Polytope of Type {18,42}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,42}*1512b
Also Known As : {18,42|2}. if this polytope has another name.
Group : SmallGroup(1512,560)
Rank : 3
Schlafli Type : {18,42}
Number of vertices, edges, etc : 18, 378, 42
Order of s0s1s2 : 126
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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) :
3-fold quotients : {18,14}*504, {6,42}*504b
7-fold quotients : {18,6}*216a
9-fold quotients : {6,14}*168, {2,42}*168
18-fold quotients : {2,21}*84
21-fold quotients : {18,2}*72, {6,6}*72a
27-fold quotients : {2,14}*56
42-fold quotients : {9,2}*36
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)( 64,128)( 65,127)( 66,129)
( 67,131)( 68,130)( 69,132)( 70,134)( 71,133)( 72,135)( 73,137)( 74,136)
( 75,138)( 76,140)( 77,139)( 78,141)( 79,143)( 80,142)( 81,144)( 82,146)
( 83,145)( 84,147)( 85,149)( 86,148)( 87,150)( 88,152)( 89,151)( 90,153)
( 91,155)( 92,154)( 93,156)( 94,158)( 95,157)( 96,159)( 97,161)( 98,160)
( 99,162)(100,164)(101,163)(102,165)(103,167)(104,166)(105,168)(106,170)
(107,169)(108,171)(109,173)(110,172)(111,174)(112,176)(113,175)(114,177)
(115,179)(116,178)(117,180)(118,182)(119,181)(120,183)(121,185)(122,184)
(123,186)(124,188)(125,187)(126,189)(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)(253,317)(254,316)(255,318)(256,320)(257,319)(258,321)(259,323)
(260,322)(261,324)(262,326)(263,325)(264,327)(265,329)(266,328)(267,330)
(268,332)(269,331)(270,333)(271,335)(272,334)(273,336)(274,338)(275,337)
(276,339)(277,341)(278,340)(279,342)(280,344)(281,343)(282,345)(283,347)
(284,346)(285,348)(286,350)(287,349)(288,351)(289,353)(290,352)(291,354)
(292,356)(293,355)(294,357)(295,359)(296,358)(297,360)(298,362)(299,361)
(300,363)(301,365)(302,364)(303,366)(304,368)(305,367)(306,369)(307,371)
(308,370)(309,372)(310,374)(311,373)(312,375)(313,377)(314,376)(315,378);;
s1 := ( 1, 64)( 2, 66)( 3, 65)( 4, 82)( 5, 84)( 6, 83)( 7, 79)( 8, 81)
( 9, 80)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)( 16, 70)
( 17, 72)( 18, 71)( 19, 67)( 20, 69)( 21, 68)( 22,106)( 23,108)( 24,107)
( 25,124)( 26,126)( 27,125)( 28,121)( 29,123)( 30,122)( 31,118)( 32,120)
( 33,119)( 34,115)( 35,117)( 36,116)( 37,112)( 38,114)( 39,113)( 40,109)
( 41,111)( 42,110)( 43, 85)( 44, 87)( 45, 86)( 46,103)( 47,105)( 48,104)
( 49,100)( 50,102)( 51,101)( 52, 97)( 53, 99)( 54, 98)( 55, 94)( 56, 96)
( 57, 95)( 58, 91)( 59, 93)( 60, 92)( 61, 88)( 62, 90)( 63, 89)(127,128)
(130,146)(131,145)(132,147)(133,143)(134,142)(135,144)(136,140)(137,139)
(138,141)(148,170)(149,169)(150,171)(151,188)(152,187)(153,189)(154,185)
(155,184)(156,186)(157,182)(158,181)(159,183)(160,179)(161,178)(162,180)
(163,176)(164,175)(165,177)(166,173)(167,172)(168,174)(190,253)(191,255)
(192,254)(193,271)(194,273)(195,272)(196,268)(197,270)(198,269)(199,265)
(200,267)(201,266)(202,262)(203,264)(204,263)(205,259)(206,261)(207,260)
(208,256)(209,258)(210,257)(211,295)(212,297)(213,296)(214,313)(215,315)
(216,314)(217,310)(218,312)(219,311)(220,307)(221,309)(222,308)(223,304)
(224,306)(225,305)(226,301)(227,303)(228,302)(229,298)(230,300)(231,299)
(232,274)(233,276)(234,275)(235,292)(236,294)(237,293)(238,289)(239,291)
(240,290)(241,286)(242,288)(243,287)(244,283)(245,285)(246,284)(247,280)
(248,282)(249,281)(250,277)(251,279)(252,278)(316,317)(319,335)(320,334)
(321,336)(322,332)(323,331)(324,333)(325,329)(326,328)(327,330)(337,359)
