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