Polytope of Type {42,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,6}*1512c
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 : {42,6}*756
   3-fold quotients : {42,6}*504b
   7-fold quotients : {6,6}*216b
   9-fold quotients : {14,6}*168, {42,2}*168
   14-fold quotients : {6,6}*108
   18-fold quotients : {21,2}*84
   21-fold quotients : {6,6}*72a
   27-fold quotients : {14,2}*56
   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 := (  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)( 10, 13)
( 11, 15)( 12, 14)( 22, 43)( 23, 45)( 24, 44)( 25, 61)( 26, 63)( 27, 62)
( 28, 58)( 29, 60)( 30, 59)( 31, 55)( 32, 57)( 33, 56)( 34, 52)( 35, 54)
( 36, 53)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 65, 66)
( 67, 82)( 68, 84)( 69, 83)( 70, 79)( 71, 81)( 72, 80)( 73, 76)( 74, 78)
( 75, 77)( 85,106)( 86,108)( 87,107)( 88,124)( 89,126)( 90,125)( 91,121)
( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,115)( 98,117)( 99,116)
(100,112)(101,114)(102,113)(103,109)(104,111)(105,110)(128,129)(130,145)
(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)(137,141)(138,140)
(148,169)(149,171)(150,170)(151,187)(152,189)(153,188)(154,184)(155,186)
(156,185)(157,181)(158,183)(159,182)(160,178)(161,180)(162,179)(163,175)
(164,177)(165,176)(166,172)(167,174)(168,173)(191,192)(193,208)(194,210)
(195,209)(196,205)(197,207)(198,206)(199,202)(200,204)(201,203)(211,232)
(212,234)(213,233)(214,250)(215,252)(216,251)(217,247)(218,249)(219,248)
(220,244)(221,246)(222,245)(223,241)(224,243)(225,242)(226,238)(227,240)
(228,239)(229,235)(230,237)(231,236)(254,255)(256,271)(257,273)(258,272)
(259,268)(260,270)(261,269)(262,265)(263,267)(264,266)(274,295)(275,297)
(276,296)(277,313)(278,315)(279,314)(280,310)(281,312)(282,311)(283,307)
(284,309)(285,308)(286,304)(287,306)(288,305)(289,301)(290,303)(291,302)
(292,298)(293,300)(294,299)(317,318)(319,334)(320,336)(321,335)(322,331)
(323,333)(324,332)(325,328)(326,330)(327,329)(337,358)(338,360)(339,359)
(340,376)(341,378)(342,377)(343,373)(344,375)(345,374)(346,370)(347,372)
(348,371)(349,367)(350,369)(351,368)(352,364)(353,366)(354,365)(355,361)
(356,363)(357,362);;
s1 := (  1, 25)(  2, 26)(  3, 27)(  4, 22)(  5, 23)(  6, 24)(  7, 40)(  8, 41)
(  9, 42)( 10, 37)( 11, 38)( 12, 39)( 13, 34)( 14, 35)( 15, 36)( 16, 31)
( 17, 32)( 18, 33)( 19, 28)( 20, 29)( 21, 30)( 43, 46)( 44, 47)( 45, 48)
( 49, 61)( 50, 62)( 51, 63)( 52, 58)( 53, 59)( 54, 60)( 64,151)( 65,152)
( 66,153)( 67,148)( 68,149)( 69,150)( 70,166)( 71,167)( 72,168)( 73,163)
( 74,164)( 75,165)( 76,160)( 77,161)( 78,162)( 79,157)( 80,158)( 81,159)
( 82,154)( 83,155)( 84,156)( 85,130)( 86,131)( 87,132)( 88,127)( 89,128)
( 90,129)( 91,145)( 92,146)( 93,147)( 94,142)( 95,143)( 96,144)( 97,139)
( 98,140)( 99,141)(100,136)(101,137)(102,138)(103,133)(104,134)(105,135)
(106,172)(107,173)(108,174)(109,169)(110,170)(111,171)(112,187)(113,188)
(114,189)(115,184)(116,185)(117,186)(118,181)(119,182)(120,183)(121,178)
(122,179)(123,180)(124,175)(125,176)(126,177)(190,214)(191,215)(192,216)
(193,211)(194,212)(195,213)(196,229)(197,230)(198,231)(199,226)(200,227)
(201,228)(202,223)(203,224)(204,225)(205,220)(206,221)(207,222)(208,217)
(209,218)(210,219)(232,235)(233,236)(234,237)(238,250)(239,251)(240,252)
(241,247)(242,248)(243,249)(253,340)(254,341)(255,342)(256,337)(257,338)
(258,339)(259,355)(260,356)(261,357)(262,352)(263,353)(264,354)(265,349)
