Polytope of Type {123,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {123,6}*1476
if this polytope has a name.
Group : SmallGroup(1476,28)
Rank : 3
Schlafli Type : {123,6}
Number of vertices, edges, etc : 123, 369, 6
Order of s₀s₁s₂ : 246
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) :
   3-fold quotients : {123,2}*492
   9-fold quotients : {41,2}*164
   41-fold quotients : {3,6}*36
   123-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope