Polytope of Type {6,42}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,42}*1512c
if this polytope has a name.
Group : SmallGroup(1512,561)
Rank : 3
Schlafli Type : {6,42}
Number of vertices, edges, etc : 18, 378, 126
Order of s0s1s2 : 42
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,42}*756
3-fold quotients : {6,42}*504b
7-fold quotients : {6,6}*216b
9-fold quotients : {6,14}*168, {2,42}*168
14-fold quotients : {6,6}*108
18-fold quotients : {2,21}*84
21-fold quotients : {6,6}*72a
27-fold quotients : {2,14}*56
54-fold quotients : {2,7}*28
63-fold quotients : {2,6}*24, {6,2}*24
126-fold quotients : {2,3}*12, {3,2}*12
189-fold quotients : {2,2}*8
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, 65)( 3, 66)( 4, 82)( 5, 83)( 6, 84)( 7, 79)( 8, 80)
( 9, 81)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)( 16, 70)
( 17, 71)( 18, 72)( 19, 67)( 20, 68)( 21, 69)( 22,108)( 23,106)( 24,107)
( 25,126)( 26,124)( 27,125)( 28,123)( 29,121)( 30,122)( 31,120)( 32,118)
( 33,119)( 34,117)( 35,115)( 36,116)( 37,114)( 38,112)( 39,113)( 40,111)
( 41,109)( 42,110)( 43, 86)( 44, 87)( 45, 85)( 46,104)( 47,105)( 48,103)
( 49,101)( 50,102)( 51,100)( 52, 98)( 53, 99)( 54, 97)( 55, 95)( 56, 96)
( 57, 94)( 58, 92)( 59, 93)( 60, 91)( 61, 89)( 62, 90)( 63, 88)(130,145)
(131,146)(132,147)(133,142)(134,143)(135,144)(136,139)(137,140)(138,141)
(148,171)(149,169)(150,170)(151,189)(152,187)(153,188)(154,186)(155,184)
(156,185)(157,183)(158,181)(159,182)(160,180)(161,178)(162,179)(163,177)
(164,175)(165,176)(166,174)(167,172)(168,173)(190,253)(191,254)(192,255)
(193,271)(194,272)(195,273)(196,268)(197,269)(198,270)(199,265)(200,266)
(201,267)(202,262)(203,263)(204,264)(205,259)(206,260)(207,261)(208,256)
(209,257)(210,258)(211,297)(212,295)(213,296)(214,315)(215,313)(216,314)
(217,312)(218,310)(219,311)(220,309)(221,307)(222,308)(223,306)(224,304)
(225,305)(226,303)(227,301)(228,302)(229,300)(230,298)(231,299)(232,275)
(233,276)(234,274)(235,293)(236,294)(237,292)(238,290)(239,291)(240,289)
(241,287)(242,288)(243,286)(244,284)(245,285)(246,283)(247,281)(248,282)
(249,280)(250,278)(251,279)(252,277)(319,334)(320,335)(321,336)(322,331)
(323,332)(324,333)(325,328)(326,329)(327,330)(337,360)(338,358)(339,359)
(340,378)(341,376)(342,377)(343,375)(344,373)(345,374)(346,372)(347,370)
(348,371)(349,369)(350,367)(351,368)(352,366)(353,364)(354,365)(355,363)
(356,361)(357,362);;
s2 := ( 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);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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, 65)( 3, 66)( 4, 82)( 5, 83)( 6, 84)( 7, 79)
( 8, 80)( 9, 81)( 10, 76)( 11, 77)( 12, 78)( 13, 73)( 14, 74)( 15, 75)
( 16, 70)( 17, 71)( 18, 72)( 19, 67)( 20, 68)( 21, 69)( 22,108)( 23,106)
( 24,107)( 25,126)( 26,124)( 27,125)( 28,123)( 29,121)( 30,122)( 31,120)
( 32,118)( 33,119)( 34,117)( 35,115)( 36,116)( 37,114)( 38,112)( 39,113)
( 40,111)( 41,109)( 42,110)( 43, 86)( 44, 87)( 45, 85)( 46,104)( 47,105)
( 48,103)( 49,101)( 50,102)( 51,100)( 52, 98)( 53, 99)( 54, 97)( 55, 95)
( 56, 96)( 57, 94)( 58, 92)( 59, 93)( 60, 91)( 61, 89)( 62, 90)( 63, 88)
(130,145)(131,146)(132,147)(133,142)(134,143)(135,144)(136,139)(137,140)
(138,141)(148,171)(149,169)(150,170)(151,189)(152,187)(153,188)(154,186)
(155,184)(156,185)(157,183)(158,181)(159,182)(160,180)(161,178)(162,179)
(163,177)(164,175)(165,176)(166,174)(167,172)(168,173)(190,253)(191,254)
(192,255)(193,271)(194,272)(195,273)(196,268)(197,269)(198,270)(199,265)
(200,266)(201,267)(202,262)(203,263)(204,264)(205,259)(206,260)(207,261)
(208,256)(209,257)(210,258)(211,297)(212,295)(213,296)(214,315)(215,313)
(216,314)(217,312)(218,310)(219,311)(220,309)(221,307)(222,308)(223,306)
(224,304)(225,305)(226,303)(227,301)(228,302)(229,300)(230,298)(231,299)
(232,275)(233,276)(234,274)(235,293)(236,294)(237,292)(238,290)(239,291)
(240,289)(241,287)(242,288)(243,286)(244,284)(245,285)(246,283)(247,281)
(248,282)(249,280)(250,278)(251,279)(252,277)(319,334)(320,335)(321,336)
(322,331)(323,332)(324,333)(325,328)(326,329)(327,330)(337,360)(338,358)
(339,359)(340,378)(341,376)(342,377)(343,375)(344,373)(345,374)(346,372)
(347,370)(348,371)(349,369)(350,367)(351,368)(352,366)(353,364)(354,365)
(355,363)(356,361)(357,362);
s2 := 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);
poly := sub<Sym(378)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope