312)(282,311)(283,307)
(284,309)(285,308)(286,304)(287,306)(288,305)(289,301)(290,303)(291,302)
(292,298)(293,300)(294,299)(317,318)(319,334)(320,336)(321,335)(322,331)
(323,333)(324,332)(325,328)(326,330)(327,329)(337,358)(338,360)(339,359)
(340,376)(341,378)(342,377)(343,373)(344,375)(345,374)(346,370)(347,372)
(348,371)(349,367)(350,369)(351,368)(352,364)(353,366)(354,365)(355,361)
(356,363)(357,362);;
s1 := (  1, 25)(  2, 27)(  3, 26)(  4, 22)(  5, 24)(  6, 23)(  7, 40)(  8, 42)
(  9, 41)( 10, 37)( 11, 39)( 12, 38)( 13, 34)( 14, 36)( 15, 35)( 16, 31)
( 17, 33)( 18, 32)( 19, 28)( 20, 30)( 21, 29)( 43, 46)( 44, 48)( 45, 47)
( 49, 61)( 50, 63)( 51, 62)( 52, 58)( 53, 60)( 54, 59)( 56, 57)( 64, 88)
( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70,103)( 71,105)( 72,104)
( 73,100)( 74,102)( 75,101)( 76, 97)( 77, 99)( 78, 98)( 79, 94)( 80, 96)
( 81, 95)( 82, 91)( 83, 93)( 84, 92)(106,109)(107,111)(108,110)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(119,120)(127,151)(128,153)
(129,152)(130,148)(131,150)(132,149)(133,166)(134,168)(135,167)(136,163)
(137,165)(138,164)(139,160)(140,162)(141,161)(142,157)(143,159)(144,158)
(145,154)(146,156)(147,155)(169,172)(170,174)(171,173)(175,187)(176,189)
(177,188)(178,184)(179,186)(180,185)(182,183)(190,214)(191,216)(192,215)
(193,211)(194,213)(195,212)(196,229)(197,231)(198,230)(199,226)(200,228)
(201,227)(202,223)(203,225)(204,224)(205,220)(206,222)(207,221)(208,217)
(209,219)(210,218)(232,235)(233,237)(234,236)(238,250)(239,252)(240,251)
(241,247)(242,249)(243,248)(245,246)(253,277)(254,279)(255,278)(256,274)
(257,276)(258,275)(259,292)(260,294)(261,293)(262,289)(263,291)(264,290)
(265,286)(266,288)(267,287)(268,283)(269,285)(270,284)(271,280)(272,282)
(273,281)(295,298)(296,300)(297,299)(301,313)(302,315)(303,314)(304,310)
(305,312)(306,311)(308,309)(316,340)(317,342)(318,341)(319,337)(320,339)
(321,338)(322,355)(323,357)(324,356)(325,352)(326,354)(327,353)(328,349)
(329,351)(330,350)(331,346)(332,348)(333,347)(334,343)(335,345)(336,344)
(358,361)(359,363)(360,362)(364,376)(365,378)(366,377)(367,373)(368,375)
(369,374)(371,372);;
s2 := (  1, 64)(  2, 66)(  3, 65)(  4, 67)(  5, 69)(  6, 68)(  7, 70)(  8, 72)
(  9, 71)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)( 16, 79)
( 17, 81)( 18, 80)( 19, 82)( 20, 84)( 21, 83)( 22, 86)( 23, 85)( 24, 87)
( 25, 89)( 26, 88)( 27, 90)( 28, 92)( 29, 91)( 30, 93)( 31, 95)( 32, 94)
( 33, 96)( 34, 98)( 35, 97)( 36, 99)( 37,101)( 38,100)( 39,102)( 40,104)
( 41,103)( 42,105)( 43,108)( 44,107)( 45,106)( 46,111)( 47,110)( 48,109)
( 49,114)( 50,113)( 51,112)( 52,117)( 53,116)( 54,115)( 55,120)( 56,119)
( 57,118)( 58,123)( 59,122)( 60,121)( 61,126)( 62,125)( 63,124)(128,129)
(131,132)(134,135)(137,138)(140,141)(143,144)(146,147)(148,149)(151,152)
(154,155)(157,158)(160,161)(163,164)(166,167)(169,171)(172,174)(175,177)
(178,180)(181,183)(184,186)(187,189)(190,253)(191,255)(192,254)(193,256)
(194,258)(195,257)(196,259)(197,261)(198,260)(199,262)(200,264)(201,263)
(202,265)(203,267)(204,266)(205,268)(206,270)(207,269)(208,271)(209,273)
