Polytope of Type {8,114}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,114}*1824
Also Known As : {8,114|2}. if this polytope has another name.
Group : SmallGroup(1824,975)
Rank : 3
Schlafli Type : {8,114}
Number of vertices, edges, etc : 8, 456, 114
Order of s0s1s2 : 456
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) :
2-fold quotients : {4,114}*912a
3-fold quotients : {8,38}*608
4-fold quotients : {2,114}*456
6-fold quotients : {4,38}*304
8-fold quotients : {2,57}*228
12-fold quotients : {2,38}*152
19-fold quotients : {8,6}*96
24-fold quotients : {2,19}*76
38-fold quotients : {4,6}*48a
57-fold quotients : {8,2}*32
76-fold quotients : {2,6}*24
114-fold quotients : {4,2}*16
152-fold quotients : {2,3}*12
228-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)(122,179)
(123,180)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)
(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)
(139,196)(140,197)(141,198)(142,199)(143,200)(144,201)(145,202)(146,203)
(147,204)(148,205)(149,206)(150,207)(151,208)(152,209)(153,210)(154,211)
(155,212)(156,213)(157,214)(158,215)(159,216)(160,217)(161,218)(162,219)
(163,220)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)(170,227)
(171,228)(229,343)(230,344)(231,345)(232,346)(233,347)(234,348)(235,349)
(236,350)(237,351)(238,352)(239,353)(240,354)(241,355)(242,356)(243,357)
(244,358)(245,359)(246,360)(247,361)(248,362)(249,363)(250,364)(251,365)
(252,366)(253,367)(254,368)(255,369)(256,370)(257,371)(258,372)(259,373)
(260,374)(261,375)(262,376)(263,377)(264,378)(265,379)(266,380)(267,381)
(268,382)(269,383)(270,384)(271,385)(272,386)(273,387)(274,388)(275,389)
(276,390)(277,391)(278,392)(279,393)(280,394)(281,395)(282,396)(283,397)
(284,398)(285,399)(286,400)(287,401)(288,402)(289,403)(290,404)(291,405)
(292,406)(293,407)(294,408)(295,409)(296,410)(297,411)(298,412)(299,413)
(300,414)(301,415)(302,416)(303,417)(304,418)(305,419)(306,420)(307,421)
(308,422)(309,423)(310,424)(311,425)(312,426)(313,427)(314,428)(315,429)
(316,430)(317,431)(318,432)(319,433)(320,434)(321,435)(322,436)(323,437)
(324,438)(325,439)(326,440)(327,441)(328,442)(329,443)(330,444)(331,445)
(332,446)(333,447)(334,448)(335,449)(336,450)(337,451)(338,452)(339,453)
(340,454)(341,455)(342,456);;
s1 := ( 1,229)( 2,247)( 3,246)( 4,245)( 5,244)( 6,243)( 7,242)( 8,241)
( 9,240)( 10,239)( 11,238)( 12,237)( 13,236)( 14,235)( 15,234)( 16,233)
( 17,232)( 18,231)( 19,230)( 20,267)( 21,285)( 22,284)( 23,283)( 24,282)
( 25,281)( 26,280)( 27,279)( 28,278)( 29,277)( 30,276)( 31,275)( 32,274)
( 33,273)( 34,272)( 35,271)( 36,270)( 37,269)( 38,268)( 39,248)( 40,266)
( 41,265)( 42,264)( 43,263)( 44,262)( 45,261)( 46,260)( 47,259)( 48,258)
( 49,257)( 50,256)( 51,255)( 52,254)( 53,253)( 54,252)( 55,251)( 56,250)
( 57,249)( 58,286)( 59,304)( 60,303)( 61,302)( 62,301)( 63,300)( 64,299)
( 65,298)( 66,297)( 67,296)( 68,295)( 69,294)( 70,293)( 71,292)( 72,291)
( 73,290)( 74,289)( 75,288)( 76,287)( 77,324)( 78,342)( 79,341)( 80,340)
( 81,339)( 82,338)( 83,337)( 84,336)( 85,335)( 86,334)( 87,333)( 88,332)
