Polytope of Type {126,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {126,6}*1512b
if this polytope has a name.
Group : SmallGroup(1512,559)
Rank : 3
Schlafli Type : {126,6}
Number of vertices, edges, etc : 126, 378, 6
Order of s₀s₁s₂ : 126
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {63,6}*756
   3-fold quotients : {126,2}*504, {42,6}*504c
   6-fold quotients : {63,2}*252, {21,6}*252
   7-fold quotients : {18,6}*216b
   9-fold quotients : {42,2}*168
   14-fold quotients : {9,6}*108
   18-fold quotients : {21,2}*84
   21-fold quotients : {18,2}*72, {6,6}*72c
   27-fold quotients : {14,2}*56
   42-fold quotients : {9,2}*36, {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 := (  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)( 10, 13)
( 11, 15)( 12, 14)( 22, 44)( 23, 43)( 24, 45)( 25, 62)( 26, 61)( 27, 63)
( 28, 59)( 29, 58)( 30, 60)( 31, 56)( 32, 55)( 33, 57)( 34, 53)( 35, 52)
( 36, 54)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 64,127)
( 65,129)( 66,128)( 67,145)( 68,147)( 69,146)( 70,142)( 71,144)( 72,143)
( 73,139)( 74,141)( 75,140)( 76,136)( 77,138)( 78,137)( 79,133)( 80,135)
( 81,134)( 82,130)( 83,132)( 84,131)( 85,170)( 86,169)( 87,171)( 88,188)
( 89,187)( 90,189)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)
( 97,179)( 98,178)( 99,180)(100,176)(101,175)(102,177)(103,173)(104,172)
(105,174)(106,149)(107,148)(108,150)(109,167)(110,166)(111,168)(112,164)
(113,163)(114,165)(115,161)(116,160)(117,162)(118,158)(119,157)(120,159)
(121,155)(122,154)(123,156)(124,152)(125,151)(126,153)(191,192)(193,208)
(194,210)(195,209)(196,205)(197,207)(198,206)(199,202)(200,204)(201,203)
(211,233)(212,232)(213,234)(214,251)(215,250)(216,252)(217,248)(218,247)
(219,249)(220,245)(221,244)(222,246)(223,242)(224,241)(225,243)(226,239)
(227,238)(228,240)(229,236)(230,235)(231,237)(253,316)(254,318)(255,317)
(256,334)(257,336)(258,335)(259,331)(260,333)(261,332)(262,328)(263,330)
(264,329)(265,325)(266,327)(267,326)(268,322)(269,324)(270,323)(271,319)
(272,321)(273,320)(274,359)(275,358)(276,360)(277,377)(278,376)(279,378)
(280,374)(281,373)(282,375)(283,371)(284,370)(285,372)(286,368)(287,367)
(288,369)(289,365)(290,364)(291,366)(292,362)(293,361)(294,363)(295,338)
(296,337)(297,339)(298,356)(299,355)(300,357)(301,353)(302,352)(303,354)
(304,350)(305,349)(306,351)(307,347)(308,346)(309,348)(310,344)(311,343)
(312,345)(313,341)(314,340)(315,342);;
s1 := (  1,277)(  2,279)(  3,278)(  4,274)(  5,276)(  6,275)(  7,292)(  8,294)
(  9,293)( 10,289)( 11,291)( 12,290)( 13,286)( 14,288)( 15,287)( 16,283)
( 17,285)( 18,284)( 19,280)( 20,282)( 21,281)( 22,256)( 23,258)( 24,257)
( 25,253)( 26,255)( 27,254)( 28,271)( 29,273)( 30,272)( 31,268)( 32,270)
( 33,269)( 34,265)( 35,267)( 36,266)( 37,262)( 38,264)( 39,263)( 40,259)
( 41,261)( 42,260)( 43,299)( 44,298)( 45,300)( 46,296)( 47,295)( 48,297)
( 49,314)( 50,313)( 51,315)( 52,311)( 53,310)( 54,312)( 55,308)( 56,307)
( 57,309)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)( 64,214)
( 65,216)( 66,215)( 67,211)( 68,213)( 69,212)( 70,229)( 71,231)( 72,230)
( 73,226)( 74,228)( 75,227)( 76,223)( 77,225)( 78,224)( 79,220)( 80,222)
