Polytope of Type {42,18}

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