Polytope of Type {6,126}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,126}*1512a
Also Known As : {6,126|2}. if this polytope has another name.
Group : SmallGroup(1512,559)
Rank : 3
Schlafli Type : {6,126}
Number of vertices, edges, etc : 6, 378, 126
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 : {2,126}*504, {6,42}*504b
6-fold quotients : {2,63}*252
7-fold quotients : {6,18}*216a
9-fold quotients : {6,14}*168, {2,42}*168
18-fold quotients : {2,21}*84
21-fold quotients : {2,18}*72, {6,6}*72a
27-fold quotients : {2,14}*56
42-fold quotients : {2,9}*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 := ( 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);;
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)( 22,107)( 23,106)( 24,108)
( 25,125)( 26,124)( 27,126)( 28,122)( 29,121)( 30,123)( 31,119)( 32,118)
( 33,120)( 34,116)( 35,115)( 36,117)( 37,113)( 38,112)( 39,114)( 40,110)
( 41,109)( 42,111)( 43, 86)( 44, 85)( 45, 87)( 46,104)( 47,103)( 48,105)
( 49,101)( 50,100)( 51,102)( 52, 98)( 53, 97)( 54, 99)( 55, 95)( 56, 94)
( 57, 96)( 58, 92)( 59, 91)( 60, 93)( 61, 89)( 62, 88)( 63, 90)(128,129)
(130,145)(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)(137,141)
(138,140)(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)(211,296)(212,295)(213,297)(214,314)(215,313)
(216,315)(217,311)(218,310)(219,312)(220,308)(221,307)(222,309)(223,305)
(224,304)(225,306)(226,302)(227,301)(228,303)(229,299)(230,298)(231,300)
(232,275)(233,274)(234,276)(235,293)(236,292)(237,294)(238,290)(239,289)
(240,291)(241,287)(242,286)(243,288)(244,284)(245,283)(246,285)(247,281)
(248,280)(249,282)(250,278)(251,277)(252,279)(317,318)(319,334)(320,336)
(321,335)(322,331)(323,333)(324,332)(325,328)(326,330)(327,329)(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)( 2,216)( 3,215)( 4,211)( 5,213)( 6,212)( 7,229)( 8,231)
( 9,230)( 10,226)( 11,228)( 12,227)( 13,223)( 14,225)( 15,224)( 16,220)
( 17,222)( 18,221)( 19,217)( 20,219)( 21,218)( 22,193)( 23,195)( 24,194)
( 25,190)( 26,192)( 27,191)( 28,208)( 29,210)( 30,209)( 31,205)( 32,207)
( 33,206)( 34,202)( 35,204)( 36,203)( 37,199)( 38,201)( 39,200)( 40,196)
( 41,198)( 42,197)( 43,236)( 44,235)( 45,237)( 46,233)( 47,232)( 48,234)
( 49,251)( 50,250)( 51,252)( 52,248)( 53,247)( 54,249)( 55,245)( 56,244)
( 57,246)( 58,242)( 59,241)( 60,243)( 61,239)( 62,238)( 63,240)( 64,277)
( 65,279)( 66,278)( 67,274)( 68,276)( 69,275)( 70,292)( 71,294)( 72,293)
( 73,289)( 74,291)( 75,290)( 76,286)( 77,288)( 78,287)( 79,283)( 80,285)
( 81,284)( 82,280)( 83,282)( 84,281)( 85,256)( 86,258)( 87,257)( 88,253)
( 89,255)( 90,254)( 91,271)( 92,273)( 93,272)( 94,268)( 95,270)( 96,269)
( 97,265)( 98,267)( 99,266)(100,262)(101,264)(102,263)(103,259)(104,261)
(105,260)(106,299)(107,298)(108,300)(109,296)(110,295)(111,297)(112,314)
(113,313)(114,315)(115,311)(116,310)(117,312)(118,308)(119,307)(120,309)
(121,305)(122,304)(123,306)(124,302)(125,301)(126,303)(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);;
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,
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*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*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)!( 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);
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)( 22,107)( 23,106)
( 24,108)( 25,125)( 26,124)( 27,126)( 28,122)( 29,121)( 30,123)( 31,119)
( 32,118)( 33,120)( 34,116)( 35,115)( 36,117)( 37,113)( 38,112)( 39,114)
( 40,110)( 41,109)( 42,111)( 43, 86)( 44, 85)( 45, 87)( 46,104)( 47,103)
( 48,105)( 49,101)( 50,100)( 51,102)( 52, 98)( 53, 97)( 54, 99)( 55, 95)
( 56, 94)( 57, 96)( 58, 92)( 59, 91)( 60, 93)( 61, 89)( 62, 88)( 63, 90)
(128,129)(130,145)(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)
(137,141)(138,140)(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)(211,296)(212,295)(213,297)(214,314)
(215,313)(216,315)(217,311)(218,310)(219,312)(220,308)(221,307)(222,309)
(223,305)(224,304)(225,306)(226,302)(227,301)(228,303)(229,299)(230,298)
(231,300)(232,275)(233,274)(234,276)(235,293)(236,292)(237,294)(238,290)
(239,289)(240,291)(241,287)(242,286)(243,288)(244,284)(245,283)(246,285)
(247,281)(248,280)(249,282)(250,278)(251,277)(252,279)(317,318)(319,334)
(320,336)(321,335)(322,331)(323,333)(324,332)(325,328)(326,330)(327,329)
(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)( 2,216)( 3,215)( 4,211)( 5,213)( 6,212)( 7,229)
( 8,231)( 9,230)( 10,226)( 11,228)( 12,227)( 13,223)( 14,225)( 15,224)
( 16,220)( 17,222)( 18,221)( 19,217)( 20,219)( 21,218)( 22,193)( 23,195)
( 24,194)( 25,190)( 26,192)( 27,191)( 28,208)( 29,210)( 30,209)( 31,205)
( 32,207)( 33,206)( 34,202)( 35,204)( 36,203)( 37,199)( 38,201)( 39,200)
( 40,196)( 41,198)( 42,197)( 43,236)( 44,235)( 45,237)( 46,233)( 47,232)
( 48,234)( 49,251)( 50,250)( 51,252)( 52,248)( 53,247)( 54,249)( 55,245)
( 56,244)( 57,246)( 58,242)( 59,241)( 60,243)( 61,239)( 62,238)( 63,240)
( 64,277)( 65,279)( 66,278)( 67,274)( 68,276)( 69,275)( 70,292)( 71,294)
( 72,293)( 73,289)( 74,291)( 75,290)( 76,286)( 77,288)( 78,287)( 79,283)
( 80,285)( 81,284)( 82,280)( 83,282)( 84,281)( 85,256)( 86,258)( 87,257)
( 88,253)( 89,255)( 90,254)( 91,271)( 92,273)( 93,272)( 94,268)( 95,270)
( 96,269)( 97,265)( 98,267)( 99,266)(100,262)(101,264)(102,263)(103,259)
(104,261)(105,260)(106,299)(107,298)(108,300)(109,296)(110,295)(111,297)
(112,314)(113,313)(114,315)(115,311)(116,310)(117,312)(118,308)(119,307)
(120,309)(121,305)(122,304)(123,306)(124,302)(125,301)(126,303)(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);
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,
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*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*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