(338,358)(339,360)(340,377)(341,376)(342,378)(343,374)(344,373)(345,375)
(346,371)(347,370)(348,372)(349,368)(350,367)(351,369)(352,365)(353,364)
(354,366)(355,362)(356,361)(357,363);;
s2 := ( 1,214)( 2,215)( 3,216)( 4,211)( 5,212)( 6,213)( 7,229)( 8,230)
( 9,231)( 10,226)( 11,227)( 12,228)( 13,223)( 14,224)( 15,225)( 16,220)
( 17,221)( 18,222)( 19,217)( 20,218)( 21,219)( 22,193)( 23,194)( 24,195)
( 25,190)( 26,191)( 27,192)( 28,208)( 29,209)( 30,210)( 31,205)( 32,206)
( 33,207)( 34,202)( 35,203)( 36,204)( 37,199)( 38,200)( 39,201)( 40,196)
( 41,197)( 42,198)( 43,235)( 44,236)( 45,237)( 46,232)( 47,233)( 48,234)
( 49,250)( 50,251)( 51,252)( 52,247)( 53,248)( 54,249)( 55,244)( 56,245)
( 57,246)( 58,241)( 59,242)( 60,243)( 61,238)( 62,239)( 63,240)( 64,277)
( 65,278)( 66,279)( 67,274)( 68,275)( 69,276)( 70,292)( 71,293)( 72,294)
( 73,289)( 74,290)( 75,291)( 76,286)( 77,287)( 78,288)( 79,283)( 80,284)
( 81,285)( 82,280)( 83,281)( 84,282)( 85,256)( 86,257)( 87,258)( 88,253)
( 89,254)( 90,255)( 91,271)( 92,272)( 93,273)( 94,268)( 95,269)( 96,270)
( 97,265)( 98,266)( 99,267)(100,262)(101,263)(102,264)(103,259)(104,260)
(105,261)(106,298)(107,299)(108,300)(109,295)(110,296)(111,297)(112,313)
(113,314)(114,315)(115,310)(116,311)(117,312)(118,307)(119,308)(120,309)
(121,304)(122,305)(123,306)(124,301)(125,302)(126,303)(127,340)(128,341)
(129,342)(130,337)(131,338)(132,339)(133,355)(134,356)(135,357)(136,352)
(137,353)(138,354)(139,349)(140,350)(141,351)(142,346)(143,347)(144,348)
(145,343)(146,344)(147,345)(148,319)(149,320)(150,321)(151,316)(152,317)
(153,318)(154,334)(155,335)(156,336)(157,331)(158,332)(159,333)(160,328)
(161,329)(162,330)(163,325)(164,326)(165,327)(166,322)(167,323)(168,324)
(169,361)(170,362)(171,363)(172,358)(173,359)(174,360)(175,376)(176,377)
(177,378)(178,373)(179,374)(180,375)(181,370)(182,371)(183,372)(184,367)
(185,368)(186,369)(187,364)(188,365)(189,366);;
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*s2*s1*s0*s1*s2*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,
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)( 64,128)( 65,127)
( 66,129)( 67,131)( 68,130)( 69,132)( 70,134)( 71,133)( 72,135)( 73,137)
( 74,136)( 75,138)( 76,140)( 77,139)( 78,141)( 79,143)( 80,142)( 81,144)
( 82,146)( 83,145)( 84,147)( 85,149)( 86,148)( 87,150)( 88,152)( 89,151)
( 90,153)( 91,155)( 92,154)( 93,156)( 94,158)( 95,157)( 96,159)( 97,161)
( 98,160)( 99,162)(100,164)(101,163)(102,165)(103,167)(104,166)(105,168)
(106,170)(107,169)(108,171)(109,173)(110,172)(111,174)(112,176)(113,175)
(114,177)(115,179)(116,178)(117,180)(118,182)(119,181)(120,183)(121,185)
(122,184)(123,186)(124,188)(125,187)(126,189)(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)(253,317)(254,316)(255,318)(256,320)(257,319)(258,321)
(259,323)(260,322)(261,324)(262,326)(263,325)(264,327)(265,329)(266,328)
(267,330)(268,332)(269,331)(270,333)(271,335)(272,334)(273,336)(274,338)
(275,337)(276,339)(277,341)(278,340)(279,342)(280,344)(281,343)(282,345)
(283,347)(284,346)(285,348)(286,350)(287,349)(288,351)(289,353)(290,352)
(291,354)(292,356)(293,355)(294,357)(295,359)(296,358)(297,360)(298,362)
(299,361)(300,363)(301,365)(302,364)(303,366)(304,368)(305,367)(306,369)
(307,371)(308,370)(309,372)(310,374)(311,373)(312,375)(313,377)(314,376)
(315,378);
s1 := Sym(378)!( 1, 64)( 2, 66)( 3, 65)( 4, 82)( 5, 84)( 6, 83)( 7, 79)
( 8, 81)( 9, 80)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)
( 16, 70)( 17, 72)( 18, 71)( 19, 67)( 20, 69)( 21, 68)( 22,106)( 23,108)