( 81,221)( 82,217)( 83,219)( 84,218)( 85,193)( 86,195)( 87,194)( 88,190)
( 89,192)( 90,191)( 91,208)( 92,210)( 93,209)( 94,205)( 95,207)( 96,206)
( 97,202)( 98,204)( 99,203)(100,199)(101,201)(102,200)(103,196)(104,198)
(105,197)(106,236)(107,235)(108,237)(109,233)(110,232)(111,234)(112,251)
(113,250)(114,252)(115,248)(116,247)(117,249)(118,245)(119,244)(120,246)
(121,242)(122,241)(123,243)(124,239)(125,238)(126,240)(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)
(169,362)(170,361)(171,363)(172,359)(173,358)(174,360)(175,377)(176,376)
(177,378)(178,374)(179,373)(180,375)(181,371)(182,370)(183,372)(184,368)
(185,367)(186,369)(187,365)(188,364)(189,366);;
s2 := ( 64,127)( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)
( 72,135)( 73,136)( 74,137)( 75,138)( 76,139)( 77,140)( 78,141)( 79,142)
( 80,143)( 81,144)( 82,145)( 83,146)( 84,147)( 85,148)( 86,149)( 87,150)
( 88,151)( 89,152)( 90,153)( 91,154)( 92,155)( 93,156)( 94,157)( 95,158)
( 96,159)( 97,160)( 98,161)( 99,162)(100,163)(101,164)(102,165)(103,166)
(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)
(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)
(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(253,316)
(254,317)(255,318)(256,319)(257,320)(258,321)(259,322)(260,323)(261,324)
(262,325)(263,326)(264,327)(265,328)(266,329)(267,330)(268,331)(269,332)
(270,333)(271,334)(272,335)(273,336)(274,337)(275,338)(276,339)(277,340)
(278,341)(279,342)(280,343)(281,344)(282,345)(283,346)(284,347)(285,348)
(286,349)(287,350)(288,351)(289,352)(290,353)(291,354)(292,355)(293,356)
(294,357)(295,358)(296,359)(297,360)(298,361)(299,362)(300,363)(301,364)
(302,365)(303,366)(304,367)(305,368)(306,369)(307,370)(308,371)(309,372)
(310,373)(311,374)(312,375)(313,376)(314,377)(315,378);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*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*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*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, 44)( 23, 43)( 24, 45)( 25, 62)( 26, 61)
( 27, 63)( 28, 59)( 29, 58)( 30, 60)( 31, 56)( 32, 55)( 33, 57)( 34, 53)
( 35, 52)( 36, 54)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)
( 64,127)( 65,129)( 66,128)( 67,145)( 68,147)( 69,146)( 70,142)( 71,144)
( 72,143)( 73,139)( 74,141)( 75,140)( 76,136)( 77,138)( 78,137)( 79,133)
( 80,135)( 81,134)( 82,130)( 83,132)( 84,131)( 85,170)( 86,169)( 87,171)
( 88,188)( 89,187)( 90,189)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)
( 96,183)( 97,179)( 98,178)( 99,180)(100,176)(101,175)(102,177)(103,173)
(104,172)(105,174)(106,149)(107,148)(108,150)(109,167)(110,166)(111,168)
(112,164)(113,163)(114,165)(115,161)(116,160)(117,162)(118,158)(119,157)
(120,159)(121,155)(122,154)(123,156)(124,152)(125,151)(126,153)(191,192)
(193,208)(194,210)(195,209)(196,205)(197,207)(198,206)(199,202)(200,204)
(201,203)(211,233)(212,232)(213,234)(214,251)(215,250)(216,252)(217,248)
(218,247)(219,249)(220,245)(221,244)(222,246)(223,242)(224,241)(225,243)
(226,239)(227,238)(228,240)(229,236)(230,235)(231,237)(253,316)(254,318)
(255,317)(256,334)(257,336)(258,335)(259,331)(260,333)(261,332)(262,328)
(263,330)(264,329)(265,325)(266,327)(267,326)(268,322)(269,324)(270,323)