( 89,331)( 90,330)( 91,329)( 92,328)( 93,327)( 94,326)( 95,325)( 96,305)
( 97,323)( 98,322)( 99,321)(100,320)(101,319)(102,318)(103,317)(104,316)
(105,315)(106,314)(107,313)(108,312)(109,311)(110,310)(111,309)(112,308)
(113,307)(114,306)(115,400)(116,418)(117,417)(118,416)(119,415)(120,414)
(121,413)(122,412)(123,411)(124,410)(125,409)(126,408)(127,407)(128,406)
(129,405)(130,404)(131,403)(132,402)(133,401)(134,438)(135,456)(136,455)
(137,454)(138,453)(139,452)(140,451)(141,450)(142,449)(143,448)(144,447)
(145,446)(146,445)(147,444)(148,443)(149,442)(150,441)(151,440)(152,439)
(153,419)(154,437)(155,436)(156,435)(157,434)(158,433)(159,432)(160,431)
(161,430)(162,429)(163,428)(164,427)(165,426)(166,425)(167,424)(168,423)
(169,422)(170,421)(171,420)(172,343)(173,361)(174,360)(175,359)(176,358)
(177,357)(178,356)(179,355)(180,354)(181,353)(182,352)(183,351)(184,350)
(185,349)(186,348)(187,347)(188,346)(189,345)(190,344)(191,381)(192,399)
(193,398)(194,397)(195,396)(196,395)(197,394)(198,393)(199,392)(200,391)
(201,390)(202,389)(203,388)(204,387)(205,386)(206,385)(207,384)(208,383)
(209,382)(210,362)(211,380)(212,379)(213,378)(214,377)(215,376)(216,375)
(217,374)(218,373)(219,372)(220,371)(221,370)(222,369)(223,368)(224,367)
(225,366)(226,365)(227,364)(228,363);;
s2 := ( 1, 21)( 2, 20)( 3, 38)( 4, 37)( 5, 36)( 6, 35)( 7, 34)( 8, 33)
( 9, 32)( 10, 31)( 11, 30)( 12, 29)( 13, 28)( 14, 27)( 15, 26)( 16, 25)
( 17, 24)( 18, 23)( 19, 22)( 39, 40)( 41, 57)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58, 78)( 59, 77)( 60, 95)( 61, 94)
( 62, 93)( 63, 92)( 64, 91)( 65, 90)( 66, 89)( 67, 88)( 68, 87)( 69, 86)
( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)( 96, 97)
( 98,114)( 99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)
(115,135)(116,134)(117,152)(118,151)(119,150)(120,149)(121,148)(122,147)
(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139)
(131,138)(132,137)(133,136)(153,154)(155,171)(156,170)(157,169)(158,168)
(159,167)(160,166)(161,165)(162,164)(172,192)(173,191)(174,209)(175,208)
(176,207)(177,206)(178,205)(179,204)(180,203)(181,202)(182,201)(183,200)
(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)(210,211)
(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)(219,221)
(229,249)(230,248)(231,266)(232,265)(233,264)(234,263)(235,262)(236,261)
(237,260)(238,259)(239,258)(240,257)(241,256)(242,255)(243,254)(244,253)
(245,252)(246,251)(247,250)(267,268)(269,285)(270,284)(271,283)(272,282)
(273,281)(274,280)(275,279)(276,278)(286,306)(287,305)(288,323)(289,322)
(290,321)(291,320)(292,319)(293,318)(294,317)(295,316)(296,315)(297,314)
(298,313)(299,312)(300,311)(301,310)(302,309)(303,308)(304,307)(324,325)
(326,342)(327,341)(328,340)(329,339)(330,338)(331,337)(332,336)(333,335)
(343,363)(344,362)(345,380)(346,379)(347,378)(348,377)(349,376)(350,375)
(351,374)(352,373)(353,372)(354,371)(355,370)(356,369)(357,368)(358,367)
(359,366)(360,365)(361,364)(381,382)(383,399)(384,398)(385,397)(386,396)
(387,395)(388,394)(389,393)(390,392)(400,420)(401,419)(402,437)(403,436)