(210,272)(211,275)(212,274)(213,276)(214,278)(215,277)(216,279)(217,281)
(218,280)(219,282)(220,284)(221,283)(222,285)(223,287)(224,286)(225,288)
(226,290)(227,289)(228,291)(229,293)(230,292)(231,294)(232,297)(233,296)
(234,295)(235,300)(236,299)(237,298)(238,303)(239,302)(240,301)(241,306)
(242,305)(243,304)(244,309)(245,308)(246,307)(247,312)(248,311)(249,310)
(250,315)(251,314)(252,313)(317,318)(320,321)(323,324)(326,327)(329,330)
(332,333)(335,336)(337,338)(340,341)(343,344)(346,347)(349,350)(352,353)
(355,356)(358,360)(361,363)(364,366)(367,369)(370,372)(373,375)(376,378);;
s3 := (  1,190)(  2,192)(  3,191)(  4,193)(  5,195)(  6,194)(  7,196)(  8,198)
(  9,197)( 10,199)( 11,201)( 12,200)( 13,202)( 14,204)( 15,203)( 16,205)
( 17,207)( 18,206)( 19,208)( 20,210)( 21,209)( 22,211)( 23,213)( 24,212)
( 25,214)( 26,216)( 27,215)( 28,217)( 29,219)( 30,218)( 31,220)( 32,222)
( 33,221)( 34,223)( 35,225)( 36,224)( 37,226)( 38,228)( 39,227)( 40,229)
( 41,231)( 42,230)( 43,232)( 44,234)( 45,233)( 46,235)( 47,237)( 48,236)
( 49,238)( 50,240)( 51,239)( 52,241)( 53,243)( 54,242)( 55,244)( 56,246)
( 57,245)( 58,247)( 59,249)( 60,248)( 61,250)( 62,252)( 63,251)( 64,316)
( 65,318)( 66,317)( 67,319)( 68,321)( 69,320)( 70,322)( 71,324)( 72,323)
( 73,325)( 74,327)( 75,326)( 76,328)( 77,330)( 78,329)( 79,331)( 80,333)
( 81,332)( 82,334)( 83,336)( 84,335)( 85,337)( 86,339)( 87,338)( 88,340)
( 89,342)( 90,341)( 91,343)( 92,345)( 93,344)( 94,346)( 95,348)( 96,347)
( 97,349)( 98,351)( 99,350)(100,352)(101,354)(102,353)(103,355)(104,357)
(105,356)(106,358)(107,360)(108,359)(109,361)(110,363)(111,362)(112,364)
(113,366)(114,365)(115,367)(116,369)(117,368)(118,370)(119,372)(120,371)
(121,373)(122,375)(123,374)(124,376)(125,378)(126,377)(127,253)(128,255)
(129,254)(130,256)(131,258)(132,257)(133,259)(134,261)(135,260)(136,262)
(137,264)(138,263)(139,265)(140,267)(141,266)(142,268)(143,270)(144,269)
(145,271)(146,273)(147,272)(148,274)(149,276)(150,275)(151,277)(152,279)
(153,278)(154,280)(155,282)(156,281)(157,283)(158,285)(159,284)(160,286)
(161,288)(162,287)(163,289)(164,291)(165,290)(166,292)(167,294)(168,293)
(169,295)(170,297)(171,296)(172,298)(173,300)(174,299)(175,301)(176,303)
(177,302)(178,304)(179,306)(180,305)(181,307)(182,309)(183,308)(184,310)
(185,312)(186,311)(187,313)(188,315)(189,314);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(378)!(  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)
( 10, 13)( 11, 15)( 12, 14)( 22, 43)( 23, 45)( 24, 44)( 25, 61)( 26, 63)
( 27, 62)( 28, 58)( 29, 60)( 30, 59)( 31, 55)( 32, 57)( 33, 56)( 34, 52)
( 35, 54)( 36, 53)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 65, 66)( 67, 82)( 68, 84)( 69, 83)( 70, 79)( 71, 81)( 72, 80)( 73, 76)
( 74, 78)( 75, 77)( 85,106)( 86,108)( 87,107)( 88,124)( 89,126)( 90,125)
( 91,121)( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,115)( 98,117)
( 99,116)(100,112)(101,114)(102,113)(103,109)(104,111)(105,110)(128,129)
(130,145)(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)(137,141)
(138,140)(148,169)(149,171)(150,170)(151,187)(152,189)(153,188)(154,184)