( 24,107)( 25,124)( 26,126)( 27,125)( 28,121)( 29,123)( 30,122)( 31,118)
( 32,120)( 33,119)( 34,115)( 35,117)( 36,116)( 37,112)( 38,114)( 39,113)
( 40,109)( 41,111)( 42,110)( 43, 85)( 44, 87)( 45, 86)( 46,103)( 47,105)
( 48,104)( 49,100)( 50,102)( 51,101)( 52, 97)( 53, 99)( 54, 98)( 55, 94)
( 56, 96)( 57, 95)( 58, 91)( 59, 93)( 60, 92)( 61, 88)( 62, 90)( 63, 89)
(127,128)(130,146)(131,145)(132,147)(133,143)(134,142)(135,144)(136,140)
(137,139)(138,141)(148,170)(149,169)(150,171)(151,188)(152,187)(153,189)
(154,185)(155,184)(156,186)(157,182)(158,181)(159,183)(160,179)(161,178)
(162,180)(163,176)(164,175)(165,177)(166,173)(167,172)(168,174)(190,253)
(191,255)(192,254)(193,271)(194,273)(195,272)(196,268)(197,270)(198,269)
(199,265)(200,267)(201,266)(202,262)(203,264)(204,263)(205,259)(206,261)
(207,260)(208,256)(209,258)(210,257)(211,295)(212,297)(213,296)(214,313)
(215,315)(216,314)(217,310)(218,312)(219,311)(220,307)(221,309)(222,308)
(223,304)(224,306)(225,305)(226,301)(227,303)(228,302)(229,298)(230,300)
(231,299)(232,274)(233,276)(234,275)(235,292)(236,294)(237,293)(238,289)
(239,291)(240,290)(241,286)(242,288)(243,287)(244,283)(245,285)(246,284)
(247,280)(248,282)(249,281)(250,277)(251,279)(252,278)(316,317)(319,335)
(320,334)(321,336)(322,332)(323,331)(324,333)(325,329)(326,328)(327,330)
(337,359)(338,358)(339,360)(340,377)(341,376)(342,378)(343,374)(344,373)
(345,375)(346,371)(347,370)(348,372)(349,368)(350,367)(351,369)(352,365)
(353,364)(354,366)(355,362)(356,361)(357,363);
s2 := Sym(378)!( 1,214)( 2,215)( 3,216)( 4,211)( 5,212)( 6,213)( 7,229)
( 8,230)( 9,231)( 10,226)( 11,227)( 12,228)( 13,223)( 14,224)( 15,225)
( 16,220)( 17,221)( 18,222)( 19,217)( 20,218)( 21,219)( 22,193)( 23,194)
( 24,195)( 25,190)( 26,191)( 27,192)( 28,208)( 29,209)( 30,210)( 31,205)
( 32,206)( 33,207)( 34,202)( 35,203)( 36,204)( 37,199)( 38,200)( 39,201)
( 40,196)( 41,197)( 42,198)( 43,235)( 44,236)( 45,237)( 46,232)( 47,233)
( 48,234)( 49,250)( 50,251)( 51,252)( 52,247)( 53,248)( 54,249)( 55,244)
( 56,245)( 57,246)( 58,241)( 59,242)( 60,243)( 61,238)( 62,239)( 63,240)
( 64,277)( 65,278)( 66,279)( 67,274)( 68,275)( 69,276)( 70,292)( 71,293)
( 72,294)( 73,289)( 74,290)( 75,291)( 76,286)( 77,287)( 78,288)( 79,283)
( 80,284)( 81,285)( 82,280)( 83,281)( 84,282)( 85,256)( 86,257)( 87,258)
( 88,253)( 89,254)( 90,255)( 91,271)( 92,272)( 93,273)( 94,268)( 95,269)
( 96,270)( 97,265)( 98,266)( 99,267)(100,262)(101,263)(102,264)(103,259)
(104,260)(105,261)(106,298)(107,299)(108,300)(109,295)(110,296)(111,297)
(112,313)(113,314)(114,315)(115,310)(116,311)(117,312)(118,307)(119,308)
(120,309)(121,304)(122,305)(123,306)(124,301)(125,302)(126,303)(127,340)
(128,341)(129,342)(130,337)(131,338)(132,339)(133,355)(134,356)(135,357)
(136,352)(137,353)(138,354)(139,349)(140,350)(141,351)(142,346)(143,347)
(144,348)(145,343)(146,344)(147,345)(148,319)(149,320)(150,321)(151,316)
(152,317)(153,318)(154,334)(155,335)(156,336)(157,331)(158,332)(159,333)
(160,328)(161,329)(162,330)(163,325)(164,326)(165,327)(166,322)(167,323)
(168,324)(169,361)(170,362)(171,363)(172,358)(173,359)(174,360)(175,376)
(176,377)(177,378)(178,373)(179,374)(180,375)(181,370)(182,371)(183,372)
(184,367)(185,368)(186,369)(187,364)(188,365)(189,366);
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*s2*s1*s0*s1*s2*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,
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