(404,435)(405,434)(406,433)(407,432)(408,431)(409,430)(410,429)(411,428)
(412,427)(413,426)(414,425)(415,424)(416,423)(417,422)(418,421)(438,439)
(440,456)(441,455)(442,454)(443,453)(444,452)(445,451)(446,450)(447,449);;
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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(456)!(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)
(122,179)(123,180)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)
(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)
(138,195)(139,196)(140,197)(141,198)(142,199)(143,200)(144,201)(145,202)
(146,203)(147,204)(148,205)(149,206)(150,207)(151,208)(152,209)(153,210)
(154,211)(155,212)(156,213)(157,214)(158,215)(159,216)(160,217)(161,218)
(162,219)(163,220)(164,221)(165,222)(166,223)(167,224)(168,225)(169,226)
(170,227)(171,228)(229,343)(230,344)(231,345)(232,346)(233,347)(234,348)
(235,349)(236,350)(237,351)(238,352)(239,353)(240,354)(241,355)(242,356)
(243,357)(244,358)(245,359)(246,360)(247,361)(248,362)(249,363)(250,364)
(251,365)(252,366)(253,367)(254,368)(255,369)(256,370)(257,371)(258,372)
(259,373)(260,374)(261,375)(262,376)(263,377)(264,378)(265,379)(266,380)
(267,381)(268,382)(269,383)(270,384)(271,385)(272,386)(273,387)(274,388)
(275,389)(276,390)(277,391)(278,392)(279,393)(280,394)(281,395)(282,396)
(283,397)(284,398)(285,399)(286,400)(287,401)(288,402)(289,403)(290,404)
(291,405)(292,406)(293,407)(294,408)(295,409)(296,410)(297,411)(298,412)
(299,413)(300,414)(301,415)(302,416)(303,417)(304,418)(305,419)(306,420)
(307,421)(308,422)(309,423)(310,424)(311,425)(312,426)(313,427)(314,428)
(315,429)(316,430)(317,431)(318,432)(319,433)(320,434)(321,435)(322,436)
(323,437)(324,438)(325,439)(326,440)(327,441)(328,442)(329,443)(330,444)
(331,445)(332,446)(333,447)(334,448)(335,449)(336,450)(337,451)(338,452)
(339,453)(340,454)(341,455)(342,456);
s1 := Sym(456)!( 1,229)( 2,247)( 3,246)( 4,245)( 5,244)( 6,243)( 7,242)
( 8,241)( 9,240)( 10,239)( 11,238)( 12,237)( 13,236)( 14,235)( 15,234)
( 16,233)( 17,232)( 18,231)( 19,230)( 20,267)( 21,285)( 22,284)( 23,283)
( 24,282)( 25,281)( 26,280)( 27,279)( 28,278)( 29,277)( 30,276)( 31,275)
( 32,274)( 33,273)( 34,272)( 35,271)( 36,270)( 37,269)( 38,268)( 39,248)
( 40,266)( 41,265)( 42,264)( 43,263)( 44,262)( 45,261)( 46,260)( 47,259)
( 48,258)( 49,257)( 50,256)( 51,255)( 52,254)( 53,253)( 54,252)( 55,251)
( 56,250)( 57,249)( 58,286)( 59,304)( 60,303)( 61,302)( 62,301)( 63,300)
( 64,299)( 65,298)( 66,297)( 67,296)( 68,295)( 69,294)( 70,293)( 71,292)
( 72,291)( 73,290)( 74,289)( 75,288)( 76,287)( 77,324)( 78,342)( 79,341)
( 80,340)( 81,339)( 82,338)( 83,337)( 84,336)( 85,335)( 86,334)( 87,333)
( 88,332)( 89,331)( 90,330)( 91,329)( 92,328)( 93,327)( 94,326)( 95,325)
( 96,305)( 97,323)( 98,322)( 99,321)(100,320)(101,319)(102,318)(103,317)
(104,316)(105,315)(106,314)(107,313)(108,312)(109,311)(110,310)(111,309)
(112,308)(113,307)(114,306)(115,400)(116,418)(117,417)(118,416)(119,415)
(120,414)(121,413)(122,412)(123,411)(124,410)(125,409)(126,408)(127,407)
(128,406)(129,405)(130,404)(131,403)(132,402)(133,401)(134,438)(135,456)