(271,319)(272,321)(273,320)(274,359)(275,358)(276,360)(277,377)(278,376)
(279,378)(280,374)(281,373)(282,375)(283,371)(284,370)(285,372)(286,368)
(287,367)(288,369)(289,365)(290,364)(291,366)(292,362)(293,361)(294,363)
(295,338)(296,337)(297,339)(298,356)(299,355)(300,357)(301,353)(302,352)
(303,354)(304,350)(305,349)(306,351)(307,347)(308,346)(309,348)(310,344)
(311,343)(312,345)(313,341)(314,340)(315,342);
s1 := Sym(378)!(  1,277)(  2,279)(  3,278)(  4,274)(  5,276)(  6,275)(  7,292)
(  8,294)(  9,293)( 10,289)( 11,291)( 12,290)( 13,286)( 14,288)( 15,287)
( 16,283)( 17,285)( 18,284)( 19,280)( 20,282)( 21,281)( 22,256)( 23,258)
( 24,257)( 25,253)( 26,255)( 27,254)( 28,271)( 29,273)( 30,272)( 31,268)
( 32,270)( 33,269)( 34,265)( 35,267)( 36,266)( 37,262)( 38,264)( 39,263)
( 40,259)( 41,261)( 42,260)( 43,299)( 44,298)( 45,300)( 46,296)( 47,295)
( 48,297)( 49,314)( 50,313)( 51,315)( 52,311)( 53,310)( 54,312)( 55,308)
( 56,307)( 57,309)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)
( 64,214)( 65,216)( 66,215)( 67,211)( 68,213)( 69,212)( 70,229)( 71,231)
( 72,230)( 73,226)( 74,228)( 75,227)( 76,223)( 77,225)( 78,224)( 79,220)
( 80,222)( 81,221)( 82,217)( 83,219)( 84,218)( 85,193)( 86,195)( 87,194)
( 88,190)( 89,192)( 90,191)( 91,208)( 92,210)( 93,209)( 94,205)( 95,207)
( 96,206)( 97,202)( 98,204)( 99,203)(100,199)(101,201)(102,200)(103,196)
(104,198)(105,197)(106,236)(107,235)(108,237)(109,233)(110,232)(111,234)
(112,251)(113,250)(114,252)(115,248)(116,247)(117,249)(118,245)(119,244)
(120,246)(121,242)(122,241)(123,243)(124,239)(125,238)(126,240)(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)(169,362)(170,361)(171,363)(172,359)(173,358)(174,360)(175,377)
(176,376)(177,378)(178,374)(179,373)(180,375)(181,371)(182,370)(183,372)
(184,368)(185,367)(186,369)(187,365)(188,364)(189,366);
s2 := Sym(378)!( 64,127)( 65,128)( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)
( 71,134)( 72,135)( 73,136)( 74,137)( 75,138)( 76,139)( 77,140)( 78,141)
( 79,142)( 80,143)( 81,144)( 82,145)( 83,146)( 84,147)( 85,148)( 86,149)
( 87,150)( 88,151)( 89,152)( 90,153)( 91,154)( 92,155)( 93,156)( 94,157)
( 95,158)( 96,159)( 97,160)( 98,161)( 99,162)(100,163)(101,164)(102,165)
(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)
(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)
(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)
(253,316)(254,317)(255,318)(256,319)(257,320)(258,321)(259,322)(260,323)
(261,324)(262,325)(263,326)(264,327)(265,328)(266,329)(267,330)(268,331)
(269,332)(270,333)(271,334)(272,335)(273,336)(274,337)(275,338)(276,339)
(277,340)(278,341)(279,342)(280,343)(281,344)(282,345)(283,346)(284,347)
(285,348)(286,349)(287,350)(288,351)(289,352)(290,353)(291,354)(292,355)
(293,356)(294,357)(295,358)(296,359)(297,360)(298,361)(299,362)(300,363)
(301,364)(302,365)(303,366)(304,367)(305,368)(306,369)(307,370)(308,371)
(309,372)(310,373)(311,374)(312,375)(313,376)(314,377)(315,378);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*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*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*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