(155,186)(156,185)(157,181)(158,183)(159,182)(160,178)(161,180)(162,179)
(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(191,192)(193,208)
(194,210)(195,209)(196,205)(197,207)(198,206)(199,202)(200,204)(201,203)
(211,232)(212,234)(213,233)(214,250)(215,252)(216,251)(217,247)(218,249)
(219,248)(220,244)(221,246)(222,245)(223,241)(224,243)(225,242)(226,238)
(227,240)(228,239)(229,235)(230,237)(231,236)(254,255)(256,271)(257,273)
(258,272)(259,268)(260,270)(261,269)(262,265)(263,267)(264,266)(274,295)
(275,297)(276,296)(277,313)(278,315)(279,314)(280,310)(281,312)(282,311)
(283,307)(284,309)(285,308)(286,304)(287,306)(288,305)(289,301)(290,303)
(291,302)(292,298)(293,300)(294,299)(317,318)(319,334)(320,336)(321,335)
(322,331)(323,333)(324,332)(325,328)(326,330)(327,329)(337,358)(338,360)
(339,359)(340,376)(341,378)(342,377)(343,373)(344,375)(345,374)(346,370)
(347,372)(348,371)(349,367)(350,369)(351,368)(352,364)(353,366)(354,365)
(355,361)(356,363)(357,362);
s1 := Sym(378)!(  1, 25)(  2, 27)(  3, 26)(  4, 22)(  5, 24)(  6, 23)(  7, 40)
(  8, 42)(  9, 41)( 10, 37)( 11, 39)( 12, 38)( 13, 34)( 14, 36)( 15, 35)
( 16, 31)( 17, 33)( 18, 32)( 19, 28)( 20, 30)( 21, 29)( 43, 46)( 44, 48)
( 45, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 58)( 53, 60)( 54, 59)( 56, 57)
( 64, 88)( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70,103)( 71,105)
( 72,104)( 73,100)( 74,102)( 75,101)( 76, 97)( 77, 99)( 78, 98)( 79, 94)
( 80, 96)( 81, 95)( 82, 91)( 83, 93)( 84, 92)(106,109)(107,111)(108,110)
(112,124)(113,126)(114,125)(115,121)(116,123)(117,122)(119,120)(127,151)
(128,153)(129,152)(130,148)(131,150)(132,149)(133,166)(134,168)(135,167)
(136,163)(137,165)(138,164)(139,160)(140,162)(141,161)(142,157)(143,159)
(144,158)(145,154)(146,156)(147,155)(169,172)(170,174)(171,173)(175,187)
(176,189)(177,188)(178,184)(179,186)(180,185)(182,183)(190,214)(191,216)
(192,215)(193,211)(194,213)(195,212)(196,229)(197,231)(198,230)(199,226)
(200,228)(201,227)(202,223)(203,225)(204,224)(205,220)(206,222)(207,221)
(208,217)(209,219)(210,218)(232,235)(233,237)(234,236)(238,250)(239,252)
(240,251)(241,247)(242,249)(243,248)(245,246)(253,277)(254,279)(255,278)
(256,274)(257,276)(258,275)(259,292)(260,294)(261,293)(262,289)(263,291)
(264,290)(265,286)(266,288)(267,287)(268,283)(269,285)(270,284)(271,280)
(272,282)(273,281)(295,298)(296,300)(297,299)(301,313)(302,315)(303,314)
(304,310)(305,312)(306,311)(308,309)(316,340)(317,342)(318,341)(319,337)
(320,339)(321,338)(322,355)(323,357)(324,356)(325,352)(326,354)(327,353)
(328,349)(329,351)(330,350)(331,346)(332,348)(333,347)(334,343)(335,345)
(336,344)(358,361)(359,363)(360,362)(364,376)(365,378)(366,377)(367,373)
(368,375)(369,374)(371,372);
s2 := Sym(378)!(  1, 64)(  2, 66)(  3, 65)(  4, 67)(  5, 69)(  6, 68)(  7, 70)
(  8, 72)(  9, 71)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)
( 16, 79)( 17, 81)( 18, 80)( 19, 82)( 20, 84)( 21, 83)( 22, 86)( 23, 85)
( 24, 87)( 25, 89)( 26, 88)( 27, 90)( 28, 92)( 29, 91)( 30, 93)( 31, 95)
( 32, 94)( 33, 96)( 34, 98)( 35, 97)( 36, 99)( 37,101)( 38,100)( 39,102)
