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Polytope of Type {3,6,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,42}*1512a
if this polytope has a name.
Group : SmallGroup(1512,561)
Rank : 4
Schlafli Type : {3,6,42}
Number of vertices, edges, etc : 3, 9, 126, 42
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 : {3,2,42}*504
   6-fold quotients : {3,2,21}*252
   7-fold quotients : {3,6,6}*216a
   9-fold quotients : {3,2,14}*168
   14-fold quotients : {3,6,3}*108
   18-fold quotients : {3,2,7}*84
   21-fold quotients : {3,2,6}*72
   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)(  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, 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);;
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 := (  1,193)(  2,195)(  3,194)(  4,190)(  5,192)(  6,191)(  7,208)(  8,210)
(  9,209)( 10,205)( 11,207)( 12,206)( 13,202)( 14,204)( 15,203)( 16,199)
( 17,201)( 18,200)( 19,196)( 20,198)( 21,197)( 22,235)( 23,237)( 24,236)
( 25,232)( 26,234)( 27,233)( 28,250)( 29,252)( 30,251)( 31,247)( 32,249)
( 33,248)( 34,244)( 35,246)( 36,245)( 37,241)( 38,243)( 39,242)( 40,238)
( 41,240)( 42,239)( 43,214)( 44,216)( 45,215)( 46,211)( 47,213)( 48,212)
( 49,229)( 50,231)( 51,230)( 52,226)( 53,228)( 54,227)( 55,223)( 56,225)
( 57,224)( 58,220)( 59,222)( 60,221)( 61,217)( 62,219)( 63,218)( 64,256)
( 65,258)( 66,257)( 67,253)( 68,255)( 69,254)( 70,271)( 71,273)( 72,272)
( 73,268)( 74,270)( 75,269)( 76,265)( 77,267)( 78,266)( 79,262)( 80,264)
( 81,263)( 82,259)( 83,261)( 84,260)( 85,298)( 86,300)( 87,299)( 88,295)
( 89,297)( 90,296)( 91,313)( 92,315)( 93,314)( 94,310)( 95,312)( 96,311)
( 97,307)( 98,309)( 99,308)(100,304)(101,306)(102,305)(103,301)(104,303)
(105,302)(106,277)(107,279)(108,278)(109,274)(110,276)(111,275)(112,292)
(113,294)(114,293)(115,289)(116,291)(117,290)(118,286)(119,288)(120,287)
(121,283)(122,285)(123,284)(124,280)(125,282)(126,281)(127,319)(128,321)
(129,320)(130,316)(131,318)(132,317)(133,334)(134,336)(135,335)(136,331)
(137,333)(138,332)(139,328)(140,330)(141,329)(142,325)(143,327)(144,326)
(145,322)(146,324)(147,323)(148,361)(149,363)(150,362)(151,358)(152,360)
(153,359)(154,376)(155,378)(156,377)(157,373)(158,375)(159,374)(160,370)
(161,372)(162,371)(163,367)(164,369)(165,368)(166,364)(167,366)(168,365)
(169,340)(170,342)(171,341)(172,337)(173,339)(174,338)(175,355)(176,357)
(177,356)(178,352)(179,354)(180,353)(181,349)(182,351)(183,350)(184,346)
(185,348)(186,347)(187,343)(188,345)(189,344);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*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*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, 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);
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)!(  1,193)(  2,195)(  3,194)(  4,190)(  5,192)(  6,191)(  7,208)
(  8,210)(  9,209)( 10,205)( 11,207)( 12,206)( 13,202)( 14,204)( 15,203)
( 16,199)( 17,201)( 18,200)( 19,196)( 20,198)( 21,197)( 22,235)( 23,237)
( 24,236)( 25,232)( 26,234)( 27,233)( 28,250)( 29,252)( 30,251)( 31,247)
( 32,249)( 33,248)( 34,244)( 35,246)( 36,245)( 37,241)( 38,243)( 39,242)
( 40,238)( 41,240)( 42,239)( 43,214)( 44,216)( 45,215)( 46,211)( 47,213)
( 48,212)( 49,229)( 50,231)( 51,230)( 52,226)( 53,228)( 54,227)( 55,223)
( 56,225)( 57,224)( 58,220)( 59,222)( 60,221)( 61,217)( 62,219)( 63,218)
( 64,256)( 65,258)( 66,257)( 67,253)( 68,255)( 69,254)( 70,271)( 71,273)
( 72,272)( 73,268)( 74,270)( 75,269)( 76,265)( 77,267)( 78,266)( 79,262)
( 80,264)( 81,263)( 82,259)( 83,261)( 84,260)( 85,298)( 86,300)( 87,299)
( 88,295)( 89,297)( 90,296)( 91,313)( 92,315)( 93,314)( 94,310)( 95,312)
( 96,311)( 97,307)( 98,309)( 99,308)(100,304)(101,306)(102,305)(103,301)
(104,303)(105,302)(106,277)(107,279)(108,278)(109,274)(110,276)(111,275)
(112,292)(113,294)(114,293)(115,289)(116,291)(117,290)(118,286)(119,288)
(120,287)(121,283)(122,285)(123,284)(124,280)(125,282)(126,281)(127,319)
(128,321)(129,320)(130,316)(131,318)(132,317)(133,334)(134,336)(135,335)
(136,331)(137,333)(138,332)(139,328)(140,330)(141,329)(142,325)(143,327)
(144,326)(145,322)(146,324)(147,323)(148,361)(149,363)(150,362)(151,358)
(152,360)(153,359)(154,376)(155,378)(156,377)(157,373)(158,375)(159,374)
(160,370)(161,372)(162,371)(163,367)(164,369)(165,368)(166,364)(167,366)
(168,365)(169,340)(170,342)(171,341)(172,337)(173,339)(174,338)(175,355)
(176,357)(177,356)(178,352)(179,354)(180,353)(181,349)(182,351)(183,350)
(184,346)(185,348)(186,347)(187,343)(188,345)(189,344);
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, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*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*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