Polytope of Type {42,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,6}*1512b
if this polytope has a name.
Group : SmallGroup(1512,561)
Rank : 3
Schlafli Type : {42,6}
Number of vertices, edges, etc : 126, 378, 18
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {21,6}*756
   3-fold quotients : {42,6}*504c
   6-fold quotients : {21,6}*252
   7-fold quotients : {6,6}*216c
   9-fold quotients : {42,2}*168
   14-fold quotients : {3,6}*108
   18-fold quotients : {21,2}*84
   21-fold quotients : {6,6}*72c
   27-fold quotients : {14,2}*56
   42-fold quotients : {3,6}*36
   54-fold quotients : {7,2}*28
   63-fold quotients : {6,2}*24
   126-fold quotients : {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)( 64,127)( 65,128)
( 66,129)( 67,145)( 68,146)( 69,147)( 70,142)( 71,143)( 72,144)( 73,139)
( 74,140)( 75,141)( 76,136)( 77,137)( 78,138)( 79,133)( 80,134)( 81,135)
( 82,130)( 83,131)( 84,132)( 85,169)( 86,170)( 87,171)( 88,187)( 89,188)
( 90,189)( 91,184)( 92,185)( 93,186)( 94,181)( 95,182)( 96,183)( 97,178)
( 98,179)( 99,180)(100,175)(101,176)(102,177)(103,172)(104,173)(105,174)
(106,148)(107,149)(108,150)(109,166)(110,167)(111,168)(112,163)(113,164)
(114,165)(115,160)(116,161)(117,162)(118,157)(119,158)(120,159)(121,154)
(122,155)(123,156)(124,151)(125,152)(126,153)(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)(253,316)(254,317)(255,318)(256,334)(257,335)
(258,336)(259,331)(260,332)(261,333)(262,328)(263,329)(264,330)(265,325)
(266,326)(267,327)(268,322)(269,323)(270,324)(271,319)(272,320)(273,321)
(274,358)(275,359)(276,360)(277,376)(278,377)(279,378)(280,373)(281,374)
(282,375)(283,370)(284,371)(285,372)(286,367)(287,368)(288,369)(289,364)
(290,365)(291,366)(292,361)(293,362)(294,363)(295,337)(296,338)(297,339)
(298,355)(299,356)(300,357)(301,352)(302,353)(303,354)(304,349)(305,350)
(306,351)(307,346)(308,347)(309,348)(310,343)(311,344)(312,345)(313,340)
(314,341)(315,342);;
s1 := (  1,341)(  2,342)(  3,340)(  4,338)(  5,339)(  6,337)(  7,356)(  8,357)
(  9,355)( 10,353)( 11,354)( 12,352)( 13,350)( 14,351)( 15,349)( 16,347)
( 17,348)( 18,346)( 19,344)( 20,345)( 21,343)( 22,321)( 23,319)( 24,320)
( 25,318)( 26,316)( 27,317)( 28,336)( 29,334)( 30,335)( 31,333)( 32,331)
( 33,332)( 34,330)( 35,328)( 36,329)( 37,327)( 38,325)( 39,326)( 40,324)
( 41,322)( 42,323)( 43,361)( 44,362)( 45,363)( 46,358)( 47,359)( 48,360)
( 49,376)( 50,377)( 51,378)( 52,373)( 53,374)( 54,375)( 55,370)( 56,371)
( 57,372)( 58,367)( 59,368)( 60,369)( 61,364)( 62,365)( 63,366)( 64,278)
( 65,279)( 66,277)( 67,275)( 68,276)( 69,274)( 70,293)( 71,294)( 72,292)
( 73,290)( 74,291)( 75,289)( 76,287)( 77,288)( 78,286)( 79,284)( 80,285)
( 81,283)( 82,281)( 83,282)( 84,280)( 85,258)( 86,256)( 87,257)( 88,255)
( 89,253)( 90,254)( 91,273)( 92,271)( 93,272)( 94,270)( 95,268)( 96,269)
( 97,267)( 98,265)( 99,266)(100,264)(101,262)(102,263)(103,261)(104,259)
(105,260)(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)(127,215)(128,216)
(129,214)(130,212)(131,213)(132,211)(133,230)(134,231)(135,229)(136,227)
(137,228)(138,226)(139,224)(140,225)(141,223)(142,221)(143,222)(144,220)
(145,218)(146,219)(147,217)(148,195)(149,193)(150,194)(151,192)(152,190)
(153,191)(154,210)(155,208)(156,209)(157,207)(158,205)(159,206)(160,204)
(161,202)(162,203)(163,201)(164,199)(165,200)(166,198)(167,196)(168,197)
(169,235)(170,236)(171,237)(172,232)(173,233)(174,234)(175,250)(176,251)
(177,252)(178,247)(179,248)(180,249)(181,244)(182,245)(183,246)(184,241)
(185,242)(186,243)(187,238)(188,239)(189,240);;
s2 := (  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]);;

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

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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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*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