Polytope of Type {42,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,6,3}*1512a
if this polytope has a name.
Group : SmallGroup(1512,561)
Rank : 4
Schlafli Type : {42,6,3}
Number of vertices, edges, etc : 42, 126, 9, 3
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 : {42,2,3}*504
6-fold quotients : {21,2,3}*252
7-fold quotients : {6,6,3}*216a
9-fold quotients : {14,2,3}*168
14-fold quotients : {3,6,3}*108
18-fold quotients : {7,2,3}*84
21-fold quotients : {6,2,3}*72
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)( 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 := ( 1,214)( 2,216)( 3,215)( 4,211)( 5,213)( 6,212)( 7,229)( 8,231)
( 9,230)( 10,226)( 11,228)( 12,227)( 13,223)( 14,225)( 15,224)( 16,220)
( 17,222)( 18,221)( 19,217)( 20,219)( 21,218)( 22,193)( 23,195)( 24,194)
( 25,190)( 26,192)( 27,191)( 28,208)( 29,210)( 30,209)( 31,205)( 32,207)
( 33,206)( 34,202)( 35,204)( 36,203)( 37,199)( 38,201)( 39,200)( 40,196)
( 41,198)( 42,197)( 43,235)( 44,237)( 45,236)( 46,232)( 47,234)( 48,233)
( 49,250)( 50,252)( 51,251)( 52,247)( 53,249)( 54,248)( 55,244)( 56,246)
( 57,245)( 58,241)( 59,243)( 60,242)( 61,238)( 62,240)( 63,239)( 64,277)
( 65,279)( 66,278)( 67,274)( 68,276)( 69,275)( 70,292)( 71,294)( 72,293)
( 73,289)( 74,291)( 75,290)( 76,286)( 77,288)( 78,287)( 79,283)( 80,285)
( 81,284)( 82,280)( 83,282)( 84,281)( 85,256)( 86,258)( 87,257)( 88,253)
( 89,255)( 90,254)( 91,271)( 92,273)( 93,272)( 94,268)( 95,270)( 96,269)
( 97,265)( 98,267)( 99,266)(100,262)(101,264)(102,263)(103,259)(104,261)
(105,260)(106,298)(107,300)(108,299)(109,295)(110,297)(111,296)(112,313)
(113,315)(114,314)(115,310)(116,312)(117,311)(118,307)(119,309)(120,308)
(121,304)(122,306)(123,305)(124,301)(125,303)(126,302)(127,340)(128,342)
(129,341)(130,337)(131,339)(132,338)(133,355)(134,357)(135,356)(136,352)
(137,354)(138,353)(139,349)(140,351)(141,350)(142,346)(143,348)(144,347)
(145,343)(146,345)(147,344)(148,319)(149,321)(150,320)(151,316)(152,318)
(153,317)(154,334)(155,336)(156,335)(157,331)(158,333)(159,332)(160,328)
(161,330)(162,329)(163,325)(164,327)(165,326)(166,322)(167,324)(168,323)
(169,361)(170,363)(171,362)(172,358)(173,360)(174,359)(175,376)(176,378)
(177,377)(178,373)(179,375)(180,374)(181,370)(182,372)(183,371)(184,367)
(185,369)(186,368)(187,364)(188,366)(189,365);;
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 := ( 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);;
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*s3*s2*s3*s2*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*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*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)!( 1,214)( 2,216)( 3,215)( 4,211)( 5,213)( 6,212)( 7,229)
( 8,231)( 9,230)( 10,226)( 11,228)( 12,227)( 13,223)( 14,225)( 15,224)
( 16,220)( 17,222)( 18,221)( 19,217)( 20,219)( 21,218)( 22,193)( 23,195)
( 24,194)( 25,190)( 26,192)( 27,191)( 28,208)( 29,210)( 30,209)( 31,205)
( 32,207)( 33,206)( 34,202)( 35,204)( 36,203)( 37,199)( 38,201)( 39,200)
( 40,196)( 41,198)( 42,197)( 43,235)( 44,237)( 45,236)( 46,232)( 47,234)
( 48,233)( 49,250)( 50,252)( 51,251)( 52,247)( 53,249)( 54,248)( 55,244)
( 56,246)( 57,245)( 58,241)( 59,243)( 60,242)( 61,238)( 62,240)( 63,239)
( 64,277)( 65,279)( 66,278)( 67,274)( 68,276)( 69,275)( 70,292)( 71,294)
( 72,293)( 73,289)( 74,291)( 75,290)( 76,286)( 77,288)( 78,287)( 79,283)
( 80,285)( 81,284)( 82,280)( 83,282)( 84,281)( 85,256)( 86,258)( 87,257)
( 88,253)( 89,255)( 90,254)( 91,271)( 92,273)( 93,272)( 94,268)( 95,270)
( 96,269)( 97,265)( 98,267)( 99,266)(100,262)(101,264)(102,263)(103,259)
(104,261)(105,260)(106,298)(107,300)(108,299)(109,295)(110,297)(111,296)
(112,313)(113,315)(114,314)(115,310)(116,312)(117,311)(118,307)(119,309)
(120,308)(121,304)(122,306)(123,305)(124,301)(125,303)(126,302)(127,340)
(128,342)(129,341)(130,337)(131,339)(132,338)(133,355)(134,357)(135,356)
(136,352)(137,354)(138,353)(139,349)(140,351)(141,350)(142,346)(143,348)
(144,347)(145,343)(146,345)(147,344)(148,319)(149,321)(150,320)(151,316)
(152,318)(153,317)(154,334)(155,336)(156,335)(157,331)(158,333)(159,332)
(160,328)(161,330)(162,329)(163,325)(164,327)(165,326)(166,322)(167,324)
(168,323)(169,361)(170,363)(171,362)(172,358)(173,360)(174,359)(175,376)
(176,378)(177,377)(178,373)(179,375)(180,374)(181,370)(182,372)(183,371)
(184,367)(185,369)(186,368)(187,364)(188,366)(189,365);
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)!( 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);
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*s3*s2*s3*s2*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*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*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