(266,350)(267,351)(268,346)(269,347)(270,348)(271,343)(272,344)(273,345)
(274,319)(275,320)(276,321)(277,316)(278,317)(279,318)(280,334)(281,335)
(282,336)(283,331)(284,332)(285,333)(286,328)(287,329)(288,330)(289,325)
(290,326)(291,327)(292,322)(293,323)(294,324)(295,361)(296,362)(297,363)
(298,358)(299,359)(300,360)(301,376)(302,377)(303,378)(304,373)(305,374)
(306,375)(307,370)(308,371)(309,372)(310,367)(311,368)(312,369)(313,364)
(314,365)(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)( 22,275)( 23,274)( 24,276)
( 25,278)( 26,277)( 27,279)( 28,281)( 29,280)( 30,282)( 31,284)( 32,283)
( 33,285)( 34,287)( 35,286)( 36,288)( 37,290)( 38,289)( 39,291)( 40,293)
( 41,292)( 42,294)( 43,297)( 44,296)( 45,295)( 46,300)( 47,299)( 48,298)
( 49,303)( 50,302)( 51,301)( 52,306)( 53,305)( 54,304)( 55,309)( 56,308)
( 57,307)( 58,312)( 59,311)( 60,310)( 61,315)( 62,314)( 63,313)( 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)( 85,212)( 86,211)( 87,213)( 88,215)
( 89,214)( 90,216)( 91,218)( 92,217)( 93,219)( 94,221)( 95,220)( 96,222)
( 97,224)( 98,223)( 99,225)(100,227)(101,226)(102,228)(103,230)(104,229)
(105,231)(106,234)(107,233)(108,232)(109,237)(110,236)(111,235)(112,240)
(113,239)(114,238)(115,243)(116,242)(117,241)(118,246)(119,245)(120,244)
(121,249)(122,248)(123,247)(124,252)(125,251)(126,250)(127,316)(128,318)
(129,317)(130,319)(131,321)(132,320)(133,322)(134,324)(135,323)(136,325)
(137,327)(138,326)(139,328)(140,330)(141,329)(142,331)(143,333)(144,332)
(145,334)(146,336)(147,335)(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)
(169,360)(170,359)(171,358)(172,363)(173,362)(174,361)(175,366)(176,365)
(177,364)(178,369)(179,368)(180,367)(181,372)(182,371)(183,370)(184,375)
(185,374)(186,373)(187,378)(188,377)(189,376);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*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*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)!(  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)
( 10, 13)( 11, 15)( 12, 14)( 22, 43)( 23, 45)( 24, 44)( 25, 61)( 26, 63)
( 27, 62)( 28, 58)( 29, 60)( 30, 59)( 31, 55)( 32, 57)( 33, 56)( 34, 52)
( 35, 54)( 36, 53)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 65, 66)( 67, 82)( 68, 84)( 69, 83)( 70, 79)( 71, 81)( 72, 80)( 73, 76)
( 74, 78)( 75, 77)( 85,106)( 86,108)( 87,107)( 88,124)( 89,126)( 90,125)
( 91,121)( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,115)( 98,117)
( 99,116)(100,112)(101,114)(102,113)(103,109)(104,111)(105,110)(128,129)
(130,145)(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)(137,141)
(138,140)(148,169)(149,171)(150,170)(151,187)(152,189)(153,188)(154,184)
(155,186)(156,185)(157,181)(158,183)(159,182)(160,178)(161,180)(162,179)
(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(191,192)(193,208)
(194,210)(195,209)(196,205)(197,207)(198,206)(199,202)(200,204)(201,203)
(211,232)(212,234)(213,233)(214,250)(215,252)(216,251)(217,247)(218,249)
(219,248)(220,244)(221,246)(222,245)(223,241)(224,243)(225,242)(226,238)
(227,240)(228,239)(229,235)(230,237)(231,236)(254,255)(256,271)(257,273)
(258,272)(259,268)(260,270)(261,269)(262,265)(263,267)(264,266)(274,295)
(275,297)(276,296)(277,313)(278,315)(279,314)(280,310)(281,312)(282,311)
(283,307)(284,309)(285,308)(286,304)(287,306)(288,305)(289,301)(290,303)
(291,302)(292,298)(293,300)(294,299)(317,318)(319,334)(320,336)(321,335)
(322,331)(323,333)(324,332)(325,328)(326,330)(327,329)(337,358)(338,360)
(339,359)(340,376)(341,378)(342,377)(343,373)(344,375)(345,374)(346,370)
(347,372)(348,371)(349,367)(350,369)(351,368)(352,364)(353,366)(354,365)
(355,361)(356,363)(357,362);
s1 := Sym(378)!(  1, 25)(  2, 26)(  3, 27)(  4, 22)(  5, 23)(  6, 24)(  7, 40)
(  8, 41)(  9, 42)( 10, 37)( 11, 38)( 12, 39)( 13, 34)( 14, 35)( 15, 36)
( 16, 31)( 17, 32)( 18, 33)( 19, 28)( 20, 29)( 21, 30)( 43, 46)( 44, 47)
( 45, 48)( 49, 61)( 50, 62)( 51, 63)( 52, 58)( 53, 59)( 54, 60)( 64,151)
( 65,152)( 66,153)( 67,148)( 68,149)( 69,150)( 70,166)( 71,167)( 72,168)
( 73,163)( 74,164)( 75,165)( 76,160)( 77,161)( 78,162)( 79,157)( 80,158)
( 81,159)( 82,154)( 83,155)( 84,156)( 85,130)( 86,131)( 87,132)( 88,127)
( 89,128)( 90,129)( 91,145)( 92,146)( 93,147)( 94,142)( 95,143)( 96,144)
( 97,139)( 98,140)( 99,141)(100,136)(101,137)(102,138)(103,133)(104,134)
(105,135)(106,172)(107,173)(108,174)(109,169)(110,170)(111,171)(112,187)
(113,188)(114,189)(115,184)(116,185)(117,186)(118,181)(119,182)(120,183)
(121,178)(122,179)(123,180)(124,175)(125,176)(126,177)(190,214)(191,215)
(192,216)(193,211)(194,212)(195,213)(196,229)(197,230)(198,231)(199,226)
(200,227)(201,228)(202,223)(203,224)(204,225)(205,220)(206,221)(207,222)
(208,217)(209,218)(210,219)(232,235)(233,236)(234,237)(238,250)(239,251)
(240,252)(241,247)(242,248)(243,249)(253,340)(254,341)(255,342)(256,337)
(257,338)(258,339)(259,355)(260,356)(261,357)(262,352)(263,353)(264,354)
(265,349)(266,350)(267,351)(268,346)(269,347)(270,348)(271,343)(272,344)
(273,345)(274,319)(275,320)(276,321)(277,316)(278,317)(279,318)(280,334)
(281,335)(282,336)(283,331)(284,332)(285,333)(286,328)(287,329)(288,330)
(289,325)(290,326)(291,327)(292,322)(293,323)(294,324)(295,361)(296,362)
(297,363)(298,358)(299,359)(300,360)(301,376)(302,377)(303,378)(304,373)
(305,374)(306,375)(307,370)(308,371)(309,372)(310,367)(311,368)(312,369)
(313,364)(314,365)(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)( 22,275)( 23,274)
( 24,276)( 25,278)( 26,277)( 27,279)( 28,281)( 29,280)( 30,282)( 31,284)
( 32,283)( 33,285)( 34,287)( 35,286)( 36,288)( 37,290)( 38,289)( 39,291)
( 40,293)( 41,292)( 42,294)( 43,297)( 44,296)( 45,295)( 46,300)( 47,299)
( 48,298)( 49,303)( 50,302)( 51,301)( 52,306)( 53,305)( 54,304)( 55,309)
( 56,308)( 57,307)( 58,312)( 59,311)( 60,310)( 61,315)( 62,314)( 63,313)
( 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)( 85,212)( 86,211)( 87,213)
( 88,215)( 89,214)( 90,216)( 91,218)( 92,217)( 93,219)( 94,221)( 95,220)
( 96,222)( 97,224)( 98,223)( 99,225)(100,227)(101,226)(102,228)(103,230)
(104,229)(105,231)(106,234)(107,233)(108,232)(109,237)(110,236)(111,235)
(112,240)(113,239)(114,238)(115,243)(116,242)(117,241)(118,246)(119,245)
(120,244)(121,249)(122,248)(123,247)(124,252)(125,251)(126,250)(127,316)
(128,318)(129,317)(130,319)(131,321)(132,320)(133,322)(134,324)(135,323)
(136,325)(137,327)(138,326)(139,328)(140,330)(141,329)(142,331)(143,333)
(144,332)(145,334)(146,336)(147,335)(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)(169,360)(170,359)(171,358)(172,363)(173,362)(174,361)(175,366)
(176,365)(177,364)(178,369)(179,368)(180,367)(181,372)(182,371)(183,370)
(184,375)(185,374)(186,373)(187,378)(188,377)(189,376);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*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*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