(136,455)(137,454)(138,453)(139,452)(140,451)(141,450)(142,449)(143,448)
(144,447)(145,446)(146,445)(147,444)(148,443)(149,442)(150,441)(151,440)
(152,439)(153,419)(154,437)(155,436)(156,435)(157,434)(158,433)(159,432)
(160,431)(161,430)(162,429)(163,428)(164,427)(165,426)(166,425)(167,424)
(168,423)(169,422)(170,421)(171,420)(172,343)(173,361)(174,360)(175,359)
(176,358)(177,357)(178,356)(179,355)(180,354)(181,353)(182,352)(183,351)
(184,350)(185,349)(186,348)(187,347)(188,346)(189,345)(190,344)(191,381)
(192,399)(193,398)(194,397)(195,396)(196,395)(197,394)(198,393)(199,392)
(200,391)(201,390)(202,389)(203,388)(204,387)(205,386)(206,385)(207,384)
(208,383)(209,382)(210,362)(211,380)(212,379)(213,378)(214,377)(215,376)
(216,375)(217,374)(218,373)(219,372)(220,371)(221,370)(222,369)(223,368)
(224,367)(225,366)(226,365)(227,364)(228,363);
s2 := Sym(456)!( 1, 21)( 2, 20)( 3, 38)( 4, 37)( 5, 36)( 6, 35)( 7, 34)
( 8, 33)( 9, 32)( 10, 31)( 11, 30)( 12, 29)( 13, 28)( 14, 27)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 39, 40)( 41, 57)( 42, 56)( 43, 55)
( 44, 54)( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58, 78)( 59, 77)( 60, 95)
( 61, 94)( 62, 93)( 63, 92)( 64, 91)( 65, 90)( 66, 89)( 67, 88)( 68, 87)
( 69, 86)( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)
( 96, 97)( 98,114)( 99,113)(100,112)(101,111)(102,110)(103,109)(104,108)
(105,107)(115,135)(116,134)(117,152)(118,151)(119,150)(120,149)(121,148)
(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)
(130,139)(131,138)(132,137)(133,136)(153,154)(155,171)(156,170)(157,169)
(158,168)(159,167)(160,166)(161,165)(162,164)(172,192)(173,191)(174,209)
(175,208)(176,207)(177,206)(178,205)(179,204)(180,203)(181,202)(182,201)
(183,200)(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)
(210,211)(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)
(219,221)(229,249)(230,248)(231,266)(232,265)(233,264)(234,263)(235,262)
(236,261)(237,260)(238,259)(239,258)(240,257)(241,256)(242,255)(243,254)
(244,253)(245,252)(246,251)(247,250)(267,268)(269,285)(270,284)(271,283)
(272,282)(273,281)(274,280)(275,279)(276,278)(286,306)(287,305)(288,323)
(289,322)(290,321)(291,320)(292,319)(293,318)(294,317)(295,316)(296,315)
(297,314)(298,313)(299,312)(300,311)(301,310)(302,309)(303,308)(304,307)
(324,325)(326,342)(327,341)(328,340)(329,339)(330,338)(331,337)(332,336)
(333,335)(343,363)(344,362)(345,380)(346,379)(347,378)(348,377)(349,376)
(350,375)(351,374)(352,373)(353,372)(354,371)(355,370)(356,369)(357,368)
(358,367)(359,366)(360,365)(361,364)(381,382)(383,399)(384,398)(385,397)
(386,396)(387,395)(388,394)(389,393)(390,392)(400,420)(401,419)(402,437)
(403,436)(404,435)(405,434)(406,433)(407,432)(408,431)(409,430)(410,429)
(411,428)(412,427)(413,426)(414,425)(415,424)(416,423)(417,422)(418,421)
(438,439)(440,456)(441,455)(442,454)(443,453)(444,452)(445,451)(446,450)
(447,449);
poly := sub<Sym(456)|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*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 >;
References : None.
to this polytope