( 40,104)( 41,103)( 42,105)( 43,108)( 44,107)( 45,106)( 46,111)( 47,110)
( 48,109)( 49,114)( 50,113)( 51,112)( 52,117)( 53,116)( 54,115)( 55,120)
( 56,119)( 57,118)( 58,123)( 59,122)( 60,121)( 61,126)( 62,125)( 63,124)
(128,129)(131,132)(134,135)(137,138)(140,141)(143,144)(146,147)(148,149)
(151,152)(154,155)(157,158)(160,161)(163,164)(166,167)(169,171)(172,174)
(175,177)(178,180)(181,183)(184,186)(187,189)(190,253)(191,255)(192,254)
(193,256)(194,258)(195,257)(196,259)(197,261)(198,260)(199,262)(200,264)
(201,263)(202,265)(203,267)(204,266)(205,268)(206,270)(207,269)(208,271)
(209,273)(210,272)(211,275)(212,274)(213,276)(214,278)(215,277)(216,279)
(217,281)(218,280)(219,282)(220,284)(221,283)(222,285)(223,287)(224,286)
(225,288)(226,290)(227,289)(228,291)(229,293)(230,292)(231,294)(232,297)
(233,296)(234,295)(235,300)(236,299)(237,298)(238,303)(239,302)(240,301)
(241,306)(242,305)(243,304)(244,309)(245,308)(246,307)(247,312)(248,311)
(249,310)(250,315)(251,314)(252,313)(317,318)(320,321)(323,324)(326,327)
(329,330)(332,333)(335,336)(337,338)(340,341)(343,344)(346,347)(349,350)
(352,353)(355,356)(358,360)(361,363)(364,366)(367,369)(370,372)(373,375)
(376,378);
s3 := Sym(378)!(  1,190)(  2,192)(  3,191)(  4,193)(  5,195)(  6,194)(  7,196)
(  8,198)(  9,197)( 10,199)( 11,201)( 12,200)( 13,202)( 14,204)( 15,203)
( 16,205)( 17,207)( 18,206)( 19,208)( 20,210)( 21,209)( 22,211)( 23,213)
( 24,212)( 25,214)( 26,216)( 27,215)( 28,217)( 29,219)( 30,218)( 31,220)
( 32,222)( 33,221)( 34,223)( 35,225)( 36,224)( 37,226)( 38,228)( 39,227)
( 40,229)( 41,231)( 42,230)( 43,232)( 44,234)( 45,233)( 46,235)( 47,237)
( 48,236)( 49,238)( 50,240)( 51,239)( 52,241)( 53,243)( 54,242)( 55,244)
( 56,246)( 57,245)( 58,247)( 59,249)( 60,248)( 61,250)( 62,252)( 63,251)
( 64,316)( 65,318)( 66,317)( 67,319)( 68,321)( 69,320)( 70,322)( 71,324)
( 72,323)( 73,325)( 74,327)( 75,326)( 76,328)( 77,330)( 78,329)( 79,331)
( 80,333)( 81,332)( 82,334)( 83,336)( 84,335)( 85,337)( 86,339)( 87,338)
( 88,340)( 89,342)( 90,341)( 91,343)( 92,345)( 93,344)( 94,346)( 95,348)
( 96,347)( 97,349)( 98,351)( 99,350)(100,352)(101,354)(102,353)(103,355)
(104,357)(105,356)(106,358)(107,360)(108,359)(109,361)(110,363)(111,362)
(112,364)(113,366)(114,365)(115,367)(116,369)(117,368)(118,370)(119,372)
(120,371)(121,373)(122,375)(123,374)(124,376)(125,378)(126,377)(127,253)
(128,255)(129,254)(130,256)(131,258)(132,257)(133,259)(134,261)(135,260)
(136,262)(137,264)(138,263)(139,265)(140,267)(141,266)(142,268)(143,270)
(144,269)(145,271)(146,273)(147,272)(148,274)(149,276)(150,275)(151,277)
(152,279)(153,278)(154,280)(155,282)(156,281)(157,283)(158,285)(159,284)
(160,286)(161,288)(162,287)(163,289)(164,291)(165,290)(166,292)(167,294)
(168,293)(169,295)(170,297)(171,296)(172,298)(173,300)(174,299)(175,301)
(176,303)(177,302)(178,304)(179,306)(180,305)(181,307)(182,309)(183,308)
(184,310)(185,312)(186,311)(187,313)(188,315)(189,314);
poly := sub<Sym(378)|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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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