Polytope of Type {6,126}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,126}*1512b
if this polytope has a 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 : 6
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) :
   2-fold quotients : {6,63}*756
   3-fold quotients : {2,126}*504, {6,42}*504c
   6-fold quotients : {2,63}*252, {6,21}*252
   7-fold quotients : {6,18}*216b
   9-fold quotients : {2,42}*168
   14-fold quotients : {6,9}*108
   18-fold quotients : {2,21}*84
   21-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {2,14}*56
   42-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {2,7}*28
   63-fold quotients : {2,6}*24
   126-fold quotients : {2,3}*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)( 64,340)
( 65,342)( 66,341)( 67,337)( 68,339)( 69,338)( 70,355)( 71,357)( 72,356)
( 73,352)( 74,354)( 75,353)( 76,349)( 77,351)( 78,350)( 79,346)( 80,348)
( 81,347)( 82,343)( 83,345)( 84,344)( 85,319)( 86,321)( 87,320)( 88,316)
( 89,318)( 90,317)( 91,334)( 92,336)( 93,335)( 94,331)( 95,333)( 96,332)
( 97,328)( 98,330)( 99,329)(100,325)(101,327)(102,326)(103,322)(104,324)
(105,323)(106,362)(107,361)(108,363)(109,359)(110,358)(111,360)(112,377)
(113,376)(114,378)(115,374)(116,373)(117,375)(118,371)(119,370)(120,372)
(121,368)(122,367)(123,369)(124,365)(125,364)(126,366)(127,277)(128,279)
(129,278)(130,274)(131,276)(132,275)(133,292)(134,294)(135,293)(136,289)
(137,291)(138,290)(139,286)(140,288)(141,287)(142,283)(143,285)(144,284)
(145,280)(146,282)(147,281)(148,256)(149,258)(150,257)(151,253)(152,255)
(153,254)(154,271)(155,273)(156,272)(157,268)(158,270)(159,269)(160,265)
(161,267)(162,266)(163,262)(164,264)(165,263)(166,259)(167,261)(168,260)
(169,299)(170,298)(171,300)(172,296)(173,295)(174,297)(175,314)(176,313)
(177,315)(178,311)(179,310)(180,312)(181,308)(182,307)(183,309)(184,305)
(185,304)(186,306)(187,302)(188,301)(189,303);;
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*s0*s1*s2*s0*s1*s0*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)
( 64,340)( 65,342)( 66,341)( 67,337)( 68,339)( 69,338)( 70,355)( 71,357)
( 72,356)( 73,352)( 74,354)( 75,353)( 76,349)( 77,351)( 78,350)( 79,346)
( 80,348)( 81,347)( 82,343)( 83,345)( 84,344)( 85,319)( 86,321)( 87,320)
( 88,316)( 89,318)( 90,317)( 91,334)( 92,336)( 93,335)( 94,331)( 95,333)
( 96,332)( 97,328)( 98,330)( 99,329)(100,325)(101,327)(102,326)(103,322)
(104,324)(105,323)(106,362)(107,361)(108,363)(109,359)(110,358)(111,360)
(112,377)(113,376)(114,378)(115,374)(116,373)(117,375)(118,371)(119,370)
(120,372)(121,368)(122,367)(123,369)(124,365)(125,364)(126,366)(127,277)
(128,279)(129,278)(130,274)(131,276)(132,275)(133,292)(134,294)(135,293)
(136,289)(137,291)(138,290)(139,286)(140,288)(141,287)(142,283)(143,285)
(144,284)(145,280)(146,282)(147,281)(148,256)(149,258)(150,257)(151,253)
(152,255)(153,254)(154,271)(155,273)(156,272)(157,268)(158,270)(159,269)
(160,265)(161,267)(162,266)(163,262)(164,264)(165,263)(166,259)(167,261)
(168,260)(169,299)(170,298)(171,300)(172,296)(173,295)(174,297)(175,314)
(176,313)(177,315)(178,311)(179,310)(180,312)(181,308)(182,307)(183,309)
(184,305)(185,304)(186,306)(187,302)(188,301)(189,303);
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*s0*s1*s2*s0*s1*s0*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